arXiv:1011.4735v2 [math.AT] 8 Jul 2011 Čech cocycles for differential characteristic classes – An ∞-Lie theoretic construction Domenico Fiorenza, Urs Schreiber and Jim Stasheff July 11, 2011 Abstract What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth ∞-groups, i.e., by smooth groupal A∞ spaces. Namely, we realize differential characteristic classes as morphisms from ∞-groupoids of smooth principal ∞-bundles with connections to ∞-groupoids of higher U (1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures. 1 ∞-Chern-Weil theory July 11, 2011 2 Summary What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology. We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth ∞-groups, i.e., by smooth groupal A∞ -spaces. Namely, we realize differential characteristic classes as morphisms from ∞-groupoids of smooth principal ∞-bundles with connections to ∞-groupoids of higher U (1)-gerbes with connections. This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems. This applies in particular to the higher twisted differential spin structures called twisted differential string structures and twisted differential fivebrane structures. To that end we define for every L∞ -algebra g a smooth ∞-group G integrating it, and define smooth G-principal ∞-bundles with connection. For every L∞ -algebra cocycle of suitable degree, we give a refined ∞-Chern-Weil homomorphism that sends these ∞-bundles to classes in differential cohomology that lift the corresponding curvature characteristic classes. When applied to the canonical 3-cocycle of the Lie algebra of a simple and simply connected Lie group G this construction gives a refinement of the secondary first fractional Pontryagin class of G-principal bundles to cocycle space. Its homotopy fiber is the 2-groupoid of smooth String(G)-principal 2-bundles with 2-connection, where String(G) is a smooth 2-group refinement of the topological string group. Its homotopy fibers over non-trivial classes we identify with the 2-groupoid of twisted differential string structures that appears in the Green-Schwarz anomaly cancellation mechanism of heterotic string theory. Finally, when our construction is applied to the canonical 7-cocycle on the Lie 2-algebra of the String-2-group, it produces a secondary characteristic map for String-principal 2-bundles which refines the second fractional Pontryagin class. Its homotopy fiber is the 6-groupoid of principal 6-bundles with 6-connection over the Fivebrane 6-group. Its homotopy fibers over nontrivial classes are accordingly twisted differential fivebrane structures that have beeen argued to control the anomaly cancellation mechanism in magnetic dual heterotic string theory. Further online resources for this document can be found at http://ncatlab.org/schreiber/show/differential+characteristic+classes ∞-Chern-Weil theory 3 July 11, 2011 Contents 1 Introduction 3 2 A review of ordinary Chern-Weil theory 2.1 The Chern-Weil homomorphism . . . . . . . . . . . . . . . . . . . 2.2 Local curvature 1-forms . . . . . . . . . . . . . . . . . . . . . . . 9 10 12 3 Smooth ∞-groupoids 15 3.1 Presentation by simplicial presheaves . . . . . . . . . . . . . . . . 16 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 BG, BGconn and principal G-bundles with connection . . 21 3.2.2 BG2 , and nonabelian gerbes and principal 2-bundles . . . 25 3.2.3 Bn U (1), Bn U (1)conn , circle n-bundles and Deligne cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Differential ∞-Lie integration 4.1 Lie ∞-Algebroids: cocycles, invariant polynomials and CS-elements 4.2 Principal ∞-bundles . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Ordinary Lie group . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Line n-group . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Smooth string 2-group . . . . . . . . . . . . . . . . . . . . 4.3 Principal ∞-bundles with connection . . . . . . . . . . . . . . . . 4.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 34 42 46 48 49 50 55 5 ∞-Chern-Weil homomorphism 57 5.1 Characteristic maps by ∞-Lie integration . . . . . . . . . . . . . 57 5.2 Differential characteristic maps by ∞-Lie integration . . . . . . . 62 5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6 Homotopy fibers of Chern-Weil: twisted 6.1 Topological and smooth c-Structures . . 6.2 Twisted differential c-Structures . . . . 6.3 Examples . . . . . . . . . . . . . . . . . differential structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 70 72 76 Appendix: ∞-Stacks over the site of Cartesian spaces 82 References 86 1 Introduction Classical Chern-Weil theory (see for instance [GHV73, MS74, HS05]) provides a toolset for refining characteristic classes of smooth principal bundles from ordinary integral cohomology to differential cohomology. This can be described as follows. For G a topological group and P → X a G-principal bundle, to any characteristic class [c] ∈ H n+1 (BG, Z), there is ∞-Chern-Weil theory July 11, 2011 4 associated a characteristic class of the bundle, [c(P )] ∈ H n+1 (X, Z). This can be seen as the homotopy class of the composition P c X− → BG − → K(Z, n + 1) P of the classifying map X − → BG of the bundle with the characteristic map c BG − → K(Z, n + 1). If G is a compact connected Lie group, and with real coefficients, there is a graded commutative algebra isomorphism between H • (BG, R) and the algebra inv(g) of adG -invariant polynomials on the Lie algebra g of G. In particular, any characteristic class c will correspond to such an invariant polynomial h−i. The Chern-Weil homomorphism associates to a choice of connection ∇ on a G-principal bundle P the closed differential form hF∇ i on X, where F∇ is the curvature of ∇. The de Rham cocycle hF∇ i is a representative for the characteristic class [c(P )] in H • (X, R). This construction can be carried out at a local level: instead of considering a globally defined connection ∇, one can consider an open cover U of X and local connections ∇i on P |Ui → Ui ; then the local differential forms hF∇i i define a cocycle in the Čech-de Rham complex, still representing the cohomology class of c(P ). There is a refinement of this construction to what is sometimes called secondary characteristic classes: the differential form hF∇ i may itself be understood as the higher curvature form of a higher circle-bundle-like structure ĉ(∇) whose higher Chern-class is c(P ). In this refinement, both the original characteristic class c(P ) as well as its curvature differential form hF∇ i are unified in one single object. This single object has originally been formalized as a Cheeger-Simons differential character. It may also be conceived of as a cocycle in the Čech-Deligne complex, a refinement of the Čech-de Rham complex [HS05]. Equivalently, as we discuss here, these objects may naturally be described in terms of what we want to call circle n-bundles with connection: smooth bundles whose structure group is a smooth refinement – which we write Bn U (1) – of the topological group B n U (1) ≃ K(Z, n + 1), endowed with a smooth connection of higher order. For low n, such Bn U (1)-principal bundles are known (more or less explicitly) as (n − 1)-bundle gerbes. The fact that we may think of ĉ(∇) as being a smooth principal higher bundle with connection suggests that it makes sense to ask if there is a general definition of smooth G-principal ∞-bundles, for smooth ∞-groups G, and whether the Chern-Weil homomorphism extends on those to an ∞-Chern-Weil homomorphism. Moreover, since G-principal bundles naturally form a parameterized groupoid – a stack – and circle n-bundles naturally form a parameterized n-groupoid – an (n − 1)-stack, an n-truncated ∞-stack, it is natural to ask whether we can refine the construction of differential characteristic classes to these ∞-stacks. Motivations for considering this are threefold: 1. The ordinary Chern-Weil homomorphism only knows about characteristic classes of classifying spaces BG for G a Lie group. Already before considering the refinement to differential cohomology, this misses useful cohomological information about connected covering groups of G. ∞-Chern-Weil theory July 11, 2011 5 For instance, for G = Spin, the Spin group, there is the second Pontryagin class represented by a map p2 : BSpin → B 8 Z. But on some Spin-principal bundles P → X classified by a map g : X → BSpin, this class may be further divisible: there is a topological group String, called the String group, such that we have a commuting diagram g̃ X w w w g BString w; w  / BSpin 1 6 p2 / B8Z ·6 p2  / B8Z of topological spaces, where the morphism on the right is given on Z by multiplication with 6 [SSS09a]. This means that if P happens to admit a String structure exhibited by a lift g̃ of its classifying map g as indicated, then its second Pontryagin class [p2 (P )] ∈ H 8 (X, Z) is divisible by 6. But this refined information is invisible to the ordinary Chern-Weil homomorphism: while Spin canonically has the structure of a Lie group, String cannot have a finite-dimensional Lie group structure (because it is a BU (1)-extension, hence has cohomology in arbitrary high degree) and therefore the ordinary Chern-Weil homomorphism can not model this fractional characteristic class. But it turns out that String does have a natural smooth structure when regarded as a higher group – a 2-group in this case [BSCS07, Hen08]. We write BString for the corresponding smooth refinement of the classifying space. As we shall show, there is an ∞-Chern-Weil homomorphism that does apply and produces for every smooth String-principal 2-bundle g̃ : X → BString a smooth circle 7-bundle with connection, which we write 1 6 p̂2 (g̃). Its curvature 8-form is a representative in de Rham cohomology of the fractional second Pontryagin class. Here and in the following • boldface denotes a refinement from continuous (bundles) to smooth (higher bundles); • the hat denotes further differential refinement (equipping higher bundles with smooth connections). In this manner, the ∞-Chern-Weil homomorphism gives cohomological information beyond that of the ordinary Chern-Weil homomorphism. And this is only the beginning of a pattern: the sequence of smooth objects that we considered continues further as · · · → BFivebrane → BString → BSpin → BSO → BO to a smooth refinement of the Whitehead tower of BO. One way to think of ∞-Chern-Weil theory is as a lift of ordinary Chern-Weil theory along such smooth Whitehead towers. ∞-Chern-Weil theory July 11, 2011 6 2. Traditionally the construction of secondary characteristic classes is exhibited on single cocycles and then shown to be independent of the representatives of the corresponding cohomology class. But this indicates that one is looking only at the connected components of a more refined construction that explicitly sends cocycles to cocycles, and sends coboundaries to coboundaries such that their composition is respected up to higher degree coboundaries, which in turn satisfy their own coherence condition, and so forth. In other words: a map between the full cocycle ∞-groupoids. The additional information encoded in such a refined secondary differential characteristic map is equivalently found in the collection of the homotopy fibers of the map, over the cocycles in the codomain. These homotopy fibers answer the question: which bundles with connection have differential characteritsic class equivalent to some fixed class, which of their gauge transformations respect the choices of equivalences, which of the higher gauge of gauge transformations respect the chosen gauge transformations, and so on. This yields refined cohomological information whose knowledge is required in several applications of differential cohomology, indicated in the next item. 3. Much of the motivation for studies of differential cohomology originates in the applications this theory has to the description of higher gauge fields in physics. Notably the seminal article [HS05] that laid the basis of generalized differential cohomology grew out of the observation that this is the right machinery that describes subtle phenomena of quantum anomaly cancellation in string theory, discussed by Edward Witten and others in the 1990s, further spelled out in [Fre00]. In this context, the need for refined fractional characteristic classes and their homotopy fibers appears. In higher analogy to how the quantum mechanics of a spinning particle requires its target space to be equipped with a Spin-structure that is differentially refined to a Spin-principal bundle with connection, the quantum dynamics of the (heterotic) superstring requires target space to be equipped with a differential refinement of a String-structure. Or rather, since the heterotic string contains besides the gravitational Spin-bundle also a U (n)-gauge bundle, of a twisted String-structure for a specified twist. We had argued in [SSS09c] that these differentially refined string backgrounds are to be thought of as twisted differential structures in the above sense. With the results of the present work this argument is lifted from a discussion of local ∞-connection data to the full differential cocycles. We shall show that by standard homotopy theoretic arguments this allows a simple derivation of the properties of untwisted differential string structures that have been found in [Wal09], and generalize these to the twisted case and all the higher analogs. Namely, moving up along the Whitehead tower of O(n), one can next ask for the next higher characteristic class on String-2-bundles and its differential refinement to a secondary characteristic class. In [SSS09a] it ∞-Chern-Weil theory July 11, 2011 7 was argued that this controls, in direct analogy to the previous case, the quantum super-5-brane that is expected to appear in the magnetic dual description of the heterotic target space theory. With the tools constructed here the resulting twisted differential fivebrane structures can be analyzed in analogy to the case of string structures. Our results allow an analogous description of twisted differential structures of ever higher covering degree, but beyond the 5-brane it is currently unclear whether this still has applications in physics. However, there are further variants in low degree that do: for instance for every n there is a canonical 4-class on pairs of n-torus bundles and dual n-torus bundles. This has a differential refinement and thus we can apply our results to this situation to produce parameterized 2-groupoids of the corresponding higher twisted differential torus-bundle extensions. We find that the connected components of these 2-groupoids are precisely the differential T-duality pairs that arise in the description of differential T-duality of strings in [KV09]. This suggests that there are more applications of refined higher differential characteristic maps in string theory, but here we shall be content with looking into these three examples. In this article, we shall define connections on principal ∞-bundles and the action of the ∞-Chern-Weil homomorphism in a natural but maybe still somewhat ad hoc way, which here we justify mainly by the two main theorems about two examples that we prove, which we survey in a moment. The construction uses essentially standard tools of differential geometry. The construction can be derived from first principles as a model (in the precise sense of model category theory) for a general abstract construction that exists in ∞-topos theory. This abstract theory is discussed in detail elsewhere [Sch10]. Notice that our approach goes beyond that of [HS05] in two ways: the ∞stacks we consider remember smooth gauge transformation and thus encode smooth structure of principal ∞-bundles already on cocycles and not just in cohomology; secondly, we describe non-abelian phenomena, such as connections on principal bundles for non-abelian structure groups, and more in general ∞connections for non-abelian structure smooth ∞-groups, such as the String-2group and the Fivebrane-6-group. This is the very essence of (higher) ChernWeil theory: to characterize non-abelian cohomology by abelian characteristic classes. Since [HS05] work with spectra, nothing non-abelian is directly available there. On the other hand, the construction we describe does not as easily allow differential refinements of cohomology theories represented by non-connective spectra. We now briefly indicate the means by which we will approach these issues in the following. The construction that we discuss is the result of applying a refinement of the machine of ∞-Lie integration [Hen08, Get09] to the L∞ -algebraic structures discussed in [SSS09b, SSS09c]: ∞-Chern-Weil theory July 11, 2011 8 For g an L∞ -algebra, its Lie integration to a Lie ∞-group G with smooth classifying object BG turns out to be encoded in the simplicial presheaf given by the assignment to each smooth test manifold U of the simplicial set exp∆ (g) : (U, [k]) 7→ HomdgAlg (CE(g), Ω• (U × ∆k )vert ) , where CE(g) is the Chevalley-Eilenberg algebra of g and ‘vert’ denotes forms which see only vector fields along ∆k . This has a canonical projection exp∆ (g) → BG, hence the name exp∆ (g). One can think of this as saying that a U parameterized smooth family of k-simplices in G is given by the parallel transport over the k-simplex of a flat g-valued vertical differential form on the trivial simplex bundle U × ∆k → U . This we discuss in detail in section 4.2. The central step of our construction is a differential refinement BGdiff of BG, where the above is enhanced to   • k o  CE(g)    Ω (U ×O ∆ )vert O     , exp∆ (g)diff : (U, [k]) 7→        W(g)  Ω• (U × ∆k ) o with W(g) the Weil algebra of g. We also consider a simplicial sub-presheaf exp∆ (g)conn ֒→ exp∆ (g)diff defined by a certain horizontality constraint. This may be thought of as assigning non-flat g-valued forms on the total space of the trivial simplex bundle U × ∆k . The horizontality constraint generalizes one of the conditions of an Ehresmann connection [Ehr51] on an ordinary G-principal bundle. This we discuss in detail in 4.3. We observe that an L∞ -algebra cocycle µ ∈ CE(g) in degree n, when we equivalently regard it as a morphism of L∞ -algebras µ : g → bn−1 R to the Eilenberg-MacLane object bn−1 R, tautologically integrates to a morphism exp∆ (µ) : exp∆ (g) → exp∆ (bn−1 R) of the above structures. What we identify as the ∞-Chern-Weil homomorphism is obtained by first extending this to the differential refinement exp∆ (µ)diff : exp∆ (g)diff → exp∆ (bn−1 R)diff in a canonical way – this we shall see introduces Chern-Simons elements – and then descending the construction along the projection exp(g)diff → BGdiff . This quotients out a lattice Γ ⊂ R and makes the resulting higher bundles with connection be circle n-bundles with connection which represent classes in differential cohomology. This we discuss in section 5. Finally, in the last part of section 5 we discuss two classes of applications and obtain the following statements. Theorem 1.0.1. Let X be a paracompact smooth manifold and choose a good open cover U. ∞-Chern-Weil theory July 11, 2011 9 Let g be a semisimple Lie algebra with normalized binary Killing form h−, −i in transgression with the 3-cocycle µ3 = 12 h−, [−, −]i. Let G be the corresponding simply connected Lie group. • 1. Applied to this µ3 , the ∞-Chern-Weil homomorphism exp(µ)conn : Č(U, BGconn ) → Č(U, B3 U (1)conn) from Čech cocycles with coefficients in the complex that classifies G-principal bundles with connection to Čech-Deligne cohomology in degree 4 is a fractional multiple of the Brylinski-McLaughlin construction [BM96b] of ČechDeligne cocycles representing the differential refinement of the characteristic class corresponding to h−, −i. In particular, in cohomology it represents the refined Chern-Weil homomorphims 1 p̂1 : H 1 (X, G)conn → Ĥ 4 (X, Z) 2 induced by the Killing form and with coefficients in degree 4 differential cohomology. For g = so(n), this is the differential refinement of the first fractional Pontryagin class. Next let µ7 ∈ CE(g) be a 7-cocycle on the semisimple Lie algebra g (this is unique up to a scalar factor). Let gµ3 → g be the L∞ -algebra-extension of g classified by µ3 (the string Lie 2-algebra. Then µ7 can be seen as a 7-cocycle also on gµ3 . • 2. Applied to µ7 regarded as a cocycle on gµ , the ∞-Chern-Weil homomorphism produces a map Č(U, BString(G)conn ) → Č(U, B7 U (1)conn ) from Čech cocycles with coefficients in the complex that classifies String(G)2-bundles with connection to degree 8 Čech-Deligne cohomology. For g = so(n) this gives a fractional refinement of the ordinary refined Chern-Weil homomorphism 1 p̂2 : H 1 (X, String)conn → Ĥ 8 (X, Z) 6 that represents the differential refinement of the second fractional Pontryagin class on Spin-bundles with String-structure. These are only the first two instances of a more general statement. But this will be discussed elsewhere. 