TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
arXiv:0910.4001v2 [math.AT] 7 Jan 2012
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Abstract. In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly
cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of,
differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spinstructures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show
that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology,
and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology
as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly
cancellation problems in higher gauge theories arising in string theory. We demonstrate that the GreenSchwarz mechanism for the H3 -field, as well as its magnetic dual version for the H7 -field define cocycles
in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-,
String(n)- and Fivebrane(n)- structures on target space, where the twist in each case is provided by the
obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U (n) or
O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian)
L∞ -algebra valued differential form data provided by the differential refinements of these twisted cocycles.
Contents
1. Introduction
2. Twisted topological structures in String theory
2.1. Twisted Spinc structures and physical applications
2.2. Twisted String structures and physical applications
2.3. Twisted Fivebrane structures and physical applications
3. Twisted differential structures in String theory
3.1. Differential twisted cohomology
3.2. Twisted string(n) 2-connections
3.3. Twisted fivebrane(n) 6-connections
Appendix A. L∞ -algebraic notions
A.1. L∞ -Algebras and L∞ -Algebroids
A.2. L∞ -algebra representations and section
References
1
4
5
8
13
19
19
25
29
33
33
35
37
1. Introduction
String theory and M-theory involve various higher gauge-fields, which are locally given by differential form
fields of higher degree and which are globally modeled by higher bundles with connection (higher gerbes with
connection, higher differential characters) [22][49]. Some of these entities arise in terms of lifts through various
connected covers of orthogonal or unitary groups. For example, an orientation of a Riemannian manifold M
can be given by a lifting of the classifying map riem : M → BO for the tangent or frame bundle of M to
a map or : M → BSO. In turn, a Spin structure on M can be given by a further lifting sp : M → BSpin.
The existence of a Spin structure is an anomaly cancellation condition for fermionic particles propagating
on M . The spaces BO, BSO and BSpin are the first steps in the Whitehead tower of BO. The next step
above BSpin is known as BString, with String the topological group known as the String group.
1
2
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Originally Killingback [33] defined a String structure on M as a lift of the transgressed map LM → BLSpin
on loop space through the Kac-Moody central extension B L̂Spin(n). The existence of such a lift cancels an
anomaly of the heterotic superstring on M in the case that the gauge bundle is trivial. Later it was realized
that this is captured down on M by a lift of sp : M → BSpin(n) to str : M → BString(n) [56]. A further
lift f iv : M → BFivebrane(n) through the next step in the Whitehead tower of BO(n) is similarly related
to anomaly cancellation for the physical fivebrane on M and accordingly the corresponding space is called
BFivebrane(n) [47].
BFivebrane(n)
✂@
✂
✂
✂
✂ BString(n)
✂
②<
✂
②
✂
②
②
✂
✂ ②②
BSpin(n)
✂ ②
8
✂
qq
q
✂ ②②
q
✂ ②
qq
✂ ②
q
BSO(n)
✂ ② qq
3
❤
❤
✂ ② q
❤ ❤
✂② ②q q
❤ ❤
❤
✂② q
❤ ❤
✂q
② q❤ ❤ ❤
❤
/ BO(n)
X
target space
Fivebrane structure
String structure
Spin structure
Orientation
Riemannian
Whitehead tower of BO(n)
Figure 1. Topological structures generalizing Spin(n) structure. In application to effective
background field theories appearing in string theory, these bare structures are twisted and
moreover refined to differential structures.
While anomalies are canceled by these lifts of maps of topological spaces, the dynamics of these systems
is controlled by smooth refinements of such maps. This is well understood for the first steps: the topological
groups O(n), SO(n) and Spin(n) naturally carry Lie group structures and the differential refinement of
X → BSpin(n) is well known to be given by a differential nonabelian Spin(n)-cocycle, namely a smooth
Spin(n)-principal bundle with connection.
However, the higher connected topological groups String(n) and Fivebrane(n) cannot be finite-dimensional
Lie groups and a smooth infinite-dimensional structure for Fivebrane is not known. Moreover, even when
such infinite-dimensional Lie group structures on these higher covers exist (as was recently found for the
String-group [39]) by themselves they lead to the wrong smooth cohomological refinement (discussed in
section 4.1 of [49]). However, String(n) does have a natural incarnation as a smooth 2-group [4] [27] [5] [3]
[6] [48][39], a higher categorical version of a Lie group (see [37] for all general matters of higher category
theory needed here and [49] for smooth higher geometry). Similarly, Fivebrane(n) does naturally exist as
a smooth 6-group (see section 4.1 of [49]). Generally, there are smooth ∞-group-refinements of all higher
connected covers of Lie groups. Being smooth, these spaces have infinitesimal approximations by L∞ algebras in generalization of how any ordinary Lie group has a Lie algebra associated with it. Therefore,
after passing to the smooth ∞-groupoid incarnation of the objects in the Whitehead tower of BO there is
a chance of obtaining differential refinements of String(n)- and Fivebrane(n)-structures (and beyond) that
are expressed in terms of higher smooth bundles with smooth L∞ -algebra-valued connection forms on them,
and indeed these structures naturally exist [46][21].