2 A review of ordinary Chern-Weil theory We briefly review standard aspects of ordinary Chern-Weil theory whose generalization we consider later on. In this section we assume the reader is familiar with basic properties of Chevalley-Eilenberg and of Weil algebras; the unfamiliar reader can find a concise account at the beginning of Section 4. ∞-Chern-Weil theory 2.1 10 July 11, 2011 The Chern-Weil homomorphism For G a Lie group and X a smooth manifold, the idea of a connection on a smooth G-principal bundle P → X can be expressed in a variety of equivalent ways: as a distribution of horizontal spaces on the tangent bundle total space T P , as the corresponding family of projection operators in terms of local connection 1-forms on X or, more generally, as defined by Ehresmann [Ehr51], and, ultimately, purely algebraically, by H. Cartan [Car50a, Car50b]. Here, following this last approach, we review how the Weil algebra can be used to give an algebraic description of connections on principal bundles, and of the Chern-Weil homomorphism. We begin by recalling the classical definition of connection on a G-principal bundle P → X as a g-valued 1-form A on P which is G-equivariant and induces the Maurer-Cartan form of G on the fibers (these are known as the Cartan-Ehresmann conditions). The key insight is then the identification of A ∈ Ω1 (P, g) with a differential graded algebra morphism; this is where the Weil algebra W(g) comes in. We will introduce Weil algebras in a precise and intrinsic way in the wider context of Lie ∞-algebroids in Section 4.1, so we will here content ourselves with thinking of the Weil algebra of g as a perturbation of the Chevalley-Eilenberg cochain complex CE(g) for g with coefficients in the polynomial algebra generated by the dual g∗ . More precisely, the Weil algebra W(g) is a commutative dg-algebra freely generated by two copies of g∗ , one in degree 1 and one in degree 2; the differential dW is the sum of the ChevalleyEilenberg differential plus σ, the shift isomorphism from g∗ in degree 1 to g∗ in degree 2, extended as a derivation. A crucial property of the Weil algebra is its freeness: dgca morphisms out of the Weil algebra are uniquely and freely determined by graded vector space morphism out of the copy of g∗ in degree 1. This means that a g-valued 1-form A on P can be equivalently seen as a dgca morphism A : W(g) → Ω• (P ) to the de Rham dg-algebra of differential forms on P . Now we can read the Cartan-Ehresmann conditions on a g-connection as properties of this dgca morphism. First, the Maurer-Cartan form on G, i.e., the left-invariant g-valued form θG on G induced by the identity on g seen as a linear morphism Te G → g, is an element of Ω1 (G, g), and so it defines a dgca morphism W(g) → Ω• (G). This morphism actually factors through the Chevalley-Eilenberg algebra of g; this is the algebraic counterpart of the fact that the curvature 2-form of θG vanishes. Therefore, the first Cartan-Ehresmann condition on the behaviour of the connection form A on the fibres of P → X is encoded in the commutativity of the following diagram of differential graded commutative algebras: Avert Ω• (P )vert o CE(g) . O O Ω• (P ) o A W(g) ∞-Chern-Weil theory July 11, 2011 11 In the upper left corner, Ω• (P )vert is the dgca of vertical differential forms on P , i.e. the quotient of Ω• (P ) by the differential ideal consisting of differential forms on P which vanish when evaluated on a vertical multivector field. Now we turn to the second Cartan-Ehresmann condition. The symmetric algebra Sym• (g∗ [−2]) on g∗ placed in degree 2 is a graded commutative subalgebra of the Weil algebra W(g), but it is not a dg-subalgebra. However, the subalgebra inv(g) of Sym• (g∗ [−2]) consisting of adg -invariant polynomials is a dg-subalgebra of W(g). The composite morphism of dg-algebras A inv(g) → W(g) − → Ω• (P ) is the evaluation of invariant polynomials on the curvature 2-form of A, i.e. on the g-valued 2-form FA = dA + 21 [A, A]. Invariant polynomials are dW -closed as elements in the Weil algebra, therefore, their images in Ω• (P ) are closed differential forms. Assume now G is connected. Then, if h−i is an adg -invariant polynomial, by the G-equivariance of A it follows that the closed differential form hFA i descends to a closed differential form on the base X of the principal bundle. Thus, the second Cartan-Ehresmann condition on A implies the commutativity of the diagram A Ω• (P ) o W(g) . O O Ω• (X) o FA inv(g) Since the image of inv(g) in Ω• (X) consists of closed forms, we have an induced graded commutative algebras morphism inv(g) → H • (X, R), the Chern-Weil homomorphism. This morphism is independent of the particular connection form chosen and natural in X. Therefore we can think of elements of inv(g) as representing universal cohomology classes, hence as characteristic classes, of G-principal bundles. And indeed, if G is a compact connected finite dimensional Lie group, then we have an isomorphism of graded commutative algebras inv(g) ∼ = H • (BG, R), corresponding to H • (G, R) being isomorphic to an exterior algebra on odd dimensional generators [Che52], the indecomposable Lie algebra cohomology classes of g. The isomorphism inv(g) ∼ = H • (BG, R) is to be thought as the universal Chern-Weil homomorphism. Traditionally this is conceived of in terms of a smooth manifold version of the universal G-principal bundle on BG. We will here instead refine BG to a smooth ∞-groupoid BG. This classfies not just equivalence classes of G-principal bundles but also their automorphisms. We shall argue that the context of smooth ∞-groupoids is the natural place (and place translates to topos) in which to conceive of the Chern-Weil homomorphism. ∞-Chern-Weil theory 2.2 July 11, 2011 12 Local curvature 1-forms Next we focus on the description of g-connections in terms of local g-valued 1forms and gauge transformations. We discuss this in terms of the local transition function data from which the total space of the bundle may be reconstructed. It is this local point of view that we will explicitly generalize in section 4. More precisely, in section 4.3 we will present algebraic data which encode an ∞connection on a trivial higher bundle on a Cartesian space Rn , and will then globalize this local picture by descent/stackification. To prepare this general construction, let us show how it works in the case of ordinary g-connections on G-principal bundles. For that purpose, consider a Cartesian space U = Rn . Every G-principal bundle on U is equivalent to the trivial G-bundle U × G equipped with the evident action of G on the second factor, and under stackification this completely characterizes G-principal bundles on general spaces. A connection on this trivial G-bundle is given by a g-valued g / A′ from the trivial bundle with 1-form A ∈ Ω1 (U, g). An isomorphism A ′ connection A to that with connection A is given by a function g ∈ C ∞ (U, G) such that the equation A′ = g −1 Ag + g −1 dg (1) holds. Here the first term on the right denotes the adjoint action of the Lie group on its Lie algebra, while the second term denotes the pullback of the Maurer-Cartan form on G along g to U . We wish to amplify a specific way to understand this formula as the Lie integration of a path of infinitesimal gauge transformations: write ∆1 = [0, 1] for the standard interval regarded as a smooth manifold (with boundary) and consider a smooth 1-form A ∈ Ω1 (U × ∆1 , g) on the product of U with ∆1 . If we think of this as the trivial interval bundle U × ∆1 → U and are inspired by the discussion in section 2.1, we can equivalently conceive of A as a morphism of dg-algebras A : W(g) → Ω• (U × ∆1 ) from the Weil algebra of g into the de Rham algebra of differential forms on the total space of the interval bundle. It makes sense to decompose A as the sum of a horizontal 1-form AU and a vertical 1-form λ dt, where t : ∆1 → R is the canonical coordinate on ∆1 : A = AU + λ dt . The vertical part Avert = λ dt of A is an element of the completed tensor product ˆ 1 (∆1 , g) and can be seen as a family of g-connections on a trivial C ∞ (U )⊗Ω G-principal bundle on ∆1 , parametrized by U . At any fixed u0 ∈ U, the 1form λ(u0 , t) dt ∈ Ω1 (∆1 , g) satisfies the Maurer-Cartan equation by trivial ∞-Chern-Weil theory 13 July 11, 2011 dimensional reasons, and so we have a commutative diagram Avert Ω• (U × ∆1 )vert o O A Ω• (U × ∆1 ) o CE(g) O W(g) By the discussion in section 2.1, this can be seen as a first Cartan-Ehresmann condition in the ∆1 -direction; it precisely encodes the fact that the 1-form A on the total space of U × ∆1 → U is flat in the vertical direction. The curvature 2-form of A decomposes as FA = FAU + F∆1 , where the first term is at each point t ∈ ∆1 the ordinary curvature FAU = dU AU + 12 [AU , AU ] of AU at fixed t ∈ ∆1 and where the second term is   ∂ 1 F∆ = dU λ + [AU , λ] − AU ∧ dt . ∂t We shall require that F∆1 = 0; this is the second Ehresmann condition in the ∆1 -direction . It implies that we have a commutative diagram Ω• (∆1 × U ) o O A Ω• (U ) o FA W(g) . O inv(g) The condition F∆1 = 0 is equivalent to the differential equation ∂ AU = dU λ + [AU , λ], ∂t whose unique solution for given boundary condition AU |t=0 specifies AU |t=1 by the formula AU (1) = g −1 AU (0)g + g −1 dg , where Z g := P exp( λdt) : U → G ∆1 is, pointwise in U, the parallel transport of λdt along the interval. We may think of this as exhibiting formula (1) for gauge transformations as arising from Lie integration of infinitesimal data. Globalizing this local picture of connections on trivial bundles and gauge transformations between them now amounts to the following. For any (smooth, paracompact) manifold X, we may find a good open cover {Ui → X}, i.e., an open cover such that every non-empty n-fold intersection Ui1 ∩ · · · ∩ Uin for all ∞-Chern-Weil theory 14 July 11, 2011 n ∈ N is diffeomorphic to a Cartesian space. The cocycle data for a G-bundle with connection relative to this cover is in degree 0 and 1 given by diagrams 0O o Avert CE(g) Ω• (∆1 × Uij )vert o O O Avert CE(g) O Ω• (Ui ) o O A Ω• (Ui ) o FA W(g) O and Ω• (∆1 × Uij ) o O A Ω• (Uij ) o FA inv(g) W(g) O , (2) inv(g) where the latter restricts to the former after pullback along the two inclusions Uij → Ui , Uj and along the face maps ∆0 = {∗} ⇒ ∆1 . This gives a collection of 1-forms {Ai ∈ Ω1 (Ui , g)}i and of smooth function {gij ∈ C ∞ (Ui ∩ Uj , G)}, such that the formula −1 −1 Aj = gij Ai gij + gij dgij for gauge transformation holds on each double intersection Ui ∩ Uj . This is almost the data defining a g-connecton on a G-principal bundle P → X, but not quite yet, since it does not yet constrain the transition functions gij on the triple intersections Ui ∩ Uj ∩ Uk to obey the cocycle relation gij gjk = gik . But since each gij is the parallel transport of our connection along a vertical 1-simplex, the cocycle condition precisely says that parallel transport along the three edges of a vertical 2-simplex is trivial, i.e. that the vertical parts of our connection forms on Uijk × ∆1 are the boundary data of a connection form on Uijk × ∆2 which is flat in the vertical direction. In other words, the collection of commutative diagrams (2) is to be seen as the 0 and 1-simplices of a simplicial set whose 2-simplices are the commutative diagrams Avert CE(g) . Ω• (∆2 × Uijk )vert o O O Ω• (∆2 × Uijk ) o O A Ω• (Uijk ) o FA W(g) O inv(g) Having added 2-simplices to our picture, we have finally recovered the standard description of connections in terms of local differential form data. By suitably replacing Lie algebras with L∞ -algebras in this derivation, we will obtain a definition of connections on higher bundles in Section 4.2. As one can expect, in the simplicial description of connections on higher bundles, simplices of arbitrarily high dimension will appear. ∞-Chern-Weil theory 3 July 11, 2011 15 Smooth ∞-groupoids In this section we introduce a central concept that we will be dealing with in this paper, smooth ∞-groupoids, as a natural generalization of the classical notion of Lie groups. A Lie groupoid is, by definition, a groupoid internal to the category of smooth spaces and smooth maps. It is a widely appreciated fact in Lie groupoid theory that many features of Lie groupoids can be usefully thought of in terms of their associated groupoid-valued presheaves on the category of manifolds, called the differentiable stack represented by the Lie groupoid. This is the perspective that immediately generalizes to higher groupoids. Since many naturally appearing smooth spaces are not manifolds – particularly the spaces [Σ, X] of smooth maps Σ → X between two manifolds – for the development of the general theory it is convenient to adopt a not too strict notion of ‘smooth space’ . This generalized notion will have to be more flexible than the notion of manifold but at the same time not too far from that. The basic example to have in mind is the following: every smooth manifold X of course represents a sheaf X : SmoothManifoldsop → Sets U 7→ C ∞ (U, X). on the category of smooth manifolds. But since manifolds themselves are by definition glued from Cartesian spaces Rn , all the information about X is in fact already encoded in the restriction of this sheaf to the category of Cartesian spaces and smooth maps between them: X : CartSpop → Sets . Now notice that also the spaces [Σ, X] of smooth maps Σ → X between two manifolds naturally exist as sheaves on CartSp, given by the assignment [Σ, X] : U 7→ C ∞ (Σ × U, X) . Sheaves of this form are examples of generalized smooth spaces that are known as diffeological spaces or Chen smooth spaces. While not manifolds, these smooth spaces do have an underlying topological space and behave like smooth manifolds in many essential ways. Even more generally, we will need to consider also ‘smooth spaces’ that do not have even an underlying topological space. The central example of such is the sheaf of (real valued) closed differential n-forms U 7→ Ωncl (U ) , which we will need to consider later in the paper. We may think of these as modelling a kind of smooth Eilenberg-MacLane space that support a single (up to scalar multiple) smooth closed n-form. A precise version of this statement ∞-Chern-Weil theory July 11, 2011 16 will play a central role later in the theory of Lie ∞-integration that we will describe in Section 4. Thus we see that the common feature of generalized smooth spaces is not that they are representable in one way or other. Rather, the common feature is that they all define sheaves on the category of the archetypical smooth spaces: the Cartesian spaces. This is a special case of an old insight going back to Grothendieck, Lawvere and others: with a category C of test spaces fixed, the correct context in which to consider generalized spaces modeled on C is the category Sh(C) of all sheaves on C: the sheaf topos [Joh03]. In there we may find a hierachy of types of generalized spaces ranging from ones that are very close to being like these test spaces, to ones that are quite a bit more general. In applications, it is good to find models as close as possible to the test spaces, but for the development of the theory it is better to admit them all. Now if the manifold X happens, in addition, to be equipped with the structure of a Lie group G, then it represents more than just an ordinary sheaf of sets: from each group we obtain a simplicial set, its nerve, whose set of k-cells is the set of k-tuples of elements in the group, and whose face and degeneracy maps are built from the product operation and the neutral element in the group. Since, for every U ∈ CartSp, also the set of functions C ∞ (U, G) forms a group, this means that from a Lie group we obtain a simplicial presheaf o n // ∞ // ∞ // C (U, G × G) // ∗ , , BG : U 7→ · · · / C (U, G) where the degeneracy maps have not been displayed in order to make the diagram more readable. The simplicial presheaves arising this way are, in fact, special examples of presheaves taking values in Kan complexes, i.e., in simplicial sets in which every horn – a simplex minus its interior and minus one face – has a completion to a simplex; see for instance [GJ99] for a review. It turns out (see section 1.2.5 of [Lur09]) that Kan complexes may be thought of as modelling ∞-groupoids: the generalization of groupoids where one has not only morphisms between objects, but also 2-morphisms between morphisms and generally (k + 1)-morphisms between k-morphisms for all k ∈ N. The traditional theory of Lie groupoids may be thought of as dealing with those simplicial presheaves on CartSp that arise from nerves of Lie groupoids in the above manner This motivates the definitions that we now turn to. 3.1 Presentation by simplicial presheaves Definition 3.1.1. A smooth ∞-groupoid A is a simplicial presheaf on the category CartSp of Cartesian spaces and smooth maps between them such that, over each U ∈ CartSp, A is a Kan complex. Much of ordinary Lie theory lifts from Lie groups to this context. The reader is asked to keep in mind that smooth ∞-groupoids are objects whose smooth structure may be considerably more general than that of a Kan complex internal ∞-Chern-Weil theory July 11, 2011 17 to smooth manifolds, i.e., of a simplicial smooth manifold satisfying a horn filling condition. Kan complexes internal to smooth manifolds, such as for instance nerves of ordinary Lie groupoids, can be thought of as representable smooth ∞-groupoids. Example 3.1.2. The basic example of a representable smooth ∞-groupoids are ordinary Lie groupoids; in particular smooth manifolds and Lie groups are smooth ∞-groupoids. A particularly important example of representable smooth ∞-groupoid is the Čech ∞-groupoid : for X a smooth manifold and U = {Ui → X} an open cover, there is the simplicial manifold o n /// {U } / {U } /// {Uij } / Č(U) := · · · ijk i / which in degree k is the disjoint union of the k-fold intersections Ui ∩ Uj ∩ · · · of open subsets (the degeneracy maps are not depicted). This is a Kan complex internal to smooth manifolds in the evident way. While the notion of simplicial presheaf itself is straightforward, the correct concept of morphism between them is more subtle: we need a notion of morphisms such that the resulting category –or ∞-category as it were – of our smooth ∞-groupoids reflects the prescribed notion of gluing of test objects. In fancier words, we want simplicial presheaves to be equivalent to a higher analog of a sheaf topos: an ∞-topos [Lur09]. This may be achieved by equipping the naive category of simplicial presheaves with a model category structure [Hov99]. This provides the information as to which objects in the category are to be regarded as equivalent, and how to resolve objects by equivalent objects for purposes of mapping between them. There are some technical aspects to this this that we have relegated to the appendix. For all details and proofs of the definitions and propositions, respectively, in the remainder of this section see there. Definition 3.1.3. Write [CartSpop , sSet]proj for the global projective model category structure on simplicial presheaves: weak equivalences and fibrations are objectwise those of simplicial sets. This model structure presents the ∞-category of ∞-presheaves on CartSp. We impose now an ∞-sheaf condition. Definition 3.1.4. Write [CartSpop , sSet]proj,loc for the left Bousfield localization (see for instance section A.3 of [Lur09]) of [CartSpop , sSet]proj at the set of all Čech nerve projections Č(U) → U for U a differentiably good open cover of U , i.e., an open cover U = {Ui → U }i∈I of U such that for all n ∈ N every n-fold intersection Ui1 ∩ · · · ∩ Uin is either empty or diffeomorphic to Rdim U . This is the model structure that presents the ∞-category of ∞-sheaves or ∞-stacks on CartSp. By standard results, it is a simplicial model category with respect to the canonical simplicial enrichment of simplicial presheaves, see [Dug01]. For X, A two simplicial presheaves, we write ∞-Chern-Weil theory July 11, 2011 18 • [CartSpop , sSet](X, A) ∈ sSet for the simplicial hom-complex of morphisms; • H(X, A) := [CartSpop , sSet](Q(X), P (A)) for the right derived hom-complex (well defined up to equivalence) where Q(X) is any local cofibrant resolution of X and P (A) any local fibrant resolution of A. Notice some standard facts about left Bousfield localization: • every weak equivalence in [CartSpop , sSet]proj is also a weak equivalence in [CartSpop , sSet]proj,loc ; • the classes of cofibrations in both model structures coincide. • the fibrant objects of the local structure are precisely the objects that are fibrant in the global structure and in addition satisfy descent over all differentiably good open covers of Cartesian spaces. What this means precisely is stated in corollary A.2 in the appendix. • the localization right Quillen functor Id : [CartSpop , sSet]proj → [CartSpop , sSet]proj,loc presents ∞-sheafification, which is a left adjoint left exact ∞-functor [Lur09], therefore all homotopy colimits and all finite homotopy limits in the local model structure can be computed in the global model structure. In particular, notice that an acyclic fibration in the global model structure will not, in general, be an acyclic fibration in the local model structure; nevertheless, it will be a weak equivalence in the local model structure. Definition 3.1.5. We write • ≃ / for isomorphisms of simplicial presheaves; • ∼ / for weak equivalences in the global model structure; • ∼loc / for weak equivalences in the local model structure; (Notice that each of these generalizes the previous.) • / / for fibrations in the global model structure. We do not use notation for fibrations in the local model structure. Since the category CartSp has fewer objects than the category of all manifolds, we have that the conditions for simplicial presheaves to be fibrant in [CartSpop , sSet]proj,loc are comparatively weak. For instance BG : U 7→ N ((C ∞ (U, G) ⇒ ∗) is locally fibrant over CartSp but not over the site of all manifolds. This is discused below in section 3.2. Conversely, the condition to be cofibrant is stronger ∞-Chern-Weil theory July 11, 2011 19 over CartSp than it is over all manifolds. But by a central result by [Dug01], we have fairly good control over cofibrant resolutions: these include notably Čech nerves Č(U) of differentiably good open covers, i.e., the Čech nerve Č(U) → X of a differentiably good open cover over a paracompact smooth manifold X is a cofibrant resolution of X in [CartSpop , sSet]proj,loc , and so we write ∼ loc Č(U) −−− →X. Notice that in the present article these will be the only local weak equivalences that are not global weak equivalences that we need to consider. In the practice of our applications, all this means that much of the technology hidden in definition 3.1.4 boils down to a simple algorithm: after solving the comparatively easy tasks of finding a version A of a given smooth ∞-groupoid that is fibrant over CartSp, for describing morphisms of smooth ∞-groupoids from a manifold X to A, we are to choose a differentiably good open cover U = {Ui → X}, form the Čech nerve simplicial presheaf Č(U) and then consider spans of ordinary morphisms of simplicial presheaves of the form Č(U) g /A . ∼loc  X Such a diagram of simplicial presheaves presents an object in H(X, A), in the hom-space of the ∞-topos of smooth ∞-groupoids. As discussed below in section 3.2, here the morphisms g are naturally identified with cocycles in nonabelian Čech cohomology on X with coefficients in A. In section 4.2, we discus that we may also think of these cocycles as transition data for A-principal ∞-bundles on X [SSS09c]. For discussing the ∞-Chern-Weil homomorphism, we are crucially interested in composites of such spans: a characteristic map on a coefficient object A is nothing but a morphism c : A → B in the ∞-topos, presented itself by a span  /B . ≀  A The evaluation of this characteristic map on the A-principal bundle on X encoded by a cocycle g : Č(U) → A is the composite morphism X → A → B in ∞-Chern-Weil theory 20 July 11, 2011 the ∞-topos, which is presented by the composite span of simplicial presheaves QX ≀  Č(U) /  /B . ≀  /A ∼loc  X Here QX → Č(U) is the pullback of the acyclic fibration  → A, hence itself an acyclic fibration; moreover, since Č(U) is cofibrant, we are guaranteed that a section Č(U) → QX exists and is unique up to homotopy. Therefore the composite morphism X → A → B is encoded in a cocylce Č(U) → B as in the diagram below: /  /B . QX C ≀  Č(U) ≀  /A ∼loc  X Our main theorems will involve the construction of such span composites. 3.2 Examples In this paper we will consider three main sources of smooth ∞-groupoids • Lie groups and Lie groupoids, leading to Kan complexes via their nerves; examples of this kind will be the smooth ∞-groupoid BG associated with a Lie group G, and its refinements BGdiff and BGconn ; • complexes of abelian groups concentrated in nonnegative degrees, leading to Kan complexes via the Dold-Kan correspondence; examples of this kind will be the smooth ∞-groupoid Bn U (1) associated with the chain complex of abelian groups consisting in U (1) concentrated in degree n, and its refinements Bn U (1)diff and Bn U (1)conn ; • Lie algebras and L∞ -algebras, via flat connections over simplices; this construction will produce, for any Lie or L∞ -algebra g, a smooth ∞-groupoid exp∆ (g) integrating g; other examples of this kind are the refinements exp∆ (g)diff and exp∆ (g)conn of exp∆ (g). In the following sections, we will investigate these examples and show how they naturally combine in ∞-Chern-Weil theory. ∞-Chern-Weil theory July 11, 2011 21 Smooth ∞-groups. With a useful notion of smooth ∞-groupoids and their morphisms thus established, we automatically obtain a good notion of smooth ∞-groups. This is accomplished simply by following the general principle by which essentially all basic constructions and results familiar from classical homotopy theory lift from the archetypical ∞-topos Top of (compactly generated) topological spaces (or, equivalently, of discrete ∞-groupoids) to any other ∞topos, such as our ∞-topos H of smooth ∞-groupoids. Namely, in classical homotopy theory a monoid up to higher coherent homotopy is a topological space X ∈ Top ≃ ∞Grpd equipped with A∞ -structure [Sta63] or, equivalently, an E1 -structure, i.e., a homotopical action of the little 1-cubes operad [May70]. A groupal A∞ -space – an ∞-group – is one where this homotopy-associative product is invertible, up to homotopy. Famously, May’s recognition theorem identifies such ∞-groups as being precisely, up to weak homotopy equivalence, loop spaces. This establishes an equivalence of pointed connected spaces with ∞-groups, given by looping Ω and delooping B: ∞Grp o Ω ≃ B / ∞Grpd∗ . Lurie shows in section 6.1.2 of [Lur09] (for ∞-groups) and in theorem 5.1.3.6 of [Lur11] that these classical statements have direct analogs in any ∞-topos. We are thus entitled to think of any (pointed) connected smooth ∞-groupoid X as the delooping BG of a smooth ∞-group G ≃ ΩX Smooth∞Grp o Ω ≃ B / H∗ = Smooth∞Grpd∗ , where we use boldface B to indicate that the delooping takes place in the ∞topos H of smooth ∞-groupoids. The most basic example for this we have already seen above: for G any Lie group the Lie groupoid BG described above is precisely the delooping of G – not in Top but in our H. In this article most smooth ∞-groups G appear in the form of their smooth delooping ∞-groupoids BG. Apart from Lie groups, the main examples that we consider will be the higher line and circle Lie groups Bn U (1) and Bn R that have arbitrary many delooping, as well as the nonabelian smooth 2-group String and the nonabelian smooth 6-group Fivebrane, which are smooth refinements of the higher connected covers of the Spin-group. 3.2.1 BG, BGconn and principal G-bundles with connection The standard example of a stack on manifolds is the classifying stack BG for G-principal bundles with G a Lie group. As an illustration of our setup, we describe what this looks like in terms of simplicial presheaves over the site CartSp. Then we discuss its differential refinements BGdiff and BGconn . ∞-Chern-Weil theory July 11, 2011 22 Definition 3.2.1. Let G be a Lie group. The smooth ∞-groupoid BG is defined to associate to a Cartesian space U the nerve of the action groupoid ∗//C ∞ (U, G), i.e., of the one-object groupoid with C ∞ (U, G) as its set of morphisms and composition given by the product of G-valued functions. Remark 3.2.2. Often this object is regarded over the site of all manifolds, where it is just a pre-stack, hence not fibrant. Its fibrant replacement over that site is the stack GBund : Manfdop → Grpd that sends a manifold to the groupoid of G-principal bundles over it. We may think instead of BG as sending a space to just the trivial G-principal bundle and its automorphisms. But since the site of Cartesian spaces is smaller, we have: Proposition 3.2.3. The object BG ∈ [CartSpop , sSet]proj,loc is fibrant. On the other hand, over the site of manifolds, every manifold itself is cofibrant. This means that to compute the groupoid of G-bundles on a manifold X in terms of morphisms of stacks over all manifolds, one usually passes to the fibrant replacement GBund of BG, then considers Hom(X, GBund) and uses the 2-Yoneda lemma to identify this with the groupoid GBund(X) of principal G-bundles on X. When working over CartSp instead, the situation is the opposite: here BG is already fibrant, but the manifold X is in general no longer cofibrant! To compute the groupoid of G-bundles on X, we pass to a cofibrant replacement of X given according to proposition 2 by the Čech nerve Č(U) of a differentiably good open cover and then compute Hom[CartSpop ,sSet] (Č(U), BG). To see that the resulting groupoid is again equivalent to GBund(X) (and hence to prove the above proposition by taking X = Rn ) one proceeds as follows: The object Č(U) is equivalent to the homotopy colimit in [CartSpop , sSet]proj over the simplicial diagram of its components   ` // ` // ` U /// ij Uij // i,j,k Uijk Č(U) ≃ hocolim · · · i i Z [k]∈∆ a ≃ ∆[k] · Ui0 ,...,ik . i0 ,··· ,ik (Here in the middle we are notationally suppressing the degeneracy maps for readability and on the second line we display for the inclined reader the formal coend expression that computes this homotopy colimit as a weighted colimit [Hov99]. The dot denotes the tensoring of simplicial presheaves over simplicial sets). Accordingly Hom(Č(U), BG) is the homotopy limit   o o {BG(Uij )} oo {BG(Uijk )} oo {BG(Ui )} Hom(Č(U), BG) ≃ holim · · · ooo Z Y . Hom(∆[k], BG(Ui0 ,...,ik )) ≃ [k]∈∆ i0 ,··· ,ik The last line tells us that an element g : Č(U) → BG in this Kan complex is a ∞-Chern-Weil theory July 11, 2011 23 diagram .. O O .O O O ∆[2] O O O ∆[1] O O ∆[0] .. O O .O O O g(2) / g(1) g(0) Q / i,j,k Q / i,j Q i BG(Uijk ) O O O BG(Uij ) O O BG(Ui ) of simplicial sets. This is a collection ({gi }, {gij }, {gijk }, · · · ), where • gi is a vertex in BG(Ui ); • gij is an edge in BG(Uij ); • gijk is a 2-simplex in BG(Uijk ) • etc. such that the k-th face of the n-simplex in BG(Ui0 ,··· ,in ) is the image of the (n − 1)-simplex under the k-th face inclusion BG(Ui0 ,··· ,îk ,··· ,in ) → BG(Ui0 ,··· ,in ). (And similarly for the coface maps, which we continue to disregard for brevity.) This means that an element g : Č(U) → BG is precisely an element of the set Č(U, BG) of nonabelian Čech cocycles with coefficients in BG. Specifically, by definition of BG, this reduces to • a collection of smooth maps gij : Uij → G, for every pair of indices i, j; • the constraint gij gjk gki = 1G on Uijk , for every i, j, k (the cocycle constraint). These are manifestly the data of transition functions defining a principal Gbundle over X. Similarly working out the morphisms (i.e., the 1-simplices) in Hom(Č(U), BG), we find that their components are collections hi : Ui → G of smooth functions, ′ such that gij = h−1 i gij hj . These are precisely the gauge transformations between the G-principal bundles given by the transition functions ({gij }) and ′ ({gij }). Since the cover {Ui → X} is good, it follows that we have indeed reproduced the groupoid of G-principal bundles Hom(Č(U), BG) = Č(U, BG) ≃ GBund(X) . Two cocycles define isomorphic principal G-bundles precisely when they define the same element in Čech cohomology with coefficients in the sheaf of smooth ∞-Chern-Weil theory 24 July 11, 2011 functions with values in G. Thus we recover the standard fact that isomorphism classes of principal G-bundles are in natural bijection with H 1 (X, G). We now consider a differential refinement of BG. Definition 3.2.4. Let G be a Lie group with Lie algebra g. The smooth ∞groupoid BGconn is defined to associate with a Cartesian space U the nerve of the action groupoid Ω1 (U, g)//C ∞ (U, G). This is over U the groupoid BGconn (U ) • whose set of objects is the set of smooth g-valued 1-forms A ∈ Ω1 (U, g); • whose morphisms g : A → A′ are labeled by smooth functions g ∈ C ∞ (U, G) such that they relate the source and target by a gauge transformation A′ = g −1 Ag + g −1 dg , where g −1 Ag denotes pointwise the adjoint action of G on g and g −1 dg is the pull-back g ∗ (θ) of the Maurer-Cartan form θ ∈ Ω1 (G, g). With X and Č(U) as before we now have: Proposition 3.2.5. The smooth ∞-groupoid BGconn is fibrant and there is a natural equivalence of groupoids H(X, BGconn ) ≃ GBundconn (X) , where on the right we have the groupoid of G-principal bundles on X equipped with connection. This follows along the above lines, by unwinding the nature of the simplicial hom-set Č(U, BGconn ) := Hom(Č(U), BGconn ) of nonabelian Čech cocycles with coefficients in BGconn . Such a cocycle is a collection ({Ai }, {gij }) consisting of • a 1-form Ai ∈ Ω1 (Ui , g) for each index i; • a smooth function gij : Uij → G, for all indices i, j; −1 −1 • the gauge action constraint Aj = gij Ai gij + gij dgij on Uij , for all indices i, j; • the cocycle constraint gij gjk gki = 1G on Uijk , for all indices i, j, k. These are readily seen to be the data defining a g-connection on a principal G-bundle over X. Notice that there is an evident “forget the connection”-morphism BGconn → BG, given over U ∈ CartSp by g g (A → A′ ) 7→ (• → •) . ∞-Chern-Weil theory 25 July 11, 2011 We denote the set of isomorphism classes of principal G-bundles with connection by the symbol H 1 (X, G)conn . Thus we obtain a morphism H 1 (X, G)conn → H 1 (X, G). Finally, we introduce a smooth ∞-groupoid BGdiff in between BG and BGconn . This may seem a bit curious, but we’ll see in section 4.3 how it is the degree one case of a completely natural and noteworthy general construction. Informally, BGdiff is obtained from BG by freely decorating the vertices of the simplices in BG by elements in Ω1 (U, g). More formally, we have the following definition. Definition 3.2.6. Let G be a Lie group with Lie algebra g. The smooth ∞groupoid BGdiff is defined to associate with a Cartesian space U the nerve of the groupoid 1. whose set of objects is Ω1 (U, g); (g,a) 2. a morphism A → A′ is labeled by g ∈ C ∞ (U, G) and a ∈ Ω1 (U, g) such that A = g −1 A′ g + g −1 dg + a 3. composition of morphisms is given by (g, a) ◦ (h, b) = (gh, h−1 ah + h−1 dh + b). Remark 3.2.7. This definition intentionally carries an evident redundancy: given any A, A′ and g the element a that makes the above equation hold does exist uniquely; the 1-form a measures the failure of g to constitute a morphism from A to A′ in BGconn . We can equivalently express the redundancy of a by saying that there is a natural isomorphism between BGdiff and the direct product of BG with the codiscrete groupoid on the sheaf of sets Ω1 (−; g). Proposition 3.2.8. The evident forgetful morphism BGconn → BG factors through BGdiff by a monomorphism followed by an acyclic fibration (in the global model structure) ∼ BGconn ֒→ BGdiff ։ BG . 3.2.2 BG2 , and nonabelian gerbes and principal 2-bundles We now briefly dicuss the first case of G-principal ∞-bundles after ordinary principal bundles, the case where G is a Lie 2-group: G-principal 2-bundles. When G = AUT(H) the automorphism 2-group of a Lie group H (see below) these structures have the same classification (though are conceptually somewhat different from) the smooth version of the H-banded gerbes of [Gir71] (see around def. 7.2.2.20 in [Lur09] for a conceptually clean account in the modern context of higher toposes): both are classified by the nonabelian cohomology 1 HSmooth (−, AUT(H)) with coefficients in that 2-group. But the main examples ∞-Chern-Weil theory July 11, 2011 26 of 2-groups that we shall be interested in, namely string 2-groups, are not equivalent to AUT(H) for any H, hence the 2-bundles considered here are strictly more general than Giraud’s gerbes. The literature knows what has been called nonabelian bundle gerbes, but despite their name these are not Giraud’s gerbes, but are instead models for the total spaces of what we call here principal 2bundles. A good discussion of the various equivalent incarnations of principal 2-bundles is in [NiWa11]. To start with, note the general abstract notion of smooth 2-groups: Definition 3.2.9. A smooth 2-group is a 1-truncated group object in H = Sh∞ (CartSp). These are equivalently given by their (canonically pointed) delooping 2-groupoids BG ∈ H, which are precisely, up to equivalence, the connected 2-truncated objects of H. For X ∈ H any object, G2Bundsmooth (X) := H(X, BG) is the 2-groupoid of smooth G-principal 2-bundles on G. While nice and abstract, in applications one often has – or can get – hold of a strict model of a given smooth 2-group. The following definitions can be found recalled in any reference on these matters, for instance in [NiWa11]. Definition 3.2.10. 1. A smooth crossed module of Lie groups is a pair of homomorphisms ∂ : G1 → G0 and ρ : G0 → Aut(G1 ) of Lie groups, such that for all g ∈ G0 and h, h1 , h2 ∈ G1 we have ρ(∂h1 )(h2 ) = h1 h2 h−1 1 and ∂ρ(g)(h) = g∂(h)g −1 . 2. For (G1 → G0 ) a smooth crossed module, the corresponding strict Lie / 2-group is the smooth groupoid G0 × G1 / G0 , whose source map is given by projection on G0 , whose target map is given by applying ∂ to the second factor and then multiplying with the first in G0 , and whose composition is given by multiplying in G1 . This groupoid has a strict monoidal structure with strict inverses given by equipping G0 × G1 with the semidirect product group structure G0 ⋉ G1 induced by the action ρ of G0 on G1 . 3. The corresponding one-object strict smooth 2-groupoid we write B(G1 → G0 ). As a simplicial object (under Duskin nerve of 2-categories) this is of the form   / ×2 / ×3 /∗ . / G × G1 B(G1 → G0 ) = cosk3 G×3 / G0 0 × G1 / 0 Examples. 1. For A any abelian Lie group, A → 1 is a crossed module. Conversely, for A any Lie group A → 1 is a crossed module precisely if A is abelian. We write B2 A = B(A → 1). This case and its generalizations is discussed below in 3.2.3. ∞-Chern-Weil theory July 11, 2011 27 2. For H any Lie group with automorphism Lie group Aut(H), the morAd phism H → Aut(H) that sends group elements to inner automorphisms, together with ρ = id, is a crossed module. We write AUT(H) := (H → Aut(H)) and speak of the automorphism 2-group of H, because this is ≃ AutH (BH). 3. For G an ordinary Lie group and c : BG → B3 U (1) a morphism in H (see the 3.2.3 for a discussion of Bn U (1)), its homotopy fiber BĜ → BG is the delooping of a smooth 2-group Ĝ. If G is compact, simple and simply connected, then this is equivalent ([Sch10], section 4.1) to a strict 2-group (Ω̂G → P G) given by a U (1)-central extension of the loop group of G, as described in [BSCS07]. This is called the string 2-group extension of G by c. We come back to this in 5.1. Observation 3.2.11. For every smooth crossed module, its delooping object B(G1 → G0 ) is fibrant in [CartSpop , sSet]. Proof. Since (G1 → G0 ) induces a strict 2-group, there are horn fillers defined by the smooth operations in the 2-group: we can always solve for the missing face in a horn in terms of an expression involving the smooth composite-operations and inverse-operations in the 2-group.  Proposition 3.2.12. Suppose that the smooth crossed module (G1 → G0 ) is such that the quotient π0 G = G0 /G1 is a smooth manifold and the projection G0 → G0 /G1 is a submersion. Then B(G1 → G0 ) is fibrant also in [CartSpop , sSet]proj,loc . Proof. We need to show that for {Ui → Rn } a good open cover, the canonical descent morphism B(C ∞ (Rn , G1 ) → C ∞ (Rn , G0 )) → [CartSpop , sSet](Č(U), B(G1 → G0 )) is a weak homotopy equivalence. The main point to show is that, since the Kan complex on the left is connected by construction, also the Kan complex on the right is. To that end, notice that the category CartSp equipped with the open cover topology is a Verdier site in the sense of section 8 of [DHI04]. By the discussion there it follows that every hypercover over Rn can be refined by a split hypercover, and these are cofibrant resolutions of Rn in both the global and the local model structure [CartSpop , sSet]proj,loc . Since also Č(U) → Rn is a cofibrant resolution and since BG is fibrant in the global structure by observation 3.2.11, it follows from the existence of the global model structure that morphisms out of Č(U) into B(G1 → G0 ) capture all cocycles over any hypercover over Rn , hence that 1 π0 [CartSpop , sSet](Č(U), B(G1 → G0 )) ≃ Hsmooth (Rn , (G1 → G0 )) ∞-Chern-Weil theory July 11, 2011 28 is the standard Čech cohomology of Rn , defined as a colimit over refinements of covers of equivalence classes of Čech cocycles. Now by prop. 4.1 of [NiWa11] (which is the smooth refinement of the statement of [BS09] in the continuous context) we have that under our assumptions on (G1 → G0 ) there is a topological classifying space for this smooth Čech cohomology set. Since Rn is topologically contractible, it follows that this is the singleton set and hence the above descent morphism is indeed an isomorphism on π0 . Next we can argue that it is also an isomorphism on π1 , by reducing to the analogous local trivialization statement for ordinary principal bundles: a loop in [CartSpop , sSet](Č(U), B(G1 → G0 )) on the trivial cocycle is readily seen to be a G0 //(G0 ⋉ G1 )-principal groupoid bundle, over the action groupoid as indicated. The underlying G0 ⋉ G1 -principal bundle has a trivialization on the contractible Rn (by classical results or, in fact, as a special case of the previous argument), and so equivalence classes of such loops are given by G0 -valued smooth functions on Rn . The descent morphism exhibits an isomorphism on these classes. Finally the equivalence classes of spheres on both sides are directly seen to be smooth ker(G1 → G0 )-valued functions on both sides, identified by the descent morphism.  Corollary 3.2.13. For X ∈ SmoothMfd ⊂ H a paracompact smooth manifold, and (G1 → G0 ) as above, we have for any good open cover {Ui → X} that the 2-groupoid of smooth (G1 → G0 )-principal 2-bundles is (G1 → G0 )Bund(X) := H(X, B(G1 )) ≃ [CartSpop , sSet](Č(U), B(G1 → G0 )) and its set of connected components is naturally isomorphic to the nonabelian Čech cohomology 1 π0 H(X, B(G1 → G0 )) ≃ Hsmooth (X, (G1 → G0 )) . 3.2.3 Bn U (1), Bn U (1)conn , circle n-bundles and Deligne cohomology A large class of examples of smooth ∞-groupoids is induced from chain complexes of sheaves of abelian groups by the Dold-Kan correspondence [GJ99]. Proposition 3.2.14. The Dold-Kan correspondence is an equivalence of categories DK / Ch+ sAb , • o N• between non-negatively graded chain complexes and simplicial abelian groups, where N• forms the normalized chains complex of a simplicial abelian group A∆ . Composed with the forgetful functor sAb → sSet and prolonged to a functor on sheaves of chain complexes, the functor op DK : [CartSpop , Ch+ • ] → [CartSp , sSet] ∞-Chern-Weil theory 29 July 11, 2011 takes degreewise surjections to fibrations and degreewise quasi-isomorphisms to weak equivalences in [CartSpop , sSet]proj . We will write an element (A• , ∂) of Ch+ • as · · · → Ak → Ak−1 → · · · → A2 → A1 → A0 and will denote by [1] the “shift on the left” functor on chain complexes defined by (A• [1])k = Ak−1 , i.e., A• [1] is the chain complex · · · → Ak−1 → · · · → A2 → A1 → A0 → 0. Remark 3.2.15. The reader used to cochain complexes, and so to the shift functor (A• [1])k = Ak+1 could at first be surprised by the minus sign in the shift functor on chain complexes; but the shift rule is actually the same in both contexts, as it is evident by writing it as (A• [1])k = Ak+deg(∂) . For A any abelian group, we can consider A as a chain complex concentrated in degree zero, and so A[n] will be the chain complex consisting of A concentrated in degree n. Definition 3.2.16. Let A be an abelian Lie group. Define the simplicial presheaf Bn A to be the image under DK of the sheaf of complexes C ∞ (−, A)[n]: Bn A : U 7→ DK(C ∞ (U, A) → 0 → · · · → 0) , with C ∞ (U, A) in degree n. Similarly, for K → A a morphism of abelian groups, write Bn (K → A) for the image under DK of the complex of sheaves of abelian groups (C ∞ (−, K) → C ∞ (−, A) → 0 → · · · → 0) Id with C ∞ (−, A) in degree n; for n ≥ 1 we write EBn−1 A for Bn−1 (A → A). Proposition 3.2.17. For n ≥ 1 the object Bn A is indeed the delooping of the object Bn−1 A. Proof. This means that there is an ∞-pullback diagram [Lur09] Bn−1 A /∗  ∗  / Bn A . This is presented by the corresponding homotopy pullback in [CartSpop , sSet]. Consider the diagram Bn−1 A / EBn−1 A  ∗  / Bn A ∼ /∗ , ∞-Chern-Weil theory 30 July 11, 2011 The right vertical morphism is a replacement of the point inclusion by a fibration and the square is a pullback in [CartSpop , sSet] (the pullback of presheaves is computed objectwise and under the DK-correspondence may be computed n−1 A as the homotopy in Ch+ • , where it is evident). Therefore this exhibits B pullback, as claimed.  