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
3
The general refinement of cohomology classes to differential cohomology classes for the case of abelian
(Eilenberg-Steenrod-type) generalized cohomology theories has been discussed by Hopkins and Singer [28]
and shown by Freed [22] to encode various differential (and twisted) structures in String theory. However, the
cohomological structures that appear in the Freed-Witten [24] and in the Green-Schwarz anomaly cancellation
mechanism [26], as well as in the magnetic dual Green-Schwarz mechanism [47] themselves originate from
–and are controlled by– nonabelian structures. These are the O(n)-principal bundle underlying the tangent
bundle of spacetime and the U(n)-principal bundle underlying the gauge bundle on spacetime, as well as
their lifts to the higher connected structure groups: a map X → BSpin(n) (smooth or not) gives a cocycle
in nonabelian cohomology, and so do its lifts such as X → BString(n) (and beyond).
Therefore, here we provide applications for the theory of (twisted) differential nonabelian cohomology,
that builds on [5] [51] [52] [52] [46] and is discussed in more detail in [49]. We show that the Freed-Witten and
the Green-Schwarz mechanisms, as well as the magnetic dual Green-Schwarz mechanism, define differential
twisted nonabelian cocycles that may be interpreted as differential twisted Spinc -, String- and Fivebranestructures, respectively. Equivalent anomalies differ by a coboundary, so that they are given by cohomologous
nontrivial cocycles. Equivalence classes of anomalies are captured by the relevant cohomology. We thus have
a refinement of the treatment in [47] to the twisted, smooth and differential cases.
In particular, the various abelian background fields appearing in the theory, such as the B-field and the
supergravity 3-form field, are unified into a natural coherent structure with the nonabelian background
fields – the spin- and gauge-connections – with which they interact. For instance, the relations between the
abelian and the nonabelian differential forms that govern the Green-Schwarz mechanism [26] are realized
here as a (twisted) Bianchi identity of a single nonabelian L∞ -algebra valued connection on a smooth twisted
String(n)-principal 2-bundle (cf. [46]). The explicit derivation of the twisted Bianchi identites of L∞ -algebra
connections corresponding to the Green-Schwarz mechanism and its magnetic dual is in section 3, with more
details in section 6.2 of [21].
Summary. In this paper we achieve the following goals:
(1)
(2)
(3)
(4)
generalize a Fiverbrane structure to the twisted case, and similarly for a twisted String structure;
provide differential cohomology versions of these twisted structures;
provide a description of the Green-Schwarz anomaly cancellation and its dual using these structures;
describe the M-theory C-field and its dual in this context.
The first two are purely mathematical results that are of independent interest in developing higher (algebraic,
geometric, topological, categorical) phenomena [21]. The third and fourth are applications to (heterotic)
string theory and to M-theory, respectively. We hope this will add to a better understanding of structures
appearing in these theories, which in turn is hoped to result in identification of yet more rich mathematical
structures within them (see [43] [44] [45] for concrete examples). From a mathematical point of view, they
serve as interesting concrete examples of the formalism we have developed in the first two points.
Section 2 discusses how the anomaly cancellation mechanisms in String-theory can be understood topologically in terms of twisted higher structures given by twisted nonabelian topological cocycles. The physical
examples of most relevance here arise in various anomaly cancellations in string theory. The Freed-Witten
condition [24] in type IIA string theory says that the third integral Stiefel-Whitney class W3 of a D-brane Q
has to be trivial relative to the Neveu-Schwarz field H3 |Q restricted to the D-brane, in that the two classes
agree: W3 = [H3 |Q ].