Proposition 3.2.18. For A = Z, R, U (1) and all n ≥ 1 we have that Bn A satisfies descent over CartSp in that it is fibrant in [CartSpop , sSet]proj,loc . Proof. One sees directly in terms of Čech cocycles that the homotopy groups based at the trivial cocycle in the simplicial hom-sets [CartSpop , sSet](Č(U), Bn A) and [CartSpop , sSet](U, Bn A) are naturally identified. Therefore it is sufficient to show that ∗ ≃ π0 [CartSpop , sSet](U, Bn A) → π0 [CartSpop , sSet](Č(U), Bn A) is an isomorphism. This amounts to proving that the n-th Čech cohomology group of U with coefficients in Z, R or U (1) is trivial, which is immediate since U is contractible (for U (1) one uses the isomophism H n (U, U (1)) ≃ H n+1 (U, Z) in Čech cohomology).  Definition 3.2.19. For X a smooth ∞-groupoid and QX → X a cofibrant replacement we say that ∼ g • for X ←−loc −− QX → Bn A a span in [CartSpop , sSet], the corresponding (Bn−1 )A-principal n-bundle is the ∞-pullback /∗ P  X g  / Bn A hence the ordinary pullback in [CartSpop , sSet] / EBn−1 A . P  QX g  / Bn A ∼loc  X • the Kan complex (Bn−1 A)Bund(X) := H(X, Bn A) is the n-groupoid of smooth Bn−1 A-principal n-bundles on X. ∞-Chern-Weil theory 31 July 11, 2011 Proposition 3.2.20. For X a smooth paracompact manifold, the n-groupoid (Bn−1 A)Bund(X) is equivalent to the n-groupoid Č(U, Bn A) of degree n Čech cocycles on X with coefficients in the sheaf of smooth functions with values in A. In particular π0 (Bn−1 A)Bund(X) = π0 H(X, Bn A) ≃ H n (X, A) is the Čech cohomology of X in degree n with coefficients in A. Proof. This follows from the same arguments as in the previous section given for the more general nonabelian Čech cohomology.  We will be interested mainly in the abelian Lie group A = U (1). The exponential exact sequence 0 → Z → R → U (1) → 1 induces an acyclic fibration ∼ (in the global model structure) Bn (Z ֒→ R) ։ Bn U (1), and one has the long fibration sequence Bn (Z ֒→ R) / Bn+1 Z · · · ≀ · · · → Bn Z / Bn R  / Bn U (1) from which one recovers the classical isomorphism H n (X, U (1)) ≃ H n+1 (X, Z). Next, we consider differential refinements of these cohomology groups. Definition 3.2.21. The smooth ∞-groupoid Bn U (1)conn is the image via the Dold-Kan correspondence of the Deligne complex U (1)[n]∞ D , i.e., of the chain complex of sheaves of abelian groups   dlog d d ∞ 1 n U (1)[n]∞ := C (−, U (1)) − − → Ω (−, R) − → · · · − → Ω (−, R) D concentrated in degrees [0, n]. Similarly, the smooth ∞-groupoid Bn (Z ֒→ R)conn is the image via the Dold-Kan correspondence of the complex of sheaves of abelian groups   d d d ∞ 1 n Z[n + 1]∞ := Z ֒→ C (−, R) − → Ω (−, R) − → · · · − → Ω (−, R) , D concentrated in degrees [0, n + 1]. ∞ The natural morphism of sheaves of complexes Z[n + 1]∞ D → U (1)[n]D is an acyclic fibration and so we have an induced acyclic fibration (in the global ∼ model structure) Bn (Z ֒→ R)conn ։ Bn U (1)conn . Therefore, we find a natural isomorphism 0 ∞ H 0 (X, U (1)[n]∞ D ) ≃ H (X, Z[n + 1]D ) and a commutative diagram H 0 (X, U (1)[n]∞ D) ≃ ≃  H 0 (X, Z[n + 1]∞ D) / H n (X, U (1))  / H n+1 (X, Z) . ∞-Chern-Weil theory 32 July 11, 2011 Definition 3.2.22. We denote the cohomology group H 0 (X, Z[n]∞ D ) by the symbol Ĥ n (X, Z), and call it the n-th differential cohomology group of X (with integer coefficients). The natural morphism Ĥ n (X, Z) → H n (X, Z) will be called the differential refinement of ordinary cohomology. Remark 3.2.23. The reader experienced with gerbes and higher gerbes will have recognized that H n (X, U (1)) ≃ H n+1 (X, Z) is the set of isomorphim classes of U (1)-(n − 1)-gerbes on a manifold X, whereas H 0 (X, U (1)[n]∞ D) ≃ Ĥ n+1 (X, Z) is the set of isomorphim classes of U (1)-(n − 1)-gerbes with connection on X, and that the natural morphism Ĥ n+1 (X, Z) → H n+1 (X, Z) is ‘forgetting the connection’, see, e.g., [Gaj97]. ∞ The natural projection U (1)[n]∞ D → C (−, U (1)[n] is a fibration, so we have n n a natural fibration B U (1)conn ։ B U (1), and, as for the case of Lie groups, we have a natural factorization ∼ Bn U (1)conn ֒→ Bn U (1)diff ։ Bn U (1) into a monomorphism followed by an acyclic fibration (in the global model structure). The smooth ∞-groupoid Bn U (1)diff is best defined at the level of chain complexes, where we have the well known “cone trick” from homological algebra to get the desired factorization. In the case at hand, it works as follows: let cone(ker π ֒→ U (1)[n]∞ D (U )) be the mapping cone of the inclusion of the kernel ∞ ∞ of π : U (1)[n]∞ D → C (−, U (1)[n] into U (1)[n]D , i.e., the chain complex dlog / Ω1 (−) d / Ω2 (−) 6 mmmm ⊕ qq ⊕ ⊕ Id m m qq mm / ··· / Ω2 (−) Ω1 (−) C ∞ (−, U (1)) d d / Ωn (−) 8 p p ⊕ ppIdp p / Ωn (−) /0 / ··· d ∞ Then U (1)[n]∞ D naturally injects into cone(ker π ֒→ U (1)[n]D ), and π induces a ∞ ∞ morphism of complexes π : cone(ker π ֒→ U (1)[n]D ) → C (−, U (1)[n] which is an acyclic fibration; the composition π ∞ → C ∞ (−, U (1)[n] U (1)[n]∞ D ֒→ cone(ker π ֒→ U (1)[n]D ) − is the sought for factorization. Definition 3.2.24. Define the simplicial presheaf Bn U (1)diff = DK (cone(ker π ֒→ U (1)[n]∞ D )) to be the image under the Dold-Kan equivalence of the chain complex of sheaves of abelian groups cone(ker π ֒→ U (1)[n]∞ D ). The last smooth ∞-groupoid we introduce in this section is the natural ambient for curvature forms to live in. As above, we work at the level of sheaves of chain complexes first. So, let ♭R[n]∞ dR be the truncated de Rham complex   d d d n+1 1 2 ♭R[n + 1]∞ dR := Ω (−) → Ω (−) → · · · → Ωcl (−) ∞-Chern-Weil theory July 11, 2011 33 seen as a chain complex concentrated in degrees [0, n]. There is a natural morphism of complexes of sheaves, which we call the curvature map, ∞ curv : cone(ker π ֒→ U (1)[n]∞ D ) → ♭R[n + 1]dR given by the projection cone(ker π ֒→ U (1)[n]∞ D ) → ker π[1] in degrees [1, n] and given by the de Rham differential d : Ωn (−) → Ωn+1 cl (−) in degree zero. Note that the preimage of (0 → 0 → · · · → Ωn+1 (−)) via curv is precisely the cl complex U (1)[n]∞ , and that for n = 1 the induced morphism D 2 curv : U (1)[1]∞ D → Ωcl (−) is the map sending a connection on a principal U (1)-bundle to its curvature 2-form. Definition 3.2.25. The smooth ∞-groupoid ♭dR Bn+1 R is   d d d ♭dR Bn+1 R = DK Ω1 (−) → Ω2 (−) → · · · → Ωn+1 (−) , cl the image under DK of the truncated de Rham complex. The above discussion can be summarized as Proposition 3.2.26. In [CartSpop , sSet]proj,loc we have a natural commutative diagram / Ωn+1 (−) . Bn U (1)conn cl  Bn U (1)diff curv  / ♭dR Bn+1 R ≀  Bn U (1) whose upper square is a pullback and whose lower part presents a morphism of smooth ∞-groupoids from Bn U (1) to ♭dR Bn+1 R. We call this morphism the curvature characteristic map. Remark 3.2.27. One also has a natural (Z ֒→ R) version of Bn U (1)diff , i.e., we have a smooth ∞-groupoid Bn (Z ֒→ R)diff with a natural morphism ∼ Bn (Z ֒→ R)diff ։ Bn U (1)diff which is an acyclic fibration in the global model structure. ∞-Chern-Weil theory July 11, 2011 34 Differential ∞-Lie integration 4 As the notion of L∞ -algebra generalizes that of Lie algebra so that of Lie ∞group generalizes that of Lie group. We describe a way to integrate an L∞ algebra g to the smooth delooping BG of the corresponding Lie ∞-group by a slight variant of the construction of [Hen08]. (Recall from the introduction that we use “Lie” to indicate generalized smooth structure which may or may not be represented by smooth manifolds). Then we generalize this to a differential integration: an integration of an L∞ -algebroid g to smooth ∞-groupoids BGdiff and BGconn . Cocycles with coefficients in BG give G-principal ∞-bundles; those with coefficients in BGdiff support the ∞-Chern-Weil homomorphism, those with coefficients in BGconn give G-principal ∞-bundles with connection. 4.1 Lie ∞-Algebroids: cocycles, invariant polynomials and CS-elements We summarize the main definitions and properties of L∞ -algebroids from [LS93, LM95], and their cocycles, invariant polynomials and Chern-Simons elements from [SSS09b, SSS09c]. Definition 4.1.1. Let R be a commutative R-algebra, and let g be a chain complex of finitely generated (in each degree) R-modules, concentrated in nonnegative degree. Then a (reduced) L∞ -algebroid (or Lie ∞-algebroid) structure on g is the datum of a degree 1 R-derivation dCE(g) on the exterior algebra ∧•R g∗ := Sym•R (g∗ [−1]) (the free graded commutative algebra on the shifted dual of g), which is a differential (i.e., squares to zero) compatible with the differential of g. A chain complex g endowed with an L∞ -algebroid structure will be called a L∞ -algebroid. The differential graded commutative algebra CE(g) := (∧•R g∗ , dCE(g )) will be called the Chevalley-Eilenberg algebra of the L∞ -algebroid g. A morphism of L∞ -algebroids g1 → g2 is defined to be a dgca morphism CE(g2 ) → CE(g1 ) Since all L∞ -algebroids which will be met in this paper will be reduced, we will just say Lie ∞-algebroid to mean reduced Lie ∞-algebroid in what follows. Remark 4.1.2. • Lie ∞-algebroids could be more intrinsically defined as follows: the category L∞ Algd ⊂ dgAlgop of L∞ -algebroids is the full subcategory of the opposite of that of differential graded commutative R-algebras on those dg-algebras whose underlying graded-commutative algebra is free on a finitely generated graded module concentrated in positive degree over the commutative algebra in degree 0. ∞-Chern-Weil theory July 11, 2011 35 • The dual g∗ = Hom−• R (g, R) is a cochain complex concentrated in nonnegative degrees. In particular the shift to the right functor [−1] changes it into a cochain complex concentrated in strictly positive degrees. • The restriction to finite generation is an artifact of dualizing g rather than working with graded alternating multilinear functions on g as the masters (Chevalley-Eilenberg-Koszul) did in the original ungraded case. In particular, the direct generalization of their approach consists in working with the cofree connected cocommutative coalgebra cogenerated by g[1], see, e.g., [Sta93]. At least for L∞ -algebras, there are alternate definitions and conventions as to bounds on the grading, signs, etc. cf. [LS93, LM95] among others. • Given an L∞ -algebroid g, the degree 0 part CE(g)0 of CE(g) is is a commutative R-algebra which we think of as the formal dual to the space of objects over which the Lie ∞-algebroid is defined. If CE(g)0 = R equals the ground field, we say we have an ∞ algebroid over the point, or equivalently that we have an L∞ -algebra. • The underlying algebra in degree 0 can be generalized to an algebra over some Lawvere theory. In particular in a proper setup of higher differential geometry, we would demand CE(g)0 to be equipped with the structure of a C ∞ -ring. Example 4.1.3. • For g an ordinary (finite-dimensional) Lie algebra, CE(g) is the ordinary Chevalley-Eilenberg algebra with coefficients in R. The differential is given by the dual of the Lie bracket, dCE(g) = [−, −]∗ extended uniquely as a graded derivation. • For a dg-Lie algebra g = (g• , ∂), the differential is dCE(g) = [−, −]∗ + ∂ ∗ . • In the general case, the total differential is further determined by (and is equivalent to) a sequence of higher multilinear brackets [LS93]. • For n ∈ N, the L∞ -algebra bn−1 R is defined in terms of CE(bn−1 R) which is the dgc-algebra on a single generator in degree n with vanishing differential. • For X a smooth manifold, its tangent Lie algebroid is defined to have CE(T X) = (Ω• (X), ddR ) the de Rham algebra of X. Notice that Ω• (X) = ∧•C ∞ (X) Γ(T ∗ X). We shall extensively use the tangent Lie algebroid T (U × ∆k ) where U ∈ CartSp and ∆k is the standard k-simplex. ∞-Chern-Weil theory July 11, 2011 36 Definition 4.1.4. For g a Lie ∞-algebroid and n ∈ N, a cocycle in degree n on g is, equivalently • an element µ ∈ CE(g) in degree n, such that dCE(g) µ = 0; • a morphism of dg-algebras µ : CE(bn−1 R) → CE(g); • a morphism of Lie ∞-algebroids µ : g → bn−1 R. Example 4.1.5. • For g an ordinary Lie algebra, a cocycle in the above sense is the same as a Lie algebra cocycle in the ordinary sense (with values in the trivial module). • For X a smooth manifold, a cocycle in degree n on the tangent Lie algebroid T X is precisely a closed n-form on X. For our purposes, a particularly important Chevalley-Eilenberg algebra is the Weil algebra. Definition 4.1.6. The Weil algebra of an L∞ -algebra g is the dg-algebra W(g) := (Sym• (g∗ [−1] ⊕ g∗ [−2]), dW (g) ) , where the differential on the copy g∗ [−1] is the sum dW (g) |g∗ = dCE(g) + σ , with σ : g∗ → g∗ [−1] is the grade-shifting isomorphism, i.e. it is the identity of g∗ seen as a degree 1 map g∗ [−1] → g∗ [−2], extended as a graded derivation, and where dW (g) ◦ σ = −σ ◦ dW (g) . Proposition 4.1.7. The Weil algebra is a representative of the free differential graded commutative algebra on the graded vector space g∗ [−1] in that there exist a natural isomorphism Homdgca (W (g), Ω• ) ≃ Homgr−vect (g∗ [−1], Ω• ), for Ω• an arbitrary dgca. Moreover, the Weil algebra is precisely that algebra with this property for which the projection morphism i∗ : g∗ [−1] ⊕ g∗ [−2] → g∗ [−1] of graded vector spaces extends to a dg-algebra homomorphism i∗ : W(g) → CE(g) . Notice that the free dgca on a graded vector space is defined only up to isomorphism. The condition on i∗ is what picks the Weil algebra among all free dg-algebras. A proof of the above proposition can be found, e.g., in [SSS09b]. Equivalently, one can state the freeness of the Weil algebra by saying that the dgca-morphisms A : W(g) → Ω• are in natural bijection with the degree 1 elements in the graded vector space Ω• ⊗ g. ∞-Chern-Weil theory July 11, 2011 37 Example 4.1.8. • For g an ordinary Lie algebra, W(g) is the ordinary Weil algebra [Car50a]. In that paper, H. Cartan defines a g-algebra as an analog of the dg-algebra Ω• (P ) of differential forms on a principal bundle, i.e. as a dg-algebra equipped with operations iξ and Lξ for all ξ ∈ g satisfying the usual relations, including Lξ = diξ + iξ d. Next, Cartan introduces the Weil algebra W(g) as the universal g-algebra and identifies a g-connection A on a principal bundle P as a morphism of g-algebras A : W(g) → Ω• (P ). This can in turn be seen as a dgca morphism satisfying the CartanEhresmann conditions, and it is this latter point of view that we generalize to an arbitrary L∞ -algebra. • The dg-algebra W(bn−1 R) of bn−1 R is the free dg-algebra on a single generator in degree n. As a graded algebra, it has a generator b in degree n and a generator c in degree (n+1) and the differential acts as dW : b 7→ c. Note that, since dCE b = 0, this is equivalent to c = σb. Remark 4.1.9. Since the Weil algebra is itself a dg-algebra whose underlying graded algebra is a graded symmetric algebra, it is itself the CE-algebra of an L∞ -algebra. The L∞ -algebra thus defined we denote inn(g): CE(inn(g)) = W(g) . Note that the underlying graded vector space of inn(g) is g ⊕ g[1]. Looking at W(g) as the Chevalley-Eilenberg algebra of inn(g) we therefore obtain the following descrition of morphisms out of W(g): for any dgca Ω• , a dgca morphism W(g) → Ω• is the datum of a pair (A, FA ), where A and FA are a degree 1 and a degree 2 element in Ω• ⊗ g, respectively, such that (A, FA ) satisfies the Maurer-Cartan equation in the L∞ -algebra Ω• ⊗ inn(g). The Maurer-Cartan equation actually completely determines FA in terms of A; this is an instance of the freeness property of the Weil algebra stated in Proposition 4.1.7. Definition 4.1.10. For X a smooth manifold, a g-valued connection form on X is a morpism of Lie ∞-algebroids A : T X → inn(g), hence a morphism of dg-algebras A : W(g) → Ω• (X) . Remark 4.1.11. A g-valued connection form on X can be equivalently seen as an element A in the set Ω1 (X, g) of degree 1 elements in Ω• (X) ⊗ g, or as a pair (A, FA ), where A ∈ Ω1 (U, g), FA ∈ Ω2 (U, g), and A and FA are related by the Maurer-Cartan equation in Ω• (X, inn(g)). The element FA is called the curvature form of A. ∞-Chern-Weil theory July 11, 2011 38 Example 4.1.12. If g is an ordinary Lie algebra, then a g-valued connection form A on X is a 1-form on X with coefficients in g, i.e. it is naturally a connection 1-form on a trivial principal G-bundle on X. The element FA in Ω2 (X, g) is then given by equation 1 FA = dA + [A, A], 2 so it is precislely the usual curvature form of A. The last ingredient we need to generalize from Lie algebras to L∞ -algebroids is the algebra inv(g) of invariant polynomials. Definition 4.1.13. An invariant polynomial on g is a dW(g) -closed element h−i in Sym• (g∗ [−2]) ⊂ W(g). To see how this definition encodes the classical definition of invariant polynomials on a Lie algebra, notice that invariant polynomials are elements of Sym• (g∗ [−2]) that are both horizontal and ad-invariant (“basic forms”). Namely, for any v ∈ g we have, for an invariant polynomial h−i, the identities ιv h−i = 0 (horizontality) and Lv h−i = 0 (ad-invariance) , where ιv : W(g) → W(g) the contraction derivation defined by v and Lv := [dW(g) , ιv ] is the corresponding Lie derivative. We want to identify two indecomposable invariant polynomials which differ by a “horizontal shift”. A systematic way of doing this is to introduce the following equivalence relation on the dgca of all invariant polynomials: we say that two invariant polynomials h−i1 , h−i2 are horizontally equivalent if there exists ω in ker(W(g) → CE(g)) such that h−i1 = h−i2 + dW ω . Write inv(g)V for the quotient graded vector space of horizontal equivalence classes of invariant polynomials. Definition 4.1.14. The dgca inv(g) is defined as the free polynomial algebra on the graded vector space invV (g), endowed with the trivial differential. Remark 4.1.15. A choice of a linear section to the projection {invariant polynomials} → inv(g)V gives a morphism of graded vector spaces inv(g)V → W(g), canonical up to horizontal homotopy, that sends each equivalence class to a representative. This linear morphism uniquely extends to a dg-algebra homomorphism inv(g) → W(g) . ∞-Chern-Weil theory July 11, 2011 39 Remark 4.1.16. The algebra inv(g) is at first sight a quite abstract construction which is apparently unrelated to an equivalence relation on indecomposable invariant polynomials. A closer look shows that it is actually not so. Namely, only indecomposable invariant polynomials can be representatives for the nonzero equivalence classes. Indeed, if h−i1 and h−i2 are two nontrivial invariant polynomials, then since the cohomology of W(g) is trivial in positive degree, there is cs1 in W(g) (not necessarily in ker(W(g) → CE(g)) ) such that dW cs1 = h−i1 , but then cs1 ∧ h−i2 is a horizontal trivialization of h−i1 ∧ h−i2 . One therefore obtains a very concrete description of the algebra inv(g) as follows: one picks a representative indecomposable invariant polynomial for each horizontal equivalence class and considers the subalgebra of W(g) generated by these representatives. The morphism inv(g) → W(g) is then realized as the inclusion of this subalgebra into the Weil algebra. Different choices of representative generators lead to distinct but equivalent subalgebras: each one is isomorphic to the others via an horizontal shift in the generators. Remark 4.1.17. For g an ordinary reductive Lie algebra, definition 4.1.14 reproduces the traditional definition of the algebra of adg -invariant polynomials. Indeed, for a Lie algebra g, the condition dW(g) h−i = 0 is precisely the usual adg invariance of an element h−i in Sym• g∗ [−2]. Morover, the horizonal equivalence on indecomposables is trivial in this case and it is a classical fact (for instance theorem I on page 242 in volume III of [GHV73]) that the graded algebra of adg -invariant polynomials is indeed free on the space of indecomposables. Definition 4.1.18. For any dgca morphism A : W(g) → Ω• , the composite morphism inv(g) → W(g) → Ω• is the evaluation of invariant polynomials on the element FA . In particular, if X is a smooth manifold and A is a g-valued connection form on X, then the image of FA : inv(g) → Ω• (X) is a collection of differential forms on X, to be called the curvature characteristic forms of A. Example 4.1.19. For the L∞ -algbera bn−1 R, in the notations of Example 4.1.8, one has inv(bn−1 R) = R[c]. Definition 4.1.20. We say an invariant polynomial h−i on g is in transgression with a cocycle µ if there exists an element cs ∈ W(g) such that 1. i∗ cs = µ; 2. dW(g) cs = h−i. We call cs a Chern-Simons element for µ and h−i. For ordinary Lie algebras this reduces to the classical notion, for instance 6.13 in vol. III of [GHV73]. Remark 4.1.21. If we think of inv(g) ⊂ ker i∗ as a subcomplex of the kernel of i∗ , then this transgression exhibits the connecting homomorphism H n−1 (CE(g)) → H n (ker i∗ ) of the long sequence in cohomology induced from the short exact sei∗ quence ker i∗ → W(g) → CE(g). If we think of ∞-Chern-Weil theory July 11, 2011 40 • W(g) as differential forms on the total space of a universal principal bundle, • CE(g) as differential forms on the fiber, • inv(g) as forms on the base, then the above notion of transgression is precisely the classical one of transgression of forms in the setting of fiber bundles (for instance section 9 of [Bor55]). Example 4.1.22. • For g a semisimple Lie algebra with h−, −i the Killing form invariant polynomial, the corresponding cocycle in transgression is µ3 = 12 h−, [−, −]i. The Chern-Simons element witnessing this transgression is cs = hσ(−), −i+ 1 2 h−, [−, −]i. • For the Weil ement of the cocycle b (as the invariant algebra W(bn−1 R) of Example 4.1.8, the element b (as elWeil algebra) is a Chern-Simons element transgressing the element of the Chevalley-Eilenberg algebra CE(bn−1 R)) to polynomial c. • For g a semisimple Lie algebra, µ3 = 12 h−, [−, −]i the canonical Lie algebra 3-cocycle in transgression with the Killing form, let gµ3 be the corresponding string Lie 2-algebra given by the next definition, 4.1.23, and discussed below in 4.2.3. Its Weil algebra is given by 1 dW ta = − C a bc tb ∧ tc + ra 2 dW b = h − µ3 and the corresponding Bianchi identities, with {ta } a dual basis for g in degree 1, with b a generator in degree 2 and h its curvature generator in degree 3. We see that every invariant polynomial of g is also an invariant polynomial of gµ3 . But the Killing form h−, −i is now horizontally trivial: let cs3 be any Chern-Simons element for h−, −i in W(g). This is not horizontal. But the element cs ˜ 3 := cs3 − µ3 + h is in ker(W (gµ3 ) → CE(gµ3 )) and dW cs˜3 = h−, −i . Therefore inv(gµ3 ) has the same generators as inv(g) except the Killing form, which is discarded. ∞-Chern-Weil theory July 11, 2011 41 Definition 4.1.23. For µ : g → bn−1 R a cocycle in degree n ≥ 1, the extension that it classifies is the L∞ -algebra given by the pullback gµ  g / inn(bn−2 R) .  / bn−1 R µ Remark 4.1.24. Dually, the L∞ -algebra gµ is the pushout W(bn−2 R) O CE(gµ ) o O CE(g) o µ CE(bn−1 R) in the category dgcAlg. This means that CE(gµ ) is obtained from CE(g) by adding one more generator b in degree (n − 1) and setting dCE(gµ ) : b 7→ −µ . These are standard constructions on dgc-algebras familiar from rational homotopy theory, realizing CE(g) → CE(gµ ) as a relative Sullivan algebra. Yet, it is still worthwhile to make the ∞-Lie theoretic meaning in terms of L∞ -algebra extensions manifest: we may think of gµ as the homotopy fiber of µ or equivalently as the extension of g classified by µ. In Section 5 we discuss how these L∞ -algebra extensions are integrated to extensions of smooth ∞-groups; the homotopy fiber point of view will be emphasized in Section 6. Example 4.1.25. For g a semisimple Lie algebra and µ = 21 h−, [−, −]i the cocycle in transgression with the Killing form, the corresponding extension is the string Lie 2-algebra gµ discussed in 4.2.3, [BSCS07, Hen08]. We may summarize the situation as follows: for µ a degree n cocycle which is in transgression with an invariant polynomial h−i via a Chern-Simons element cs, the corresponding morphisms of dg-algebras fit into a commutative diagram CE(g) o O W(g) o O inv(g) o µ cs h−i CE(bn−1 ) O W(bn−1 R) O inv(bn−1 R) In section 4.3 we will see that under ∞-Lie integration this diagram corresponds to a universal circle n-bundle connection on BG. The composition of the diagrams defining the cells in exp∆ (g)diff (section 4.3) with this diagram models the ∞-Chern-Weil homomorphism for the characteristic class given by h−i. ∞-Chern-Weil theory 4.2 July 11, 2011 42 Principal ∞-bundles We describe the integration of Lie ∞-algebras g to smooth ∞-groupoids BG in the sense of section 3. The basic idea is Sullivan’s old construction [Sul77] in rational homotopy theory of a simplicial set from a dg-algebra. It was essentially noticed by Getzler [Get09], following Hinich [Hin97], that this construction may be interpreted in ∞-Lie theory as forming the smooth ∞-groupoid underlying the Lie integration of an L∞ -algebra. Henriques [Hen08] refined the construction to land in ∞groupoids internal to Banach spaces. Here we observe that the construction has an evident refinement to yield genuine smooth ∞-groupoids in the sense of section 3 (this refinement has independently also been considered by Roytenberg in [Royt09]): the integrated smooth ∞-groupoid sends each Cartesian space U to a Kan complex which in degree k is the set of smoothly U -parameterized families of smooth flat g-valued differential forms on the standard k-simplex ∆k ⊂ Rk regarded as a smooth manifold (with boundary and corners). To make this precise we need a suitable notion of smooth differential forms on the k-simplex. Recall that an ordinary smooth form on ∆k is a smooth form on an open neighbourhood of ∆n in Rn . This says that the derivatives are well behaved at the boundary. The following technical definition imposes even more restrictive conditions on the behaviour at the boundary. Definition 4.2.1. For any point p in ∆k , let ∆p be the lowest dimensional subsimplex of ∆k the point p belongs to, and let πp the orthogonal projection on the affine subspace spanned by ∆p . A smooth differential form ω on ∆k is said to have sitting instants along the boundary if for any point p in ∆k there is a neighborhood Vp of p such that ω = πp∗ (ω|∆p ) on Vp . For any U ∈ CartSp, a smooth differential form ω on U × ∆k is said to have sitting instants if for all points u : ∗ → U the pullback along (u, Id) : ∆k → U × ∆k has sitting instants. Smooth forms with sitting instants clearly form a sub-dg-algebra of all smooth forms. We shall write Ω•si (U × ∆k ) to denote this sub-dg-algebra. Remark 4.2.2. The inclusion Ω•si (∆k ) ֒→ Ω• (∆k ) is a quasi-isomorphism. Indeed, by using bump functions with sitting instants one sees that the sheaf of differential forms with sitting instants is fine, and it is imemdiate to show that the stalkwise Poincaré lemma holds for this sheaf. Hence the usual hypercohomology argument applies. We thank Tom Goodwillie for having suggested a sheaf-theoretic proof of this result. Remark 4.2.3. For a point p in the interior of the simplex ∆k the sitting instants condition is clearly empty; this justifies the name “sitting instants along the boundary”. Also note that the dimension of the normal direction to the boundary depends on the dimension of the boundary stratum: there is one perpendicular direction to a codimension-1 face, and there are k perpendicular directions to a vertex. ∞-Chern-Weil theory July 11, 2011 43 Definition 4.2.4. For a Cartesian space U , we denote by the symbol Ω•si (U × ∆k )vert ⊂ Ω• (U × ∆k ) the sub-dg-algebra on forms that are vertical with respect to the projection U × ∆k → U . Equivalently this is the completed tensor product, ˆ •si (∆k ), Ω•si (U × ∆k )vert = C ∞ (U ; R)⊗Ω where C ∞ (U ; R) is regarded as a dg-algebra concentrated in degree zero. Example 4.2.5. • A 0-form (a smooth function) has sitting instants on ∆1 if in a neighbourhood of the endpoints it is constant. A smooth function f : U × ∆1 → R is in Ω0si (U × ∆1 )vert if for each u ∈ U it is constant in a neighbourhood of the endpoints of ∆1 . • A 1-form has sitting instants on ∆1 if in a neighbourhood of the endpoints it vanishes. • Let X be a smooth manifold and ω ∈ Ω• (X) be a smooth form on X. Let φ : ∆k → X be a smooth map with sitting instants in the ordinary sense: for every r-face of ∆k there is a neighbourhood such that φ is perpendicularly constant on that neighbourhood. Then the pullback form φ∗ ω is a form with sitting instants on ∆k . Remark 4.2.6. The point of the definition of sitting instants, clearly reminiscent of the use of normal cylindrical collars in cobordism theory, is that, when glueing compatible forms on simplices along faces, the resulting differential form is smooth. Proposition 4.2.7. Let Λki ⊂ ∆k be the ith horn of ∆k , regarded naturally as a closed subset of Rk−1 . If {ωj ∈ Ω•si (∆k−1 )} is a collection of smooth forms with sitting instants on the (k − 1)-simplices of Λki that match on their coinciding faces, then there is a unique smooth form ω on Λki that restricts to ωj on the jth face. Proof. By the condition that forms with sitting instants are constant perpendicular to their value on a face in a neighbourhood of any face it follows that if two agree on an adjacent face then all derivatives at that face of the unique form that extends both exist in all directions. Hence that unique form extending both is smooth.  Definition 4.2.8. For g an L∞ -algebra, the simplicial presheaf exp∆ (g) on the site of Cartesian spaces is defined as exp∆ (g) : (U, [k]) 7→ HomdgAlg (CE(g), Ω•si (U × ∆k )vert ). ∞-Chern-Weil theory July 11, 2011 44 Note that the construction of exp∆ (g) is functorial in g: a morphism of L∞ -algebras g1 → g2 , i.e., a dg-algebra morphism CE(g2 ) → CE(g1 ), induces a morphism of simplicial presheaves exp∆ (g1 ) → exp∆ (g2 ). Remark 4.2.9. A k-simplex in exp∆ (g)(U ) may be thought of as a smooth family of flat g-valued forms on ∆n , parametrized by U . We write exp∆ (g) for this simplicial presheaf to indicate that it plays a role analogous to the formal exponentiation of a Lie algebra to a Lie group. Proposition 4.2.10. The simplicial presheaf exp∆ (g) is a smooth ∞-groupoid in that it is fibrant in [CartSpop , sSet]proj : it takes values in Kan complexes. We say that the smooth ∞-groupoid exp∆ (g) integrates the L∞ -algebra g. Proof. Since our forms have sitting instants, this follows in direct analogy to the standard proof that the singluar simplicial complex of any topological space is a Kan complex: we may use the standard retracts of simplices onto their horns to pull back forms from horns to simplices. The retraction maps are smooth except where they cross faces, but since the forms have sitting instants there, their smooth pullback exists nevertheless. Let π : ∆k → Λki be the standard retraction map of a k-simplex on its i-th horn. Since π is smooth away from the primages of the faces, the commutative diagram id×i / U × ∆k U × Λki JJ JJ JJ id×π JJ id J%  U × Λki induces a commutative diagram of dgcas (id×i)∗ ◦− HomdgAlg (CE(g), Ω•si (U × ∆k )vert ) , HomdgAlg (CE(g), Ω•si (U × Λki )vert ) o O lYYYYYY YYYYYY YYYYYY (id×π)∗ ◦− YYYYYY id Y HomdgAlg (CE(g), Ω•si (U × Λni )vert ) so that, in particular, the horn-filling map HomdgAlg (CE(g), Ω•si (U × Λki )vert ) → HomdgAlg (CE(g), Ω•si (U × ∆k )vert ) is surjective.  Example 4.2.11. We may parameterize the 2-simplex as ∆2 = {(x, y) ∈ R2 ||x| ≤ 1 , 0 ≤ y ≤ 1 − |x|} . The retraction map ∆2 → Λ21 in this parameterization is (x, y) 7→ (x, 1 − |x|) . This is smooth away from x = 0. A 1-form with sitting instants on Λ11 vanishes in a neighbourhood of x = 0, hence its pullback along this map exists and is smooth. ∞-Chern-Weil theory 45 July 11, 2011 Typically one is interested not in exp∆ (g) itself, but in a truncation thereof. For our purposes truncation is best modeled by the coskeleton operation. Write ∆≤n ֒→ ∆ for the full subcategory of the simplex category on the first n objects [k], with 0 ≤ k ≤ n. Write sSet≤n for the category of presheaves on ∆≤n . By general abstract reasoning the canonical projection trn : sSet → sSet≤n has a left adjoint skn : sSet≤n → sSet and a right adjoint coskn : sSet≤n → sSet. (skn ⊣ trn ⊣ coskn ) : sSet o o skn trn / sSet≤n . coskn The coskeleton operation on a simplicial set is the composite coskn := coskn ◦ trn : sSet → sSet . Since coskn is a functor, it extends to an operation of simplicial presheaves, which we shall denote by the same symbol coskn : [CartSpop , sSet] → [CartSpop , sSet] For X ∈ sSet or X ∈ [CartSpop,sSet ] we say coskn X is its n-coskeleton. Remark 4.2.12. Using the adjunction relations, we have that the k-cells of coskn X are images of the n-truncation of ∆[k] in the n-truncation of X: (coskn X)k = HomsSet (∆[k], coskn X) = HomsSet≤n (trn ∆[k], trn X) . A standard fact (e.g. [DK84], [GJ99]) is Proposition 4.2.13. For X a Kan complex • the simplicial homotopy groups πk of coskn X vanishing in degree k ≥ n; • the canonical morphism X → coskn X (the unit of the adjunction) is an isomorphism on all πk in degree k < n; • in fact, the sequence X → · · · → coskk X → coskk−1 X → · · · → cosk1 X → cosk0 X ≃ ∗ is a model for the Postnikov tower of X. Example 4.2.14. For G a groupoid and N G its simplicial nerve, the canonical morphism N G → cosk2 N G is an isomorphism. Definition 4.2.15. We say a Kan complex or L∞ -groupoid X is an n-groupoid if the canonical morphism X → coskn+1 X is an isomorphism. If this morphism is just a weak equivalence, we say X is an n-type. ∞-Chern-Weil theory 46 July 11, 2011 We now spell out details of the Lie ∞-integration for 1. an ordinary Lie algebra 2. the string Lie 2-algebra 3. the line Lie n-algebras bn−1 R. The basic mechanism is is that discused in [Hen08], there for Banach ∞-groupoids. We present now analogous discussions for the context of smooth ∞-groupoids that we need for the differential refinement in 4.3 and then for the construction of the ∞-Chern-Weil homomorphism in 5 4.2.1 Ordinary Lie group Let G be a Lie group with Lie algebra g. Then every smooth g-valued 1-form on the 1-simplex defines an element of G by parallel transport: tra : Ω1si ([0, 1], g) → G ω 7→ P exp Z [0,1] ω ! , where the right hand P exp(· · · ) is notation defined to be the endpoint evaluation f (1) of the unique solution f : [0, 1] → G to the differential equation df + rf ∗ (ω) = 0 with initial condition f (0) = e, where rg : G → G denotes the right action of g ∈ G on G itself. In the special case that G is simply connected, there is a unique smooth path γ : [0, 1] → G starting at the neutral element e such that ω equals the pullback γ ∗ θ of the Maurer-Cartan form on G. The value of the parallel transport is then the endpoint of this path in G. More generally, this construction works in families and produces for every Cartesian space U , a parallel transport map tra : Ω1si (U × [0, 1], g)vert → C ∞ (U, G) from smooth U -parameterized families of g-valued 1-forms on the interval to smooth functions from U to G. If we now consider a g-valued 1-form ω on the n-simplex instead, parallel transport along the sequence of edges [0, 1], [1, 2],. . . , [n − 1, n] defines an element in Gn+1 , and so we have an induced map Ω1si (U × ∆n , g)vert → BG(U )n . This map, however is not in general a map of simplicial sets: the composition of parallel transport along [0, 1] and [1, 2] is in general not the same as the parallel transport along the edge [0, 2] so parallel transport is not compatible with face maps. But precisely if the g-valued 1-form is flat does its parallel transport (over the contractible simplex) only depend on the endpoint of the path along which it is transported. Therefore we have in particular the following ∞-Chern-Weil theory 47 July 11, 2011 Proposition 4.2.16. Let G be a Lie group with Lie algebra g. • Parallel transport along the edges of simplices induces a morphism of smooth ∞-groupoids tra : exp∆ (g) → BG. • When G is simply connected, there is a canonical bijection between smooth flat g-valued 1-forms A on ∆n and smooth maps φ : ∆n → G that send the 0-vertex to the neutral element. This bijection is given by A = φ∗ θ, where θ is the Maurer-Cartan form of G. As for every morphism of Kan complexes, we can look at coskeletal approximations of parallel transport given by the morphism of coskeleta towers exp(g) / ··· / coskn+1 (exp(g)) / coskn (exp(g)) / ··· /∗ / ···  / coskn+1 (BG)  / coskn (BG) / ···  /∗ tra  BG Proposition 4.2.17. If the Lie group G is (k − 1)-connected, then the induced maps coskn (exp∆ (g)) → coskn (BG) are acyclic fibrations in [CartSpop , sSet]proj for any n ≤ k. Proof. Recall that an acyclic fibration in [CartSpop , sSet]proj is a morphism of simplicial presheaves that is objectwise an acyclic Kan fibration of simplicial sets. By standard simplicial homotopy theory [GJ99], the latter are precisely the maps that have the left lifting property against all simplex boundary inclusions ∂∆[p] ֒→ ∆[p]. Notice that for n = 0 and n = 1 the statement is trivial. For n ≥ 2 we have an isomorphism BG → coskn BG. Hence we need to prove that for 2 ≤ n ≤ k we have for all U ∈ CartSp lifts σ in diagrams of the form ∂∆[n] i  t ∆[n] σ t t / exp∆ (g) t: t tra  / BG By parallel transport and using the Yoneda lemma, the outer diagram is equivalently given by a map U × ∂∆p → G that is smooth with sitting instants on each face ∆p−1 . By proposition 4.2.7 this may be thought of as a smooth map U × S p−1 → G. The lift σ then corresponds to a smooth map with sitting instants σ : U × ∆n → G extending this, hence to a smooth map σ : U × Dp → G that in a neighbourhood of S p−1 is constant in the direction perpendicular to that boundary. ∞-Chern-Weil theory July 11, 2011 48 By the connectivity assumption on G there is a continuous map with these properties. By the Steenrod-Wockel-approximation theorem [Woc09], this delayed homotopy on a smooth function is itself continuously homotopic to a smooth such function. This smooth enhancement of the continuous extension is a lift σ.  For n = 1 the Kan complex cosk1 (BG) is equivalent to the point. For n = 2 we have an isomorphism BG → cosk2 BG (since BG is the nerve of a Lie groupoid) and so the proposition asserts that for simply connected Lie groups cosk2 exp∆ (g) is equivalent to BG. Corollary 4.2.18. If G is a compact connected and simply connected Lie group with Lie algebra g, then the natural morphism exp∆ (g) → BG induces an acyclic fibration cosk3 (exp∆ (g)) → BG in the global model structure. Proof. Since a compact connected and simply connected Lie group is automatically 2-connected, we have an induced acyclic fibration cosk3 (exp(g)) → cosk3 (BG). Now notice that BG is 2-coskeletal, i.e, its coskeleta tower stabilizes at cosk2 (BG) = BG. 4.2.2 Line n-group Definition 4.2.19. For n ≥ 1 write bn−1 R for the line Lie n-algebra: the L∞ -algebra characterized by the fact that its Chevalley-Eilenberg algebra is generated from a single generator c in degree n and has trivial differentual CE(bn−1 R) = (∧• hci, d = 0) . Proposition 4.2.20. Fiber integration over simplices induces an equivalence Z ≃ : exp∆ (bn−1 R) → Bn R. ∆• Proof. By the Dold-Kan correspondence we only need to show that integration along the simplices is a chain map from the normalized chain complex of exp∆ (bn−1 R) to C ∞ (−)[n]. The normalized chain complex N• (exp∆ (bn−1 R)) ˆ ncl (∆k ), and the differential has in degree −k the abelian group C ∞ (−)⊗Ω ∂ : N −k (exp∆ (bn−1 R)) → N −k+1 (exp∆ (bn−1 R)) maps a differential form ω to the alternating sum of its restrictions on the faces of the simplex. If ω is an element in C ∞ (−) ⊗ Ωncl (∆k ), integration of ω on ∆k is zero unless k = n, which shows that integration along the simplex maps N • (exp∆ (bn−1 R)) to C ∞ (−)[n]. Showing that this map is actually a map of chain complexes is trivial in all degrees but for k = n+1; in this degree, checking that itegration along simplices is a chain map amounts to checking that for a closed n-form ω on the (n + 1)-simplex, the integral of ω on the boundary of ∆n+1 vanishes, and this is obvious by Stokes theorem.  Remark 4.2.21. For n = 1, the morphism exp∆ (R) → BR coincides with the morphism described in Proposition 4.2.16, for G = R. ∞-Chern-Weil theory 4.2.3 49 July 11, 2011 Smooth string 2-group Definition 4.2.22. Let string := soµ3 be the extension of the Lie algebra so classified by its 3-cocycle µ3 = 21 h−, [−, −]i according to definition 4.1.23. This is called the string Lie 2-algebra. Let BString := cosk3 exp∆ (soµ3 ) be its Lie integration. We call this the delooping of the smooth string 2-group. The Banach-space ∞-groupoid version of this Lie integration is discussed in [Hen08]. Remark 4.2.23. The 7-cocycle µ7 on so is still, in the evident way, a cocycle on soµ3 µ7 : so3 → b6 R . Proposition 4.2.24. We have an ∞-pullback of smooth ∞-groupoids /∗ BString  BSpin 1 2 p1  / B3 U (1) presented by the ordinary pullback of simplicial presheaves ^ BString / EB2 (Z → R) ,   exp∆ (µ3 ) / B3 (Z → R) cosk3 exp∆ (so) ≀  BSpin ^ is induced by integrating the 2-form over simplices. where BString → BString ∼ Remark 4.2.25. In terms of definition 3.2.19, BString is the smooth B2 U (1)principal 3-bundle over BSpin classified by the smooth refinement of the first fractional Pontryagin class. Proof. Since all of cosk3 exp∆ (so), B3 (Z → R) and EB2 (Z → R) are fibrant in [CartSpop , sSet]proj and since EB2 U (1) → B3 (Z → R) is a fibration (being the image under DK of a surjection of complexes of sheaves), we have by standard facts about homotopy pullbacks that the ordinary pullback is a homotopy pullback in [CartSpop , sSet]proj . By [Lur09] this presents the ∞-pullback of ∞-presheaves on CartSp. And since ∞-stackification is left exact, this is also presents the ∞-pullback of ∞-sheaves. ∞-Chern-Weil theory 50 July 11, 2011 This ordinary pullback manifestly has 2-cells given by 2-simplices in G labeled by elements in U (1) R and 3-cells being 3-simplices in G such that the labels of their faces differ by ∆3 →G µ modZ. This is the definition of BString. That exp∆ (µ3 ) indeed presents a smooth refinement of the second fractional Pontryagin class as indicated is shown below.  Proposition 4.2.26. There is a zig-zag of equivalences cosk3 exp(soµ3 ) ≃ · · · ≃ B(Ω̂Spin → P Spin) in [CartSpop , sSet]proj , of the Lie integration, prop. 4.2.24, of soµ3 with the strict 2-group, def. 3.2.10 coming from the crossed module (Ω̂Spin → P Spin) of Fréchet Lie groups, discussed in [BSCS07], consisting of the centrally extended loop group and the path group of Spin. This is proven in section 4.2 of [Sch10]. Proposition 4.2.27. The object BString = cosk3 (exp(soµ3 )) is fibrant in [CartSpop , sSet]proj,loc . Proof. Observe first that both object are fibrant in [CartSpop , sSet]proj (the Lie integration by prop. 4.2.10, the delooped strict 2-group by observation 3.2.11). The claim then follows with prop. 4.2.26 and prop. 3.2.12, which imply that for C({Ui }) → Rn the Čech nerve of a good open cover, hence a cofibrant resultions, there is a homotopy equivalence [CartSpop , sSet](Č(U), cosk3 exp(soµ3 )) ≃ [CartSpop , sSet](Č(U), B(Ω̂Spin → P Spin)) .  Corollary 4.2.28. A Spin-principal bundle P → X can be lifted to a Stringprincipal bundle precisely if it trivializes 21 p1 , i.e., if the induced mophism H(X, BSpin) → H(X, B3 U (1)) is homotopically trivial. The choice of such a lifting is called a String structure on the Spin-bundle. We discuss string structures and their twisted versions further in 6. 4.3 Principal ∞-bundles with connection For an ordinary Lie group G with Lie algebra g, we have met in section 3.2.1 the smooth groupoids BG, BGconn and BGdiff arising from G, and in 4.2 the smooth ∞-groupoid exp∆ (g) coming from g, and have shown that they are related by a diagram exp∆ (g) BGconn  / BGdiff ∼  / / BG ∞-Chern-Weil theory July 11, 2011 51 and that BGconn is the moduli stack of G-principal bundles with connection. Now we dicuss such differential refinements exp∆ (g)diff and exp∆ (g)conn that complete the above diagram for any integrated smooth ∞-group exp∆ (g). Where a truncation of exp∆ (g) is the object that classifies G-principal ∞-bundles, the corresponding truncation of exp∆ (g)conn classifies principal ∞-bundles with connection. Between exp∆ (g)conn and exp∆ (g)diff , we will also meet the Chern-Weil ∞-groupoid exp∆ (g)CW which is the natural ambient for ∞-Chern-Weil theory to live in. For the following, let g be any Lie ∞-algebra. Definition 4.3.1. The differential refinement exp∆ (g)diff of exp∆ (g) is the simplicial presheaf on the site of Cartesian spaces given by the assignment   Avert   • k o  CE(g)  Ωsi (U × ∆ )vert     O O     , (U, [k]) 7→       A  • k   W (g)    Ωsi (U × ∆ ) o where on the right we have the set of commuting diagrams in dgcAlg as indicated. Remark 4.3.2. This means that a k-cell in exp∆ (g)diff over U ∈ CartSp is a g-valued form A on U × ∆k that satisfies the condition that its curvature forms FA vanish when restricted in all arguments to vectors on the simplex. This is the analog of the first Ehresmann condition on a connection form on an ordinary principal bundle: the form A on the trivial simplex bundle U × ∆k → U is flat along the fibers. Proposition 4.3.3. The evident morphism of simplicial presheaves ∼ exp∆ (g)diff / / exp∆ (g) is an acyclic fibration of smooth ∞-groupoids in the global model structure. Proof. We need to check that, for all U ∈ CartSp and [k] ∈ ∆ and for all diagrams ∂∆[k] A|∂ / exp∆ (g)diff (U ) p8 pp pA  p p Avert  / exp∆ (g)(U ) ∆[k] we have a lift as indicated by the dashed morphism. For that we need to extend the composite Avert Ω•si (U × ∆n )vert W(g) → CE(g) → ∞-Chern-Weil theory 52 July 11, 2011 to an element in Ω•si (U ×∆k )⊗g with fixed boudary value A∂ in Ω•si (U ×∂∆k )⊗g. To see that this is indeed possible, use the decomposition