Higher versions of this example –within our point of view– are the Green-Schwarz mechanism and its
magnetic dual version. Recall the notion of String structures, e.g. from [47], as maps from a space X
to BString(n), the 3-connected cover of BSpin(n). In [61] the notion of twist for a String structure was
considered: a space X can have a twisted String structure without having a String structure, i.e. the fractional
Pontrjagin class 21 p1 (T X) of the tangent bundle can be nonzero while the modified class is 12 p1 (T X)+[β] = 0,
where β : X → K(Z, 4) is a fixed twist for the String structure. The Green-Schwarz mechanism in string
4
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
ĝE
ĝT X
BString o❴ ❴ ❴ ❴ ❴ ❴ ❴ X ❋❴ ❴ ❴ ❴ ❴ ❴ ❴/ BU h8i
❋
✇
❋❋
✇
❋❋
✇✇
❋❋
✇✇
✇
❋❋
✇
✇✇
ks H3 +3
gT X
gE❋
❋❋
✇
1
1
❋❋
✇✇
c
(E)
p
(T
X)
✇
2
1
6
2
❋❋
✇
❋❋
✇✇
✇
1
1
#
{✇
p
c
2
2 1
6
/ B 3 U (1) o
BSpin
BU h6i
|
|
{z
}
abelian cohomology
{z
nonabelian cohomology
}
Figure 2. Abelian versus nonabelian cohomology. Since the groups String(n) as
well as Fivebrane(n) are shifted central extension of nonabelian groups, cohomology with
coefficients in these groups has abelian components but also components in nonabelian
cohomology [58]. This appears as abelian cohomology twisted by nonabelian cocycles in a
certain way. The Green-Schwarz mechanism implies that two classes in ordinary abelian
cohomology, namely in degree four differential integral cohomology, coincide. But these
classes are particularly obstruction classes to String-lifts in nonabelian cohomology. The
middle part of Figure 1 , labeled “abelian cohomology”, identifies the cocycle representative
in H 4 (X, Z) and the coboundary between them, but does not specify where these cocycles
come from. The outer part of the diagram, labeled “nonabelian cohomology” does specify
the object whose class is the one identified by the middle part. We can interpret this in
ordinary homotopy theory, where it describes topological obstruction theory, but we can
also interpret this after differential refinement in the ∞-topos [37] of smooth ∞-groupoids
[49]. In any case the morphisms in the above diagram may be interpreted as cocycles. The
smooth and differential refinement we discuss in section 3.
theory may be understood as defining a twisted String structure on target space, with twist given in terms
of a classifying map for the gauge bundle.
Since a String structure is refined by a Fivebrane structure in analogy to how a String structure itself
refines a Spin structure, it is natural to consider twists of Fivebrane structures in the above sense. In this
paper we give a definition of twisted Fivebrane structures and show that the dual Green-Schwarz mechanism
in heterotic String theory, reviewed and formalized in detail in [47], provides an example. Hence, variations
on the twisted Fivebrane condition do in fact appear in string theory and in M-theory and correspond, as
we will see, to anomaly cancellation conditions for the heterotic fivebrane [18] [36] and for the M-fivebrane
[64] [66] [23] [16], respectively. We discuss these two cases in section 2.2.1 and section 2.3.1 in terms of
topological cocycles (maps to the appropriate classifying spaces) and describe their differential refinements
in section 3.2 and 3.3.
Some time has passed between the original inception of the discussion presented in this article and the
present form. The differential structures discussed here have motivated us with collaborators to further
expand the general theory of higher smooth stacks and the formulation of differential cohomology in terms
of these. In section 3 we survey this context and refer to various technical results obtained meanwhile.
2. Twisted topological structures in String theory
We discuss here cohomological conditions arising from anomaly cancellation in String theory, for various
σ-models. In each case, we introduce a corresponding notion of topological twisted structures in terms of
which we interpret the corresponding anomaly cancellation condition. This prepares the ground for the
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
5
material in section 3, where the differential refinement of these twisted structures is considered leading to
the derivation of the differential anomaly-free field configurations.
The physics of all the cases we consider involves a manifold X –the target space– or a submanifold Q ֒→ X
thereof –a brane– equipped with
• two principal bundles with their canonically associated vector bundles:
– a Spin-principal bundle underlying the tangent bundle T X (and we will write T X also to denote
that Spin-principal bundle),
– and a complex vector bundle E → X – the “gauge bundle” – associated to a SU (n)-principal
bundle or to an E8 -principal bundle with respect to a unitary representation of E8 ;
• an n-gerbe / circle (n + 1)-bundle with class H n+2 (X, Z) representing the higher background gauge
field and denoted [Hi ] or [Gi ] or similar in the following.
All these structures are equipped with a suitable notion of connetions, locally given by some differential-form
data. The connection on the Spin-bundle encodes the field of gravity, that on the gauge bundle a Yang-Mills
field and that on the n-gerbe a higher analog of the electromagnetic field.