ˆ •si (∆k ) Ω•si (U × ∆k ) = Ω•si (U × ∆n )vert ⊕ Ω>0 (U )⊗Ω >0 >0 ˆ •si (∆k ). to write A∂ = Avert |∂∆k +A>0 in Ω>0 (U )⊗Ω ∂ . Extend A∂ to an element A This is a trivial extension problem: any smooth differential form on the boundary of an k-simplex can be extended to a smooth differential form on the whole simplex. Then the degree 1 element Avert + A>0 is a solution to our original extension problem.  Remark 4.3.4. This means that exp∆ (g)diff is a certain resolution of exp∆ (g). In the full abstract theory [Sch10], the reason for its existence is that it serves to model the canonical curvature characteristic map BG → ♭dR Bn U (1) in the ∼ ∞-topos of smooth ∞-groupoids by a truncation of the zig-zag exp∆ (g) ← n−1 exp∆ (g)diff → exp∆ (b R) of simplicial presheaves. By the nature of acyclic g ∼ fibrations, we have that for every exp∆ (g)-cocycle X ←−loc −− Č(U) → exp∆ (g) there is a lift gdiff to an exp∆ (g)diff -cocycle QX gdiff / exp∆ (g)diff . ≀ ≀  Č(U) g  / exp∆ (g) ∼loc  X For the abstract machinery of ∞-Chern-Weil theory to work, it is only the existence of this lift that matters. However, in practice it is useful to make certain nice choices of lifts. In particular, when X is a paracompact smooth manifold, there is always a choice of lift with the property that the corresponding curvature characteristic forms are globally defined forms on X, instead of more general (though equivalent) cocycles in total Čech-de Rham cohomology. Moreover, in this case the local connection forms can be chosen to have ∆-horizontal curvature. Lifts with this special property are genuine ∞-connections on the ∞-bundles classified by g. The following definitions formalize this. But it is important to note that genuine ∞-connections are but a certain choice of gauge among all differential lifts. Notably when the base X is not a manifold but for instance a non-trivial orbifold, then genuine ∞-connections will in general not even exist, whereas the differential lifts always do exist, and always support the ∞-Chern-Weil homomorphism. Proposition 4.3.5. If g is the Lie algebra of a Lie group G, then there is a ∞-Chern-Weil theory 53 July 11, 2011 natural commutative diagram exp∆ (g)diff  BGdiff ∼ / / exp∆ (g) ∼  / / BG In particular, if G is (k − 1)-connected, with k ≥ 2, then the induced morphism coskk (exp∆ (g)diff ) → BGdiff is an acyclic fibration in the global model structure. Proof. We have seen in Remark 3.2.7 that there is a natural isomorphism BGdiff ∼ = BG×Codisc(Ω1 (−; g), so in order to give the morphism exp∆ (g)diff → BGdiff making the above diagram commute we only need to give a natural morphism exp∆ (g)diff → Codisc(Ω1 (−; g); this is evaluation of the connection form A on the vertices of the simplex. Assume now G is (k−1)-connected, with k ≥ 2. Then, by Propositions 4.2.17 and 4.3.3, both coskk (exp∆ (g)diff ) → coskk (exp∆ (g)) and coskk (exp∆ (g)) → BG are acyclic fibrations. We have a commutative diagram coskk (exp∆ (g)diff ) ∼ / / coskk (exp∆ (g)) ≀  BGdiff  / / BG, ∼ so, by the “two out of three” rule, also coskk (exp∆ (g)diff ) → BGdiff is an acyclic fibration.  Definition 4.3.6. The simplicial presheaf exp∆ (g)CW ⊂ exp∆ (g)diff is the subA presheaf of exp∆ (g)diff on those k-cells Ω•si (U × ∆k ) ← W(g) that make also the bottom square in the diagram Ω•si (U × ∆k )vert o O Avert Ω•si (U × ∆k ) o O A Ω• (U ) o FA CE(g) O W (g) O inv(g) commute. Remark 4.3.7. A k-cell in exp(g)CW parameterised by a Cartesian space U is a g-valued differential form A on the total space U × ∆k such that ∞-Chern-Weil theory 54 July 11, 2011 1. its restriction to the fiber ∆k of U × ∆k → U is flat, and indeed equal to the canonical g-valued form there as encoded by the cocycle Avert (which, recall, is the datum in exp(g) that determines the G-bundle itself); this we may think of as the first Ehresmann condition on a connection; 2. all its curvature characteristic forms hFA i descend to the base space U of U × ∆k → U ; this we may think of as a slightly weakened version of the second Ehresmann condition on a connection: this is the main consequence of the second Ehresmann condition. These are the structures that have been considered in [SSS09b] and [SSS09c]. Proposition 4.3.8. exp∆ (g)CW is fibrant in [CartSpop , sSet]proj . Proof. As in the proof of Propositition 4.2.10 we find horn fillers σ by pullback along the standard retracts, which are smooth away from the loci where our forms have sitting instants. Ω•si (U × ∆k )vert o O Ω•si (U × Λki ) o O Avert Ω•si (U × ∆k ) o O Ω•si (U × Λki ) o O A Ω• (U ) o Ω• (U ) o FA CE(g) O W(g) O :σ inv(g)  We say that exp(gµ )CW is the Chern-Weil ∞-groupoid of g. Definition 4.3.9. Write exp∆ (g)conn for the simplicial sub-presheaf of exp∆ (g)diff given in degree k by those g-valued forms satisfying the following further horizontality condition: • for all vertical (i.e., tangent to the simplex) vector fields v on U × ∆k , we have ιv FA = 0 . Remark 4.3.10. This extra condition is the direct analog of the second Ehresmann condition. For ordinary Lie algebras we have discussed this form of the second Ehresmann condition in section 2.2. Remark 4.3.11. If we decompose differential forms on the products U × ∆k as M ˆ qsi (∆k ) Ω•si (U × ∆k ) = Ωp (U )⊗Ω p,q∈N then ∞-Chern-Weil theory 55 July 11, 2011 1. k-simplices in exp∆ (g)diff are those connection forms A : W (g) → Ω•si (U × ∆k ) whose curvature form has only the (p, q)-components with p > 0; 2. k-simplices in exp∆ (g)conn are those k-simplices in exp∆ (g)diff whose curvature is furthermore constrained to have precisely only the (p, 0)-components, with p > 0. Proposition 4.3.12. We have a sequence of inclusions of simplicial presheaves exp∆ (g)conn ֒→ exp∆ (g)CW ֒→ exp∆ (g)diff . Proof. Let h−i be an invariant polynomial on g, and A a k-cell of exp∆ (g)conn . Since dW (g) h−i = 0, we have dhFA i = 0, and since ιv FA = 0 we also have ιv hFA i = 0 for v tangent to the k-simplex. Therefore by Cartan’s formula also the Lie derivatives Lv hFA i are zero. This implies that the curvature characteristic forms on exp∆ (g)conn descend to U and hence that A defines a k-cell in exp∆ (g)CW .  4.3.1 Examples We consider the special case of the above general construction again for the special case that g is an ordinary Lie algebra and for g of the form bn−1 R. Proposition 4.3.13. Let G be a Lie group with Lie algebra g. Then, for any k ∈ N there is a pullback diagram coskk exp∆ (g)conn / coskk exp∆ (g)  BGconn  / BG in the category of simplicial presheaves. Proof. The result is trivial for n = 0. For n = 1 we have to show that given two 1-forms A0 , A1 ∈ Ω1 (U, g), a gauge transformation g : U → G between them, and any lift λ(u, t)dt of g to a 1-form in Ω1 (U × ∆1 , g), there exists a unique 1-form A ∈ Ω1 (U × ∆1 , g) whose vertical part is λ, whose curvature is of type (2,0), and such that A U×{0} = A0 ; A U×{1} = A1 . We may decompose such A into its vertical and horizontal components A = λ dt + AU , where λ ∈ C ∞ (U ×∆1 ) and AU in the image of Ω1 (U, g). Then the horizontality condition ι∂t FA = 0 on A is the differential equation ∂ AU = dU λ + [AU , λ] . ∂t ∞-Chern-Weil theory 56 July 11, 2011 For the given initial condition AU (t = 0) = A0 this has a unique solution, given by AU (t) = g(t)−1 A0 g(t) + g(t)−1 dt g(t), where g(t) ∈ G is the parallel transport for the connection λ dt along the path [0, t] in the 1-simplex ∆1 . Evaluating at t = 1, and using g(1) = g, we find A(1) = g −1 A0 g + g −1 dg = A1 , as required. These computations carry on without substantial modification to higher simplices: using that λ dt is required to be flat along the simplex, it follows the value of AU at any point in the simplex is determined by a differential equation as above, for parallel transport along any path from the 0-vertex to that point. Accordingly we find unique lifts A, which concludes the proof.  Corollary 4.3.14. If G is a compact simply connected Lie group, there is a ∼ → BGconn in [CartSpop , sSet]proj . weak equivalence cosk3 exp(g)conn − Proof. By proposition 4.2.17 we have that cosk3 exp ∆(g) → BG is an acyclic fibration in the global model structure. Since these are preserved under pullback, the claim follows by the above proposition.  Proposition 4.3.15. Integration along simplices gives a morphism of smooth ∞-groupoids Z diff : exp∆ (bn−1 R)diff → Bn Rdiff . ∆• Proof. By means of the Dold-Kan correspondence we only need to show that integration along simplices is a morphism of complexes from the normalized chain complex of exp∆ (bn−1 R) to the cone / Ω1 (U ) d / Ω2 (U ) p pp7 ⊕ pppIdp ⊕ ppp ⊕ 2 1 / / ··· Ω (U ) Ω (U ) C ∞ (U ) d d d d / Ωn (U ) 8 p p ⊕ ppIdp p / Ωn (U ) /0 / ··· (3) The normalized chain complex N • (exp∆ (bn−1 R)) has in degree −k the subspace of Ωn (U × ∆k ) consisting of those n-forms whose (0, n)-component ω0,n lies in ˆ ncl (∆k ); the differential C ∞ (U )⊗Ω ∂ : N −k (exp∆ (bn−1 R)) → N −k+1 (exp∆ (bn−1 R)) maps an n-form ω on U × ∆k to the alternate sum of its restrictions to the faces R dif f of U × ∂∆k . For k 6= 0, let ∆• be the map Z dif f : N −k (exp∆ (bn−1 R)) → Ωn−k (−) ⊕ Ωn−k+1 (−) k ∆ Z  Z ω 7→ ddR ω , ω, ∆k ∆k ∞-Chern-Weil theory and, for k = 0 let R dif f ∆0 Z 57 July 11, 2011 be the identity dif f ∆0 = id : N 0 (exp∆ (bn−1 R)) → Ωn (−). R dif f The map ∆• actually takes its values in the cone (3). Indeed, if k > n + 1, then both the integral of ω and of ddR ω are zero by dimensional reasons; for k = n + 1, the only possibly nontrivial contribution to the integral over ∆n+1 comes from d∆n+1 ω0,n , which is zero by hypothesis (where we have written ddR = d∆k + dU for the decomposition of the de Rham differential associated with the product structure of U × ∆k ). R dif f The fact that ∆• is a chain map immediately follows by the Stokes formula: Z Z Z ∂ω = ω= d∆k ω ∆k−1 ∂∆k ∆k and by the identity ddR = d∆k + dU .  Corollary 4.3.16. Integration along simplices induces a morphism of smooth ∞-groupoids Z conn : exp∆ (bn−1 R)conn → Bn Rconn . ∆• Proof. By Proposition 3.2.26, we only need to check that the image of the R diff composition curv ◦ ∆• lies in the subcomplex (0 → 0 → · · · → Ωn+1 cl (−)) of ♭dR Bn+1 R, and this is trivial since by definition of exp∆ (bn−1 R)conn the curvature of ω, i.e., the de Rham differential ddR ω, is 0 along the simplex.  5 ∞-Chern-Weil homomorphism With the constructions that we have introduced in the previous sections, there is an evident Lie integration of a cocycle µ : g → bn−1 R on a L∞ -algebra g to a morphism exp∆ (g) → exp∆ (bn−1 R) that truncates to a characteristic map BG → Bn R/Γ. Moreover, this has an evident lift to a morphism exp∆ (g)diff → exp∆ (bn−1 R)diff between the differential refinements. Truncations of this we shall now identify with the Chern-Weil homomorphism and its higher analogs. 5.1 Characteristic maps by ∞-Lie integration We have seen in section 4 how L∞ -algebras g, bn−1 R integrate to smooth ∞groupoids exp∆ (g), exp∆ (bn−1 R) and their differential refinements exp∆ (g)diff , exp∆ (bn−1 R)diff as well as to various truncations and quotients of these. We remarked at the end of 4.1 that a degree n cocycle µ on g may equivalently be thought of as a morphism µ : g → bn−1 R, i.e., as a dg-algebra morphism µ : CE(bn−1 R) → CE(g). ∞-Chern-Weil theory 58 July 11, 2011 Definition 5.1.1. Given an L∞ -algebra cocycle µ : g → bn−1 R as in section 4.1, define a morphism of simplicial presheaves exp∆ (µ) : exp∆ (g) → exp∆ (bn−1 R) by componentwise composition with µ:   Avert CE(g) 7→ exp∆ (µ)k : Ω•si (U × ∆k )vert ← .  µ Avert CE(g) ← CE(bn−1 R) : µ(Avert ) Ω•si (U × ∆k )vert ← Write Bn R/µ for the pushout exp∆ (µ) / exp (bn−1 R) ∆ exp∆ (g) R ∆• ∼  coskn exp∆ (g) / Bn R .  / Bn R/µ By slight abuse of notation, we shall denote also the bottom morphism by exp∆ (µ) and refer to it as the Lie integration of the cocycle µ. Remark 5.1.2. The object Bn R/µ is typically equivalent to the n-fold delooping Bn (Λµ → R) of the reals modulo a lattice Λµ ⊂ R of periods of µ, as discussed below. Moreover, as discussed in section 4, we will be considering ∼ weak equivalences coskn exp∆ (g) → BG. Therefore exp∆ (µ) defines a characteristic morphism of smooth ∞-groupoids BG → Bn (Λµ → R), presented by the span of morphisms of simplicial presheaves coskn exp∆ (g) exp∆ (µ) / Bn R/µ . ≀  BG Proposition 5.1.3. Let G be a Lie group with Lie algebra g and µ : g → bn−1 R a degree n Lie algebra cocycle. Then there is a smallest subgroup Λµ of (R, +) such that we have a commuting diagram exp∆ (g) exp∆ (µ)  coskn exp∆ (g) / exp (bn−1 R) ∆ R ∆• ∼ / Bn R .  / Bn (Λµ → R) Proof. We exhibit the commuting diagram naturally over each Cartesian space U . The vertical map Bn R(U ) → Bn (Λµ → R)(U ) is the obvious quotient map of simplicial abelian groups. Since Bn (Λµ → R) is (n − 1)-connected ∞-Chern-Weil theory 59 July 11, 2011 and coskn exp∆ (g) is n-coskeletal, it is sufficient to define the horizontal map coskn exp∆ (g) → Bn (Λµ → R) on n-cells. For the diagram to commute, the 1 n Rbottom morphism must send a form Avert ∈ Ωsi (U × ∆ , g) to the image of ∆n µ(Avert ) ∈ R under the quotient map. For this assignment to constitute a morphism ofRsimplicial sets, it must be true that for all Avert ∈ Ω1si (U ×∂∆n−1 , g) the integral ∂∆n+1 Avert ∈ R lands in Λµ ⊂ R. Recall that we may identify flat g-valued forms on ∂∆n+1 with based smooth R n+1 maps ∂∆ → G. We observe that ∂∆n+1 Avert only depends on the homotopy class of such a map: if we have two homotopic n-spheres Avert and A′vert then by the arguments as in the proof of proposition 4.2.17, using [Woc09], there is a smooth homotopy interpolating between them, hence a corresponding extension ′ of Âvert . Since this is closed, the fiber integrals of R Avert and Avert coincide. Therefore we have a group homomorphism ∂∆n+1 : πn (G, eG ) → R. Take Λµ to be the subgroup of R generated by its image. This is the minimal subgroup of R for which we have a commutative diagram as stated.  Remark 5.1.4. If G is compact and simply connected, then its homotopy groups are finitely generated and so is Λµ . Example 5.1.5. Let G be a compact, simple and simply connected Lie group and µ3 the canonical 3-cocycle on its semisimple Lie algebra, normalized such that its left-invariant extension to a differential 3-form on G represents a generator of H 3 (G, Z) ≃ Z in de Rham cohomology. In this case we have Λµ3 ≃ Z and the diagram of morphisms discussed above is exp∆ (g)  cosk3 (exp∆ (g)) ≀  BG R ∆• exp∆ (µ) / B3 R  / B3 (Z → R) ≀  B3 U (1) This presents a morphism of smooth ∞-groupoids BG → B3 U (1). Let X → BG ∼ be a morphism of smooth ∞-groupoids presented by a Čech-cocycle X ←−loc −− Č(U) → BG as in section 3. Then the composite X → BG → B3 U (1) is a cocycle for a B2 U (1)-principal 3-bundle presented by a span of simplicial ∞-Chern-Weil theory July 11, 2011 60 presheaves ĝ QX C / cosk3 exp∆ (g) exp∆ (µ)/ B3 (Z → R) . ≀ ≀  Č(U)  / BG ∼loc  X Here the acyclic fibration QX → Č(U) is the pullback of the acyclic fibration cosk3 exp∆ (g) → BG from proposition 4.2.17 and Č(U) → QX is any choice of section, guaranteed to exist, uniquely up to homotopy, since Č(U) is cofibrant according to proposition 2. This span composite encodes a morphism of 3-groupoids of Čech-cocycles cµ : Č(U, BG) → Č(U, B3 (Z → R)) given as follows 1. it reads in a Čech-cocycle (gij ) for a G-principal bundle; 2. it forms a lift ĝ of this Čech-cocycle of the following form: • over double intersections we have that ĝij : (Ui ∩ Uj ) × ∆1 → G is a smooth family of based paths in G, with ĝij (1) = gij ; • over triple intersections we have that ĝijk : (Ui ∩ Uj ∩ Uk ) × ∆2 → G is a smooth family of 2-simplices in G with boundaries labeled by the based paths on double overlaps: gij ~ CCC gij ·ĝjk ~ C ~ ~~ ĝijk CCCC ~ ! ~~ gik e ĝij ĝik • on quadruple intersections we have that ĝijkl : (Ui ∩ Uj ∩ Uk ∩ Ul ) : ∆3 → G is a smooth family of 3-simplices in G, cobounding the union of the 2-simplices corresponding to the triple intersections. 3. The morphism exp∆ (µ) : cosk3 exp∆ (g) → exp∆ (b2 R) takes these smooth families of 3-simplices and integrates over them the 3-form µ3 (θ ∧ θ ∧ θ) to obtain the Čech-cocycle Z ∗ ĝijkl µ(θ ∧ θ ∧ θ) mod Z) ∈ Č(U, B3 U (1)) . ( ∆3 Note that µ3 (θ ∧θ ∧θ) is the canonical 3-form representative of a generator of H 3 (G, Z). ∞-Chern-Weil theory July 11, 2011 61 In total the composite of spans therefore encodes a map that takes a Čechcocycle (gij ) for a G-principal bundle to a degree 3 Čech-cocycle with values in U (1). Remark 5.1.6. The map of Čech cocycles obtained in the above example from a composite of spans of simplicial presheaves is seen to be the special case of the construction considered in [BM96b] that is discussed in section 4 there, where an explicit Čech cocycle for the second Chern class of a principal SU (n)-bundle is described. See [BM93] for the analogous treatment of the first Pontryagin class of a principal SO(n)-bundle and also [BM94, BM96a]. Proposition 5.1.7. For G = Spin the morphism cµ3 from example 5.1.5 is a smooth refinement of the first fractional Pontryagin class exp∆ (µ3 ) = 1 ∼ p1 : BSpin ← cosk3 exp∆ (g) → B3 U (1) 2 in that postcomposition with this characteruistic map induces the morphism 1 p1 : H 1 (X, Spin) → H 4 (X, Z) . 2 Proof. Using the identification from example 5.1.5 of the composite of spans with the construction in [BM96b] this follows from the main theorem there. The strategy there is to refine to a secondary characteristic class with values in Deligne cocycles that provide the differential refinement of H 4 (X, Z). The proof is completed by showing that the curvature 4-form of the refining Deligne  cocycle is the correct de Rham image of 12 p1 . Below we shall rederive this theorem as a special case of the more general ∞Chern-Weil homomorphism. We now turn to an example that genuinely lives in higher Lie theory and involves higher principal bundles. Proposition 5.1.8. The canonical projection cosk7 exp∆ (soµ3 ) → BString is an acyclic fibration in the global model structure. Proof. The 3-cells in BString are pairs consisting of 3-cells in exp∆ (so), together with labels on their boundary, subject to a condition that guarantees that the boundary of a 4-cell in String never wraps a 3-cycle in Spin. Namely, a morphism ∂∆4 → BString is naturally identified with a smooth map φ : S 3 → Spin equipped with a 2-form ω ∈ Ω2 (S 3 ) such that dω = φ∗ µ3 (θ ∧ θ ∧ θ). But since µ3 (θ ∧ θ ∧ θ) is the image in de Rham cohomology of the generator of H 3 (Spin, Z) ≃ Z this means that such φ must represent the trivial element in π3 (Spin). Using this, the proof of the claim proceeds verbatim as that of proposition 4.2.17, using that the next non-vanishing homotopy group of Spin after π3 is π7  and that the generator of H 8 (BString, Z) is 61 p2 . ∞-Chern-Weil theory 62 July 11, 2011 Remark 5.1.9. Therefore the Lie integration of the 7-cocycle exp∆ (µ7 ) cosk7 exp∆ (soµ3 ) / B7 (Z → R) ≀  BString presents a characteristic map BString → B7 U (1). Proposition 5.1.10. The Lie integration of µ7 : soµ3 → b6 R is a smooth refinement 1 p2 : BString → B7 U (1) . 6 of the second fractional Pontryagin class [SSS09a] in that postcomposition with this morphism represents the top horizontal morphism in H 1 (X, String) 1 6 p2 / H 8 (X, Z) . ·6  H 1 (X, Spin) p2  / H 8 (X, Z) Proof. As in the above case, we shall prove this below by refining to a morphism of differential cocycles and showing that the corresponding curvature 8-form represents the fractional Pontryagin class in de Rham cohomology.  5.2 Differential characteristic maps by ∞-Lie integration We wish to lift the integration in section 5.1 of Lie ∞-algebra cocycles from exp∆ (g) to its differential refinement exp∆ (g)diff in order to obtain differential characteristic maps with coefficients in differential cocycles such that postcomposition with these is the ∞-Chern-Weil homomorphism. We had obtained exp∆ (µ) essentially by postcomposition of the k-cells in exp∆ (g) with the cocyµ cle g → bn−1 R. Since the k-cells in exp∆ (g)diff are diagrams, we need to extend the morphism µ accordingly to a diagram. We had discussed in section 4.1 how transgressive cocycles extend to a diagram CE(g) o O W(g) o O inv(g) o µ CE(bn−1 R) , O cs W(bn−1 R) O h−i inv(bn−1 R) ∞-Chern-Weil theory 63 July 11, 2011 where h−i is an invariant polynomial in transgression with µ and cs is a ChernSimons element witnessing that transgression. Definition 5.2.1. Define the morphism of simplicial presheaves exp∆ (cs)diff : exp∆ (g)diff → exp∆ (bn−1 R)diff degreewise by pasting composition with this diagram:   Avert • k o CE(g) Ω (U × ∆ ) vert  si  O O     exp∆ (cs)k :       A W(g) Ω•si (U × ∆k ) o     7→    Ω•si (U k ×∆ O Avert )vert o A Ω•si (U × ∆k ) o CE(g) o O W(g) o µ CE(bn−1 R) O cs W(bn−1 R)  : µ(Avert )       : cs(A) Write (Bn R/cs)diff for the pushout exp∆ (g)diff exp∆ (cs)  coskn exp∆ (g)diff / exp (bn−1 R)diff ∆ R ∆• ∼ / Bn Rdiff .  / (Bn R/cs)diff Remark 5.2.2. This induces a corresponding morphism on the Chern-Weil subobjects exp∆ (cs)CW : exp∆ (g)CW → exp∆ (bn−1 R)CW ∞-Chern-Weil theory 64 July 11, 2011 degreewise by pasting composition with the full transgression diagram    Ω• (U × ∆k )vertAvert o  si O     exp∆ (cs)k :   Ω• (U × ∆k ) o A si  O     FA Ω• (U ) o   Ω• (U × ∆k )vertAvert o  si O     7→   Ω• (U × ∆k ) o A si  O     FA Ω• (U ) o CE(g)   O      W(g)   O     inv(g) CE(g) o O W(g) o O inv(g) o µ CE(bn−1 R) O cs W(bn−1 R) O h−i inv(bn−1 R)  : µ(Avert )        : cs(A)       : hFA i Moreover, this restricts further to a morphism of the genuine ∞-connection subobjects exp∆ (cs)conn : exp∆ (g)conn → exp∆ (bn−1 R)conn . Indeed, the commutativity of the lower part of the diagram encodes the classical equation dcs(A) = hFA i stating that the curvature of the connection cs(A) is the horizontal differential form hFA i in Ω(U ). This shows that the image of exp∆ (cs)CW is actually contained in exp∆ (bn−1 R)conn , and so the restriction to exp∆ (g)conn defines a morphism between the genuine ∞-connection subobjects. Remark 5.2.3. In the typical application – see the examples discussed below – we have that (Bn R/cs)diff is Bn (Λµ → R)diff and usually even Bn (Z → R)diff . ∞-Chern-Weil theory 65 July 11, 2011 The above constructions then yield a sequence of spans in [CartSpop , sSet]: R exp∆ (µ,cs) ∆• / Bn (Z → R)conn coskn (exp∆ (g)conn ) EE EE yy R |yy " exp (µ,cs) ∆ ∆• / Bn (Z → R)diff coskn (exp∆ (g)diff ) EE EE yy R " y| y exp (µ) • ∆ / Bn (Z → R) coskn (exp∆ (g)) ∆ ≀ ≀  BGconn ≀  BGdiff < yy yy ≀  BG < yy yy c ĉ ĉ ≀ ≀  / Bn U (1) Eb E EE   / Bn U (1)diff bEE EE  / Bn U (1)conn Here we have • the innermost diagram presents the morphism of smooth ∞-groupoids cµ : BG → Bn U (1) that is the characteristic map obtained by Lie integration from µ. Postcomposition with this is the morphism cµ : H(X, BG) → H(X, Bn U (1)) that sends G-principal ∞-bundles to the corresponding circle n-bundles. In cohomology/on equivalence classes, this is the ordinary characteristic class cµ : H 1 (X, G) → H n+1 (X, Z) . • The middle diagram is the differential refinement of the innermost diagram. By itself this is weakly equivalent to the innermost diagram and hence presents the same characteristic map cµ . But the middle diagram does support also the projection ∼ BG ← BGdiff → Bn (Z → R)diff → ♭dR Bn+1 R onto the curvature characteristic classes. This is the simple version of the ∞-Chern-Weil homomorphism that takes values in de Rham cohomology n+1 HdR (X) = π0 H(X, ♭dR Bn+1 R) H(X, BG) → H(X, ♭dR Bn+1 R) . • The outermost diagram restrict the innermost diagram to differential refinements that are genuine ∞-connections. These map to genuine ∞connections on circle n-bundles and hence support the map to secondary characteristic classes H(X, BGconn) → H(X, Bn U (1)conn ) . ∞-Chern-Weil theory 5.3 66 July 11, 2011 Examples We spell out two classes of examples of the construction of the ∞-Chern-Weil homomorphism: The Chern-Simons circle 3-bundle with connection In example 5.1.5 we had considered the canonical 3-cocycle µ3 ∈ CE(g) on the semisimple Lie algebra g of a compact, simple and simply connected Lie group G and discussed how its Lie integration produces a map from Čech-cocycles for G-principal bundles to Čech-cocycles for circle 3-bundles. This map turned out to coincide with that given in [BM96b]. We now consider its differential refinement. From example 4.1.22 we have a Chern-Simons element cs3 for µ3 whose invariant polynomial is the Killing form h−, −i on g. By definition 5.2.1 this induces a differential Lie integration exp∆ (cs) of µ. As a consequence of all the discussion so far, we now simply read off the following corollary. Corollary 5.3.1. Let cosk3 exp∆ (g)conn R ∆• exp∆ (cs3 ) / B3 (Z → R)conn ≀  BGconn be the span of simplicial presheaves obtained from the Lie integration of the differential refinement of the cocycle from example 5.1.5. Composition with this span QX C ˆ (ĝ,∇) / cosk3 exp∆ (g)conn ≀  Č(U) R ∆• exp∆ (cs3 ) / B3 (Z → R)conn ≀ (g,∇)  / BGconn ∼loc  X (where QX → Č(U) is the pullback acyclic fibration and Č(U) → QX any choice of section from the cofibrant Č(U) through this acyclic fibration) produces a map from Čech cocycles for smooth G-principal bundles with connection to degree 4 Čech-Deligne cocycles ĉcs : Č(U, BGconn ) → Č(U, B3 U (1)conn ) on a paracompact smooth manifold X as follows: ∞-Chern-Weil theory 67 July 11, 2011 • the input is a set of transition functions and local connection data (gij , Ai ) on a differentiably good open cover {Ui → X} as in section 2.2; (notice that there is a G-principal bundle P → X with Ehresmann connection 1-form A ∈ Ω1 (P, g) and local sections {σi : Ui → P |Ui } such that σi |Uij = σj |Uij gij and Ai = σi∗ A) • the span composition produces a lift of this data: – on double intersections a smooth family ĝij : (Ui ∩ Uj ) × ∆1 → G ∗ of based paths in G, together with a 1-form Aij := ĝij Ai ∈ Ω1 (Uij × 1 ∆ , g); – on triple intersections a smooth family ĝijk : (Ui ∩ Uj ∩ Uk ) × ∆2 → G ∗ of based 2-simplices in G, together with a 1-form Aijk := ĝijk Ai ∈ 1 1 Ω (Uijk × ∆ , g); – on quadruple intersections a smooth family ĝijkl : (Ui ∩ Uj ∩ Uk ∩ Ul ) × ∆3 → G of based 2-simplices in G, together with a 1-form ∗ Aijkl := ĝijkl Ai ∈ Ω1 (Uijkl × ∆1 , g); • this lifted cocycle data is sent to the Čech-Deligne cocycle Z Z Z (cs(Ai ), cs(Âij ), cs(Âijk ), µ(Âijkl )) = ∆1 ∆2 ∆3 Z Z Z ∗ ∗ ∗ (cs(Ai ), ĝij cs(A), ĝijk cs(A), ĝijkl µ(A)) , ∆1 ∆2 ∆3 where cs(A) is the Chern-Simons 3-form obtained by evaluating a g-valued 1-form in the chosen Chern-Simons element cs. Proof. That we obtain Čech-Deligne data as indicated is a straightforward matter of inserting the definitions of the various morphisms. That the data indeed satisifies the Čech-cocycle condition follows from the very fact that by construction these are components of a morphism Č(U) → B3 (Z → R)conn , as discussed in section 3. The curvature 4-form of the resulting Čech-Deligne cocycle is (up to a scalar factor) the Pontryagin form hFA ∧ FA i. By the general properties of Deligne cohomology this represents in de Rham cohomology the integral class in H 4 (X, Z) of the cocycle, so that we find that this is a multiple of the class of the G-bundle P → X corresponding to the Killing form invariant polynomial. In the case that G = Spin, we have that H 3 (G, Z) ≃ Z. By proposition 5.1.3 it follows that the above construction produces a generator of this cohomology group: there cannot be a natural number ≥ 2 by which this R/Z-cocycle is divisible, since that would mean that µ3 (θ ∧θ ∧θ) had a period greater than 1 around the generator of π3 (G), which by construction it does not. But this generator is the fractional Pontryagin class 21 p1 (see the review in [SSS09a] for instance).  ∞-Chern-Weil theory 68 July 11, 2011 Definition 5.3.2. We write 1 p̂1 : BSpinconn → B3 U (1)conn 2 in H for the morphism of smooth ∞-groupoids given by the above corollary and call this the differential first fractional Pontryagin map. Remark 5.3.3. The Čech-Deligne cocycles produced by the span composition in the above corollary are again those considered in section 4 of [BM96b]. We may regard the above corollary as explaining the deeper origin of that construction. But the full impact of the construction in the above corollary is that it applies more generally in cases where standard Chern-Weil theory is not applicable, as discussed in the introduction. We now turn to the first nontrivial example for the ∞-Chern-Weil homomorphism beyond the traditional ChernWeil homomorphism. The Chern-Simons circle 7-bundle with connection Recall from proposition 5.1.10 the integration of the 7-cocycle µ7 on the String 2-group. We find a Chern-Simons element cs7 ∈ W (soµ3 ) and use this to obtain the differential refinement of this characteristic map. Corollary 5.3.4. Let R cosk7 exp∆ (soµ3 )conn ∆• exp∆ (cs7 ) / B7 (Z → R)conn ≀  BStringconn be the span of simplicial presheaves obtained from the Lie integration of the differential refinement of the cocycle from proposition 5.1.10. Composition with this span QX C ˆ (ĝ,∇) ≀  Č(U) R / cosk7 exp∆ (soµ3 )conn ∆• exp∆ (cs7 ) / B7 (Z → R)conn ≀ (g,∇)  / BStringconn ∼loc  X (where QX → Č(U) is the pullback acyclic fibration and Č(U) → QX any choice of section from the cofibrant Č(U) through this acyclic fibration) produces a map from Čech cocycles for smooth principal String 2-bundles with connection to degree 8 Čech-Deligne cocycles ĉcs7 : Č(U, BStringconn ) → Č(U, B7 U (1)conn ) ∞-Chern-Weil theory July 11, 2011 69 on a paracompact smooth manifold X. For P → X a principal Spin bundle with String-structure, i.e. with a trivialization of 21 p1 (P ), the integral part of ĉcs7 (P ) is the second fractional Pontryagin class 16 p2 (P ). Proof. As above.  This completes the proof of theorem 1.0.1. Definition 5.3.5. We write 1 pˆ2 : BStringconn → B7 U (1)conn 6 in H for the morphism of smooth ∞-groupoids presented by the above construction, and speak of the differential second fractional Pontryagin map. Remark 5.3.6. Notice how the fractional differential class 16 p̂2 comes out as compared to the construction in [BM96b], where a Čech cocycle representing −2p2 is obtained . There, in order to be able to fill the simplices in the 7coskeleton one works with chains in the Stiefel manifold SO(n)/SO(q) and multiplies these with the cardinalities of the torsion homology groups in order to ensure that they they become chain boundaries that may be filled. On the other hand, in the construction above the lift to the Čech cocycle of a String 2-bundle ensures that all the simplices of the cocycle in Spin(n) can already be filled genuinely, without passing to multiples. Therefore the cocycle constructed here is a fraction of the cocycle constructed there by these integer factors. 6 Homotopy fibers of Chern-Weil: twisted differential structures Above we have shown how to construct refined secondary characteristic maps as morphisms of smooth ∞-groupoids of differential cocycles. This homotopical refinement of secondary characteristic classes gives access to their homotopy fibers. Here we discuss general properties of these and indicate how the resulting twisted differential structures have applications in string physics. Some of the computations necessary for the following go beyond the scope of this article and will not be spelled out. Details on these are in the followup [FSS]. See also section 4.2 of [Sch10]. In 6.1 below we consider some basic concepts of obstruction theory in order to set the scene for the its differential refinement further below in 6.2. Before we get to that, it may be worthwhile to note the following subtlety. There are two different roles played by topological spaces in the homotopy theory of higher bundles: ∞-Chern-Weil theory July 11, 2011 70 1. they serve as a model for discrete ∞-groupoids via the standard Quillen equivalence Top o |−| Sing / sSet ≃ ∞Grpd , where the ∞-groupoids on the right are “discrete” in direct generalization to the sense in which a discrete group is discrete, 2. and they also serve to model actual geometric structure in the sense of “continuous cohesion”, that for instance distinguishes a non-discrete topological group from the underlying discrete group. Therefore a topological group and more generally a simplicial topological group is a model for something that pairs these two aspects of topological spaces. To make this precise, let Top be a small category of suitably nice topological spaces and continuous maps between them, equipped with the standard Grothendieck topology of open covers. Then we can consider the the ∞-topos of ∞-sheaves over Top, presented by simplicial presheaves over Top, and this is the context that contains topological ∞-groupoids in direct analogy to the smooth ∞-groupoids that we considered in the bulk of the article. ∞Grpd ≃ Sh∞ (∗) ≃ (sSet)op Top∞Grpd ≃ Sh∞ (Top) ≃ ([Topop , sSet]loc )op H := Smooth∞Grpd ≃ Sh∞ (CartSp) ≃ ([CartSpop , sSet]loc )op We have geometric realization functors (see 3.2 and 3.3 in [Sch10]) Π : Top∞Grpd → ∞Grpd and Π : Smooth∞Grpd → ∞Grpd , which on objects represented by simplicial topological spaces are given by the traditional geometric realization operation. For G a topological group or topological ∞-group, we write BG for its delooping in Top∞Grpd. Under geometric realization this becomes the standard classifying space BG := Π(BG), which, while naturally presented by a topological space, is really to be regarded as a presentation for a discrete ∞-groupoid. 6.1 Topological and smooth c-Structures An important fact about the geometric realization of topological ∞-groupoids is Milnor’s theorem [Mil56]: Theorem 6.1.1. For every connected ∞-groupoid (for instance presented by a connected homotopy type modeled on a topological space) there is a topological group such that its topological delooping groupoid BG has a geometric realization wealy equivalent to it. ∞-Chern-Weil theory July 11, 2011 71 This has the following simple, but important consequence. Let G be a topological group and consider some characteristic map c : BG → K(Z, n + 1), representing a characteristic class [c] ∈ H n+1 (BG, Z). Then consider the homotopy fiber /∗ BGc  BG c  / K(Z, n + 1) formed in ∞Grpd. While this homotopy pullback takes place in discrete ∞groupoids, Milnor’s theorem ensures that there is in fact a topological group Gc such that BGc is indeed its classifying space. For X ∈ Topsm , the set of homotopy classes [X, BG] is in natural bijection with equivalence classes of G-principal topological bundles P → X. One says that P has c-structure if it is in the image of [X, BGc ] → [X, BG]. Remark 6.1.2. By the defining universal property of homotopy fibers, the datum of a (equivalence class of a) principal Gc -bundle over X is equivalent to the datum of a principal G-bundle P over X whose characteristic class [c(P )] vanishes. Example 6.1.3. Classical examples of this construction are Ow1 = SO and U c1 = SU . Indeed is well known that the structure group of an O-bundle can be reduced to SO if and only if its first Stiefel-Withney class vanishes. More precisely, an principal SO-bundle can be seen as a principal O-bundle with a trivializiation of the associated orientation Z/2Z-bundle. Similarly, an SU bundle is a U bundles with a trivialization of the associated determinant bundle, and such a trivialization exists if and only if the first Chern class of the given U (n)-bundle vanishes. 1 A more advanced example is the one described in Section 5: Spin 2 p1 = String, i.e., String-bundles are Spin-bundles with a trivialization of the associated 2-gerbe. For a more refined description of c-structures, we need to consider not just the set of equivalence classes of bundles, but the full cocycle ∞-groupoids: whose objects are such bundles, whose morphisms are equivalences between such bundles, whose 2-morphisms are equivalaneces between such equivalences, and so on. But for this purposes it matters whether we form homotopy fibers in discrete or in topological ∞-groupoids. We shall be interested in homotopy fibers of topological ∞-groupoids. Definition 6.1.4. Let G be a simplicial topological group and c : BG → Bn U (1) a classifying map. Write BGc for the homotopy fiber /∗ BGc  BG c  / Bn U (1) ∞-Chern-Weil theory July 11, 2011 72 of topological ∞-groupoids. Then for X a paracompact topological space, we say that the ∞-groupoid cStruc(X) := Top∞Grpd(X, BGc ) is the ∞-groupoid of topological c-structures on X. Analogously, for G a smooth ∞-group and c : BG → Bn U (1) a morphism of smooth ∞-groupoids as in 3, we write BGc for its homotopy fiber in H = Smooth∞Grpd and says that cStruc(X) := H(X, BGc ) is the ∞-groupoid of smooth c-structures on X. Among the first nontrivial examples for these notions is the following Definition 6.1.5. Let 1 p1 : BSpin → B3 U (1) 2 be the smooth refinement of the first fractional Pontryagin class, from corollary 5.3.1. We write 1 BString := BSpin 2 p1 and call String the smooth String 2-group. By prop. 4.2.24 the smooth 2-groupoid BString is presented by the simplicial presheaf cosk3 exp(soµ3 ). Proposition 6.1.6. Under geometric realization the delooping of the smooth String 2-group yields the classifying space of the topological string group ΠBString ≃ BString . Moreover, in cohomology smooth 21 p1 -structures on a manifold X are equivalent to ordinary String-structures, hence 21 p1 -structures. Proof. The first statement is proven in section 4.2 of [Sch10]. The second statement follows with proposition 4.2.26 from proposition 4.1 in [NiWa11].  6.2 Twisted differential c-Structures By the universal property of the homotopy pullback, the ∞-groupoid of topological c-structures on X, def. 6.1.4, can be equivalently described as the homotopy pullback /∗ cStruc(X)  Top∞Grpd(X, BG) c  / Top∞Grpd(X, Bn U (1)) ∞-Chern-Weil theory 73 July 11, 2011 of ∞-groupoids of cocycles over X, where the right vertical morphism picks any cocycle representing the trivial class. From this point of view, there is no reason to restrict one’s attention to the fiber of c Top∞Grpd(X, BG) − → Top∞Grpd(X, Bn U (1)) over the distinguished point in Top∞Grpd(X, Bn U (1) corresponding to the trivial Bn U (1)-bundle over X. Rather, it is more natural and convenient to look at all homotopy fibers at once, i.e. to consider all possible (isomorphism classes of) Bn U (1)-bundles over X. Definition 6.2.1. For c : BG → Bn U (1) a characteristic map in either H = Top∞Grpd or H = Smooth∞Grpd, and for X a paracompact topological space or paracompact smooth manifold, respectively, let cStructw (X) be the ∞-groupoid defined by the homotopy pullback cStructw (X) tw / H n+1 (X; Z) , χ  H(X, BG) c  / H(X, Bn U (1)) where the right vertical morphism from the cohomology set into the cocycle n-groupoid picks one basepoint in each connected component, i.e., picks a representative U (1)-(n − 1)-gerbe for each degree n + 1 integral cohomology class. We call cStructw (X) the ∞-groupoid of (topological or smooth) twisted cstructures. For τ ∈ cStructw (X) we say tw(τ ) ∈ H n+1 (X; Z) is its twist and χ(τ ) ∈ Top∞Grpd(X, BG) is the (topological or smooth) underlying Gprincipal ∞-bundle of τ , or that τ is a tw(τ )-twisted lift of χ(τ ). For [ω] ∈ H n+1 (X; Z) a cohomology class, cStructw=[ω] (X) is the full sub∞-groupoid of cStructw (X) on those twisted structures with twist [ω]. The following list basic properties of cStructw (X) that follow directly on general abstract grounds. Proposition 6.2.2. 1. The definition of cStructw (X) is independent, up to equivalence, of the choice of the right vertical morphism. Indeed, all choices of such are (non canonically) equivalent as ∞-functors. 2. For BG a topological k-groupoid for k ≤ n−1, the ∞-groupoid cStructw (X) is an (n − 1)-groupoid. 3. The following pasting diagram of homotopy pullbacks shows how cStructw=[ω] (X) c can be equivalently seen as the homotopy fiber of Top∞Grpd(X, BG) − → Top∞Grpd(X, Bn U (1)) over a representative U (1)-(n − 1)-gerbe for the ∞-Chern-Weil theory 74 July 11, 2011 cohomology class [ω]: cStructw=[ω] (X) /∗ , [ω]  cStructw (X) tw  / H n+1 (X; Z) c  / Top∞Grpd(X, Bn U (1) χ  Top∞Grpd(X, BG) In particular one has cStructw=0 (X) ∼ = cStruc(X). We consider the following two examples, being the direct differential refinement of those of def. 6.2: Definition 6.2.3. For 21 p1 : BSpin → B3 U (1) the smooth first fractional Pontryagin class from prop. 5.1.7, we call 1 p1 Structw (X) 2 the 2-groupoid of smoth twisted String-structures on X. For 61 p2 : BSpin → B7 U (1) the smooth second fractional Pontryagin class from prop. 5.1.10, we call 1 p2 Structw (X) 6 the 6-groupoid of twisted differential Fivebrane-structures on X. The terminology here arises from the applications in string theory that originally motivated these constructions, as described in [SSS09a]. In order to explicity compute simplicial sets modelling ∞-groupoids of smooth twisted c-structures, the usual recipe for computing homotopy fibers applies: it is sufficient to present the smooth cocycle c by a fibration of simplicial presheaves and then form an ordinary pullback of simplicial presheaves. We shall discuss now how to obtain such fibrations by Lie integration of factorizations of the L∞ -cocycles µ3 : so → b2 R and µ7 : soµ3 → b6 R. These factorizations at the L∞ -algebra level are due to [SSS09c]. The full proofs that their Lie integration produces the desired fibration is due to [FSS] and can be found in section 4.2 of [Sch10]. Definition 6.2.4. Let string := soµ3 be the string Lie 2-algebra from Definition 4.2.22, and let (bR → string) be the Lie 3-algebra defined by the fact that its Chevalley-Eilenberg algebra is that of so with two additional generators, b in degree 2 and c in degree 3, and with the differential extended to these as dCE b = c − µ3 dCE c = 0 . ∞-Chern-Weil theory 75 July 11, 2011 There is an evident sequence of morphisms of L∞ -algbras so → (bR → string) → b2 R factoring the 3-cocycle µ3 : so → b2 R. Proposition 6.2.5. The Lie integration, according to definition 4.2.8, of this sequence of L∞ -algebra morphisms is a factorization 1 ∼ / / B3 U (1) p1 : cosk3 exp(so) → cosk3 exp(bR → string) 2 of the smooth refinement of the first fractional Pontryagin class from proposition 5.1.7 into a weak equivalence followed by a fibration in [CartSpop , sSet]proj . Corollary 6.2.6. The 2-groupoid of twisted string structures on a smooth manifold X is presented by the ordinary fibers of [CartSpop , sSet](Č(U), cosk3 exp(bR → soµ3 )) → [CartSpop , sSet](Č(U)), B3 U (1)) . We spell out the explicit presentation for 12 p1 Structw (X) further below, after passing to the following differential refinement. Recall that when an L∞ -algebra cocycle µ : g → bn R can be transgressed to an invariant polynomial by a Chern-Simons element, as in section 5.2, then the smooth characteristic map c = exp(µ) refines to a differential characteristic map ĉ : BGconn → Bn U (1)conn , where BGconn := coskn+1 exp∆ (g)conn . In terms of this there is a straightforward refinement of 6.2.1: Definition 6.2.7. For X a smooth manifold, let ĉStructw (X) be the ∞-groupoid defined by the homotopy pullback ĉStructw (X) tw / Ĥ n+1 (X; Z) diff , χ  H(X, BGconn ) ĉ  / H(X, Bn U (1)conn ) where the right vertical morphism from the cohomology set into the cocycle n-groupoid picks one basepoint in each connected component. We call ĉStructw (X) the ∞-groupoid of twisted differential ĉ-structures on X. Such twisted differential structures enjoy the analogous properties listed in prop. 6.2.2. In particular, also for differential refinements one has a natural interpretation of untwisted ĉ-structures: the component of ĉStruc(X) over the 0-twist is the ∞-groupoid of Ĝ-∞-connections ĉStructw=0 (X) ≃ Smooth∞Grpd(X, BĜconn ) , where Bn−2 U (1) → Ĝ → G is the extension of ∞-groups classified by c : BG → Bn U (1). This is shown in detail in [FSS], see also section 4.2 of [Sch10]. ∞-Chern-Weil theory 6.3 76 July 11, 2011 Examples We consider the following two examples: Definition 6.3.1. For 21 p̂1 : BSpinconn → B3 U (1)conn the differential first fractional Pontryagin class from