The σ-model quantum field theory of a super-brane propagating in such a background (for instance the
superstring, or the super 5-brane) has an effective action functional on its bosonic worldvolume fields that
takes values, in general, in the fibers of the Pfaffian line bundle of a worldvolume Dirac operator, tensored
with a line bundle that encodes the electric and magnetic charges of the higher gauge field. Only if this
tensor product anomaly line bundle is trivializable is the effective bosonic action a well-defined starting
point for quantization of the σ-model. Therefore, the Chern class of this line bundle over the bosonic
configuration space is called the global anomaly of the system. Conditions on the background gauge fields
that ensure that this class vanishes are called global anomaly cancellation conditions. These turn out to be
conditions on cohomology classes that are characteristic of the above background fields. This is what we
discuss in this section. Moreover, the anomaly line bundle is canonicaly equipped with a connection, induced
from the connections of the background gauge fields, hence induced from their differential cohomology data.
The curvature 2-form of this connection over the bosonic configuration space is called the local anomaly
of the σ-model. Conditions on the differential data of the background gauge field that canonically induce
a trivialization of this 2-fom are called local anomaly cancellation conditions. We consider these below in
section 3.
The phenomenon of anomaly line bundles of σ-models induced from background field differential cohomology is classical in the physics literature, albeit in broad terms. A clear exposition is in [22]. Only recently
has the special case of the heterotic string σ-model for trivial background gauge bundle been made fully
precise in [13], using a certain model [60] for the differential string structures that we discuss in section 3.
2.1. Twisted Spinc structures and physical applications. As a preparation for the twisted String- and
Fivebrane structures discussed in the following sections, we consider twisted Spinc -structures and their role
in anomaly cancellation for the open type II string.
2.1.1. Type II superstring on D-branes. The open type II string propagating on a Spin-manifold X in the
presence of a) a background B-field with class [H3 ] ∈ H 3 (X, Z) and b) with endpoints fixed on a D-brane
given by an oriented submanifold Q ֒→ X, has a global worldsheet anomaly that vanishes if [24] the condition
(2.1)
W3 (Q) + [H3 ]|Q = 0 ∈ H 3 (Q; Z) ,
holds. Here W3 (Q) is the third integral Stiefel-Whitney class of the tangent bundle T Q of the brane and
[H3 ]Q denotes the restriction of [H3 ] to Q. Sufficiency of the condition is discussed in [20].
6
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Notice that W3 (Q) is the obstruction to lifting the orientation structure on Q to a Spinc structure. We
can formulate this in terms of homotopy theory as follows. There is a homotopy pullback diagram
BSpinc
/∗
⌋⌋
(2.2)
BSO
W3
/ B 2 U (1)
of topological spaces, where BSO is the classifying space of the (stable) special orthogonal group, B 2 U (1) ≃
K(Z, 3) is the Eilenberg-MacLane space that classifies degree-3 integral cohomology, and the continuous map
denoted W3 is a representative of the universal class W3 under this classification.
Homotopy pullbacks and their properties play a central role in all of our discussion here, first in the
category of topological spaces, and then later, in section 3, in categories of higher stacks. The reader
unfamiliar with the basics of abstract homotopy theory might consult the review in section A.2 of [37]. A
basic fact is that a homotopy fiber of connected spaces as in (2.2), where the right vertical morphism is the
point inclusion, may be computed, up to weak homotopy equivalence, as an ordinary pullback (an ordinary
fiber product) after replacing the point inclusion by the path fibration. Specifically, for X any pointed
topological space, write P X for the space of continuous paths in X that end at the base point x ∈ X, and
write P X → X for the projection to the other endpoint of the path. Then the ordinary pullback
(2.3)
Ωx X
/ PX
⌋
∗
x
/X
is a model for the loop space of X. Moreover, for φ : Y → X any map, the ordinary pullback
(2.4)
hofib(φ)
/ PX
⌋
Y
/X
φ
is a model for the homotopy fiber hofib(φ) of φ.
The homotopy pullback (2.2) exhibits the classifying space of the group Spinc as the homotopy fiber of
W3 . The universal property of the homotopy pullback says that the space of continuous maps Q → BSpinc
is the same (is weak homotopy equivalent to) the space of maps oQ : Q → BSO that are equipped with
o
Q
/ BSO W3 / B 3 U (1) to the trivial cocycle Q → ∗ → B 3 U (1). In
a homotopy from the composite Q
other words, for every choice of homotopy filling of the outer diagram
(2.5)
Q❋
❋
❋
❋
❋"
BSpinc
"/
∗
oQ
#
BSO
W3
/ B 2 U (1)
there is a contractible space of choices for the dashed arrow such that everything commutes up to homotopy.