definition 5.3.2 and 61 p̂2 : BStringconn → B7 U (1)conn the differential second fractional Pontryagin class from definition 5.3.5, we call 1 p̂1 Structw (X) 2 the 2-groupoid of twisted differential String-structures on X and 1 p̂2 Structw (X) 6 the 6-groupoid of twisted differential Fivebrane-structures on X. We indicate now explicit constructions of these higher groupoids of twisted structures. Twisted differential String-structures. The factorization 1 ∼ p1 : cosk3 exp(so) → cosk3 exp(bR → string) 2 / / B3 U (1) of the smooth first fractional Pontryagin class from prop. 6.2.5 has a differential refinement, from which we can compute the 2-groupoid of twisted differential string structures by an ordinary pullback of simplicial sets. This is achieved by factoring the commutative diagram CE(so) o O W(so) o O inv(so) o µ CE(b2 R) O cs W(b2 R) O h−i inv(b2 R) as a commutative diagram CE(so) o O ∼ CE(bR → string) o O CE(b2 R) O W(so) o O ∼ W̃(bR → string) o O W(b2 R) O inv(so) o = inv(bR → string) o inv(b2 R) ∞-Chern-Weil theory 77 July 11, 2011 as in [SSS09c]. In the above diagram the Weil algebra W(bR → string) is replaced by the modified Weil algebra W̃(bR → string) presented by 1 dta = − C a bc tb ∧ tc + ra 2 db = c − cs3 + h dc = g dra = −C a bc tb ∧ rc dh = h−, −i − g dg = 0. Here {ta } are the coordinates on so relative to a basis {ea }, C a bc are the structure constants of the Lie brackets of so with respect to this basis, b and c are the additional generators of the Chevalley-Eilenberg algebra CE(bR → string), the generators ra , h, g are the images of of ta , b, c via the shift isomorphism, and cs3 is a Chern-Simons element transgressing the cocycle µ3 to the Killing form h−, −i. The modified Weil algebra W̃(bR → string) is ismorphic (via a distinguished isomorphism) to the Weil algebra W(bR → string) as a dgca, but the isomorphism between the two does not preserves the graded subspaces of polynomials in the shifted generators. In particuar the modified algebra takes care of realizing the horizontal homotopy between h−, −i and g as a polynomial in the shifted generators, see the third item in example 4.1.22. Since the notion of curvature forms depends on the splitting of the generators of the Weil algebra into shifted and unshifted generators (see Remark 4.1.9), the modified Weil algebra will lead to a modified version of exp(bR → string)conn which we will denote by exp(bR → string)conn ] . This is a resolution of exp(so)conn that is naturally adapted to the computation of the homotopy fiber of 21 p1 . As we will show below, it is precisely this resolution that is the relevant one for applications to the Green-Schwarz mechanism. Proposition 6.3.2. Lie integration of the above diagram of differential L∞ algebra cocycles provides a factorization 1 ∼ p̂1 : cosk3 exp(so)conn → cosk3 exp(bR → string)conn ] 2 / / B3 U (1)conn of the differential first fractional Pontryagin class from definition 5.3.2 into a weak equivalence followed by a fibration in [CartSpop , sSet]proj . This is due to [FSS]. Details can be found in section 4.2 of [Sch10]. Corollary 6.3.3. The 2-groupoid of twisted differential string structures on a smooth manifold X with respect to a differentiably good open cover U = {Ui → X} is presented by the ordinary fibers of the morphism of simplicial sets op 3 [CartSpop , sSet](Č(U), cosk3 exp∆ (bR → string)conn ] ) → [CartSp , sSet](Č(U), B U (1)conn ) . A k-simplex for k ≤ 3 in the simplicial set of local differential forms data describing a differential twisted string structure consists, for any k-fold intersection ∞-Chern-Weil theory 78 July 11, 2011 UI := Ui0 ,··· ,ik in the cover U, of a triple (ω, B, C)I of connection data such the corresponding curvature data (Fω , H, G)I are horizontal. Here ωI ∈ Ω1si (UI × ∆k ; so), BI ∈ Ω2si (UI × ∆k ; R), CI ∈ Ω3si (UI × ∆k ; R) and 1 FωI = dωI + [ωI , ωI ], 2 HI = dBI + cs(ωI ) − CI , GI = dCI . Remark 6.3.4. The curvature forms of a twisted string structure obey the Bianchi identities dFωI = −[ωI , FωI ], dHI = hFωI ∧ FωI i − GI , dGI = 0. Twisted differential String-structures and the Green-Schwarz mechanism. The above is the local differential form data governing what in string theory is called the Green-Schwarz mechanism. We briefly indicate what this means and how it is formalized by the notion of twisted differential Stringstructures (for background and references on the physics story see for instance [SSS09a]). The standard action functionals of higher dimensional supergravity theories are generically anomalous in that instead of being functions on the space of field configurations, they are just sections of a line bundle over these spaces. In order to get a well defined action principle as input for a path-integral quantization to obtain the corresponding quantum field theories, one needs to prescribe in addition the data of a quantum integrand. This is a choice of trivialization of these line bundles, together with a choice of flat connection. For this to be possible, the line bundle has to be trivializable and flat in the first place. Its failure to be tivializable – its Chern class – is called the global anomaly, and its failure to be flat – its curvature 2-form – is called its local anomaly. But moreover, the line bundle in question is the tensor product of two different line bundles with connection. One is a Pfaffian line bundle induced from the fermionic degrees of freedom of the theory, the other is a line bundle induced from the higher form fields of the theory in the presence of higher electric and magnetic charge. The Pfaffian line bundle is fixed by the requirement of supersymmetry, but there is freedom in choosing the background higher electric and magnetic charge. Choosing these appropriately such as to ensure that the tensor product of the two anomaly line bundles produces a flat trivializable line bundle is called an anomaly cancellation by a Green-Schwarz mechanism. Concretely, the higher gauge background field of 10-dimensional heterotic supergravity is the Kalb-Ramond field, which in the absence of fivebrane magnetic charge is modeled by a circle 2-bundle (a bundle gerbe) with connection and curvature 3-form H ∈ Ω3 (X), satisfying the higher Maxwell equation dH = 0 . In order to cancel the relevant quantum anomaly it turns out that a magnetic background charge density is to be added to the system, whose differential form ∞-Chern-Weil theory July 11, 2011 79 representative is the difference jmag := hF∇Spin ∧F∇Spin i−hF∇SU ∧F∇SU i between the Pontryagin forms of the Spin-tangent bundle and of a given SU-gauge bundle (here we leave normalization constants implict in the definition of the invariant polynomials h−, −i). This modifies the above Maxwell equation locally, on a patch Ui ⊆ X to dHi = hFωi ∧ Fωi i − hFAi ∧ FAi i . Comparing with proposition 6.3.3 we see that, while such Hi is no longer be the local curvature 3-forms of a circle 2-bundle (2-gerbe), they are that of a twisted circle 3-bundle – a Čech-Deligne 2-cochain that trivializes the difference of the two Chern-Simons Čech-Deligne 3-cocycles – that is part of the data of a twisted differential string-structure with Gi = hFAi ∧ FAi i. Note that the above differential form equation exhibits a de Rham homotopy between the two Pontryagin forms. This is the local differential aspect of the very definition of a twisted differential string-structure: a homotopy from the Chern-Simons circle 3-bundle of the Spin-tangent bundle to a given twisting circle 3-bundle, which here is itself a Chern-Simons 3-bundle, coming from an SU-bundle. This anomaly cancellation has been known in the physics literture since the seminal article [Ki87]. Recently [Bun09] has given a rigorous proof in the special case that underlying topological class of the twisting gauge bundle is trivial. This proof used the model of twisted differential string structures with topologically tivial twist given in [Wal09]. This model is constructed in terms of bundle 2-gerbes and does not exhibit the homotopy pullback property of definition 6.2.7 explicitly. However, the author shows that his model satisfies the properties 6.2.2 satisfied by the abstract homotopy pullback. Twisted differential fivebrane structures. The construction of an explicit Kan complex model for the 6-groupoid of twisted differential fivebrane structures proceeds in close analogy to the above discussion for twisted differential string structures, by adding throughout one more layer of generators in the CE-algebra. Definition 6.3.5. Write fivebrane := (soµ3 )µ7 for the L∞ -algebra extension of the string Lie 2-algebra (def. 4.2.22) by the 7-cocycle µ7 : soµ3 → b6 R (remark 4.2.23) according to prop. 4.1.23. Following [SSS09b] we call this the fivebrane Lie 6-algebra. Remark 6.3.6. The Chevelley-Eilenberg algebra CE(fivebrane) is given by 1 dta = − C a bc tb ∧ tc 2 1 db2 = −µ3 := − h−, [−, −]i 2 1 db6 = −µ7 := − h−, [−, −], [−, −], [−, −]i 8 for {ta } and b2 generators of degree 1 and 2, respectively, as for the string Lie 2-algebra, and b6 a new generator in degree 6. ∞-Chern-Weil theory 80 July 11, 2011 Definition 6.3.7. Let (b5 R → fivebrane) be the Lie 7-algebra defined by having CE-algebra given by 1 dta = − C a bc tb ∧ tc 2 db2 = c3 − µ3 db6 = c7 − µ7 dc3 = 0 dc7 = 0 . Proposition 6.3.8. In the evident factorization ∼ µ7 : CE(string) o CE(b5 → fivebrane) o CE(b6 R) of the 7-cocycle µ7 , the first morphism is a quasi-isomorphism. As before, it is convenient to lift this factorization to the differential refinement by using a slightly modified Weil algebra to collect horizontal generators W̃ (b5 → fivebrane) ≃ W (b5 → fivebrane) given by 1 dta = − C a bc tb ∧ tc + ra 2 db2 = c3 − cs3 + h3 db6 = c7 − cs7 + h7 dc3 = g4 dc7 = g8 dh3 = h−, −i − g4 dh7 = h−, −, −, −i − g8 , where h−, −, −, −i is the second Pontryagin polynomial for so, to obtain a factorization CE(string) o O ∼ CE(b5 R → fivebrane) o O CE(b6 R) . O W(string) o O ∼ W̃(b5 R → fivebrane) o O W(b6 R) O inv(string) o = inv(b5 R → fivebrane) o inv(b6 R) This is the second of the big diagrams in [SSS09c]. Using this and following through the same steps as for twisted differential string-structures above, one ∞-Chern-Weil theory 81 July 11, 2011 finds that the 6-groupoid of twisted differential fivebrane structures over some X with respect to a diffrentiably good open cover U has k-cells for k ≤ 7 given by differential form data ωI ∈ Ω1si (UI ×∆k ; so), (B2 )I ∈ Ω2si (UI ×∆k ; R), (C3 )I ∈ Ω3si (UI × ∆k ; R), (B6 )I ∈ Ω6si (UI ×∆k ; R), (C7 )I ∈ Ω7si (UI × ∆k ; R) with horizontal curvature forms 1 FωI = dωI + [ωI , ωI ], 2 (H3 )I = d(B2 )I + cs3 (ωI ) − (C3 )I , (G4 )I = d(C4 )I , (H7 )I = d(B6 )I + cs7 (ωI ) − (C7 )I , (G8 )I = d(C8 )I . And Bianchi identities dFωI = −[ωI , FωI ], d(H3 )I = hFωI ∧FωI i− (G4 )I , d(H7 )I = hFωI ∧FωI ∧FωI ∧FωI i− (G8 )I , d(G4 )I = 0, d(G8 )I = 0 Twisted differential fivebrane structures and the dual Green-Schwarz mechanism. On a 10-dimensional smooth manifold X a (twisted) circle 2bundle with local connection form {(B2 )I } and (local) curvature forms {(H3 )I } is the electric/magnetic dual of a (twisted) circle 6-bundle with local connection 6-forms {(B2 )I } and (local) curvature forms {(H7 )I }. It is expected (see the references in [SSS09a]) that there is a magnetic dual quantum heterotic string theory where the string – electrically charged under B2 – is replaced by the fundamental fivebrane – magnetically charged under B6 . While the understanding of the 6-dimensional fivebrane sigma-model is rudimentary, its fermionic worldvolume quantum anomaly can and has been computed and the corresponding anomaly cancelling Green-Schwarz mechanism has been written down (all reviewed in [SSS09a]). If X does have differential string structure then its local differential expression is the relation d(H7 )I = hFωI ∧ FωI ∧ FωI ∧ FωI i − hFAI ∧ FAI ∧ FAI ∧ FAI i for some normalization of invariant polynomials, where the second term is the curvature characteristic form of the next higher Chern class of the background SU-principal gauge bundle. Comparing with the above formula, we find that this is indeed modeled by twisted differential fivebrane structures. ∞-Chern-Weil theory July 11, 2011 82 Appendix: ∞-Stacks over the site of Cartesian spaces Here we give a formal description of simplicial presheaves over the site of Cartesian spaces and prove several statements mentioned in Section 3. Definition 6.3.9. For X a d-dimensional paracompact smooth manifold, a differentiably good open cover is an open cover U = {Ui → X}i∈I such that for all n ∈ N every n-fold intersection Ui1 ∩· · ·∩Uin is either empty or diffeomorphic to Rd . Notice that this is asking a little more than that the intersections are contractible, as for ordinary good open covers. Proposition A.1. Differentiably good open covers always exist. Proof. By [Gre79] every paracompact manifold admits a Riemannian metric with positive convexity radius rconv ∈ R. Choose such a metric and choose an open cover consisting for each point p ∈ X of the geodesically convex open subset Up := Bp (rconv ) given by the geodesic rconv -ball at p. Since the injectivity radius of any metric is at least 2rconv [Ber76] it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection Up1 ∩ · · · ∩ Upn the geodesic flow around any point u is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image. Moreover, the preimage of the intersection region under the geometric flow is a star-shaped region in the tangent space Tu X: the intersection of geodesically convex regions is itself geodesically convex, so that for any v ∈ Tu X with exp(v) ∈ Up1 ∩ · · · ∩ Upn the whole geodesic segment t 7→ exp(tv) for t ∈ [0, 1] is also in the region. So we have that every finite non-empty intersection of the Up is diffeomorphic to a star-shaped region in a Euclidean space. It is then a folk theorem that every star-shaped region is diffeomorphic to an Rn ; an explicit proof of this fact is in theorem 237 of [Fer07].  Recall the following notions [Joh03]. Definition A.1. A coverage on a small category C is for each object U ∈ C a choice of collections of morphisms U = {Ui → U } – called covering families – such that whenever U is a covering family and V → U any morphism in C there exists a covering family {Vj → V } such that all diagrams Vj  V ∃ / Ui  /U exist as indicated. The covering sieve corresponding to a covering family U is the colimit Č(U)k ∈ [C op , Set] S(U) = lim → [k]∈∆ ∞-Chern-Weil theory July 11, 2011 83 of the Čech-nerve, formed after Yoneda embedding in the category of presheaves on C. Definition A.2. A site is a small category C equipped with a coverage. A sheaf on a site is a presheaf A : C op → Set such that for each covering sieve S(U) → U the morphism A(U ) ≃ [C op , Set](U, A) → [C op , Set](S(U), A) is an isomorphism. Remark A.1. Often this is formulated in terms of Grothendieck topologies instead of coverages. But every coverage induces a unique Gorthendieck topology such that the corresponding notions of sheaf coincide. An advantage of using coverages is that there are fewer morphisms to check the sheaf condition against. In the language of left exact reflective localizations: the coverage sieve projections of a covering family form a small set such that localizing the presheaf category at this set produces the category of sheaves. This localization however inverts more morphisms than just the coverage sieves. This saturated class of inverted morphisms contains also the sieve projections of the corresponding Grothendieck topology. Below we use this for obtaining the ∞-stacks/∞-sheaves by left Bousfield localization just at a coverage. Corollary A.1. Differentiably good open covers form a coverage on the category CartSp. Proof. The pullback of a differentiably good open cover always exists in the category of manifolds, where it is an open cover. By the above, this may always be refined again by a differentiably good open cover.  Definition A. 3. We consider CartSp as a site by equipping it with this differentiably-good-open-cover coverage. Definition A.4. Write [CartSpop , sSet]proj for the global projective model category structure on simplicial presheaves whose weak equivalences and fibrations are objectwise those of simplicial sets. Write [CartSpop , sSet]proj,loc for the left Bousfield localization of [CartSpop , sSet]proj at at the set of coverage Čech-nerve projections Č(U) → U . This is a simplicial model category with respect to the canonical simplicial enrichment of simplicial presheaves, see [Dug01]. For X, A two objects, we write [CartSpop , sSet](X, A) ∈ sSet for the simplicial homcomplex of morphisms between them. Proposition A. 2. In [CartSpop , sSet]proj,loc the Čech-nerve Č(U) → X of a differentiably good open cover over a paracompact smooth manifold X is a cofibrant resolution of X. ∞-Chern-Weil theory July 11, 2011 84 Proof. By assumption Č(U) is degreewise a coproduct of representables (this is what the definition of differentiably good open cover formulates). Clearly its degeneracies split off as a direct summand in each degree (the summand of intersections Ui0 ∩· · · Uin where at least one index repeats). With this it follows from corollary 9.4 in [Dug01] that Č(U) is cofibrant in the global projective model structure. Since left Bousfield localization keeps the cofibrations unchanged, it follows that it is also cofibrant in the local structure. That the projection Č(U ) → X is a weak equivalence in the local structure follows by using our theorem A.1 below in proposition A.4 of [DHI04].  Corollary A.2. The fibrant objects of [CartSpop , sSet]proj,loc are precisely those simplicial presheaves A that are objectwise Kan complexes and such that for all differentiably good open covers U of a Cartesian space U the induced morphism ≃ A(U ) → [CartSpop , sSet](U, A) → [CartSpop , sSet](Č(U), A) is a weak equivalence of Kan complexes. This is the descent condition or ∞-sheaf/∞-stack condition on A. Proof. By standard facts about left Bousfield localizations we have that the fibrant objects are the degreewise fibrant object such that the morphisms RHom(U, A) → RHom(Č(U), A) are weak equivalences of Kan complexes, where RHom denotes the right derived simplicial hom-complex in the global projective model structure. Since every representable U is cofibrant and since Č(U) is cofibrant by the above proposition, these hom-complexes are equivalent to the hom-complexes in [CartSpop , sSet] as indicated.  Finally we establish the equivalence of the localization at a coverage that we are using to the localization at the corresponding Grothendieck topology, which is the one commonly found discussed in the literature. Theorem A.1. Let C be any small category equipped with a coverage given by covering families {Ui → U }. Then the ∞-topos presented by the left Bousfield localization of [C op , CartSp]proj at the coverage covering families is equivalent to that presented by the left Bousfield localization at the covers for the corresponding Grothendieck topology. We prove this for the injective model structure on simplicial presheaves. The result then follows since that is Quillen equivalent to the projective one and so presents the same ∞-topos. Write S(U) → j(U ) for the sieve corresponding to a covering family, regarded as a subfunctor of the representable functor j(U ) (the Yoneda embedding of U ), which we both regard as simplicially discrete objects in [C op , sSet]. Write [C op , sSet]inj,cov for the left Bousfield localization of the injective model structure at the morphisms S(U) → j(U ) corresponding to covering families. ∞-Chern-Weil theory 85 July 11, 2011 Lemma A.1. A subfunctor inclusion S̃ ֒→ j(U ) corresponding to a sieve that contains a covering sieve S(U) is a weak equivalence in [C op , sSet]inj,cov Proof. Let J be the set of morphisms in the bigger sieve that are not in the smaller sieve. By assumption we can find for each j ∈ J a covering family {Vj,k → Vj } such that for all j, i the diagrams Vj,k  Vj / Ui f  /U commute. Consider then the commuting diagram `  / S({Ui } ∪ {Vj,i }) j S({Vj,k }) . ≀ `  j j(Vj )  / S({Ui } ∪ {Vj }) = S̃ Observe that this is a pushout in [C op , sSet], that the top morphism is a cofibration in [C op , sSet]inj and hence in [C op , sSet]inj,cov , that the left morphism a weak equivalence in the local structure and that by general properties of left Bousfield localization the localization is left proper. Therefore the pushout morphism S({Ui } ∪ {Vj,k }) → S({Ui } ∪ {Vj }) = S̃ is a weak equivalence. Then observe that from the horizontal morphisms of the above commuting diagrams that defined the covers {Vj,k → Vj } we have an induced morphism S({Ui } ∪ {Vj,k }) → S({Ui }) that exhibit S({Ui }) as a retract S({Ui })  S̃ / S({Ui } ∪ {Vj,k }) =  / S̃ / S({Ui }) . =  / S̃ By closure of weak equivalences under retracts, this shows that the inclusion S({Ui }) → S̃ is a weak equivalence. By 2-out-of-3 this finally means that S̃ ֒→ j(U ) is a weak equivalence.  Corollary A.3. For S({Ui }) → j(U ) a covering sieve, its pullback f ∗ S({Ui }) → j(V ) in [C, sSet] along any morphism j(f ) : j(V ) → j(U ) f ∗ S({Ui })  j(V ) is also a weak equivalence. / S({Ui }) j(f )  / j(U ) ∞-Chern-Weil theory 86 July 11, 2011 Lemma A.2. If S({Ui }) → j(U ) is the sieve of a covering family and S̃ ֒→ j(U ) is any sieve such that for every fi : Ui → U the pullback f ∗ S̃ is a weak equivalence, then S̃ → j(U ) becomes an isomorphism in the homotopy category. Proof. First notice that if fi∗ S̃ is a weak equivalence for every i, then the pullback of S̃ to any element of the sieve S({Ui }) is a weak equivalence. Use the Yoneda lemma to write S({Ui }) ≃ j(V ) . lim → V →Ui →U Then consider these objects in the ∞-category of ∞-presheaves that is presented by [C op , sSet]inj [Lur09]. Since that has universal colimits we have the pullback square ∼ / S̃ / lim fV∗ S̃ i∗ lim j(V ) → → i S({Ui }) ∼ / lim →  fV :V →Ui →U j(V )  / j(U ) (fV ) and the left vertical morphism is a colimit over morphisms that are weak equivalences in [C op , sSet]inj,loc . By the general properties of reflective sub-∞categories this means that the total left vertical morphism becomes an isomorphism in the homotopy category of [C op , sSet]inj,cov . Also the bottom morphism is an isomorphism there, and hence the right vertical one is.  Proof of the theorem. 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