Since a choice of map oQ : Q → BSO is an orienation structure on Q, and a choice of map Q → BSpinc
is a Spinc structure, this implies that W3 (oQ ) is the obstruction to the existence of a Spinc structure on
Q (equipped with oQ ). Moreover, since Q is a manifold (hence a CW-complex), the functor Maps(Q, −)
that forms mapping spaces out of Q preserves homotopy pullbacks. Since Maps(Q, BSO) is the space of
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
7
orientation structures, we can refine the discussion so far by noticing that the space of Spinc structures on
Q, denoted Maps(Q, BSpinc ), is itself the homotopy pullback in the diagram
Maps(Q, BSpinc )
/∗
⌋⌋
(2.6)
.
Maps(Q, BSO)
Maps(Q,W3 )
/ Maps(Q, B 2 U (1))
A variant of this characterization will be crucial for the definition of (spaces of) twisted such structures
below.
These kinds of arguments, although elementary in homotopy theory, are of importance for the interpretation of anomaly cancellation conditions that we consider here. Variants of these arguments (first for other
topological structures, then with twists, then refined to smooth and differential structures) will appear over
and over again in our discussion.
In the case that the class of the B-field vanishes on the D-brane, [H3 ]|Q = 0, hence that its representative
H3 : Q → K(Z, 3) factors through the point up to homotopy, condition (2.1) states that the oriented D-brane
Q must admit a Spinc structure, namely a choice of null-homotopy η in 1
oQ
(2.7)
/ BSO
Q ❖❖
❖❖❖
✇
✇
✇
✇
✇
❖❖❖ w ✇η
❖❖
W3 .
H3 |Q ≃∗ ❖❖❖'
/ K(Z, 3)
X
H3
If, more generally, [H3 ]|Q does not necessarily vanish, then condition (2.1) still is equivalent to the existence
of a homotopy η in a diagram of the above form:
oQ
(2.8)
/ BSO
Q ❖❖
❖❖❖
✇
❖❖❖ w ✇✇✇✇η✇
❖❖
W3 .
H3 |Q ❖❖❖'
/ K(Z, 3)
X
H3
We may think of this as saying that η still “trivializes” W3 (oQ ), but not with respect to the canonical
trivial cocycle, but with respect to the given reference background cocycle H3 |Q of the B-field. Accordingly,
following [61], we may say that such an η exhibits not a Spinc -structure on Q, but an [H3 ]Q -twisted Spinc
structure.
For this notion to be useful, we need to say what an equivalence or homotopy between two twisted Spinc
structures is, what a homotopy between such homotopies is, etc., and hence what the space of twisted Spinc
structures is. However, by generalization of (2.6) we naturally do have such a space.
Definition 2.1. For X a manifold and [c] ∈ H 3 (X, Z) a degree-3 cohomology class, we say that the space
W3 Struc(Q)[c] defined as the homotopy pullback
W3 Struc(Q)[H3 ]|Q
(2.9)
⌋⌋
/∗
c
Maps(Q, BSO)
Maps(Q,W3 )
,
/ Maps(Q, B 2 U (1))
is the space of [c]-twisted Spinc structures on X, where the right vertical morphism picks any representative
c : X → B 2 U (1) ≃ K(Z, 3) of [c].
1Beware that there are homotopies filling all our diagrams, but only in some cases, such as here, do we want to make them
explicit and give them a name.
8
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
In terms of this notion, the anomaly cancellation condition (2.1) is now read as being a requirement of
existence of structure:
Observation 2.2. On an oriented manifold Q, condition (2.1) implies the existence of [H3 ]|Q -twisted W3
structure, provided by a lift of the orientation structure oQ on T Q through the left vertical morphism in def.
2.1.
This makes good sense, because that extra structure is the extra structure of the background field of the
σ-model background, subjected to the condition of anomaly freedom. We will see this in more detail in the
following examples, and then again in section 3.
2.2. Twisted String structures and physical applications. We discuss twisted String structures and
their role in anomaly cancellation of
(1) the heterotic string;
(2) M-theory in the bulk;
(3) the boundary in M-theory.
2.2.1. The heterotic string. The heterotic/type I string, propagating on a Spin-manifold X and coupled to
a gauge field given by a Hermitean complex vector bundle E → X, has a global anomaly that vanishes if
the Green-Schwarz anomaly cancellation condition [26]
(2.10)
1
p1 (T X) − ch2 (E) = 0 ∈ H 4 (X; Z)
2
holds. Here ch2 (E) is the second Chern character of E which reduces to the second Chern class c2 (E) in the
cases we consider, and 12 p1 is given by the following classical fact (see [9]).
Fact 2.3. The first Pontrjagin class p1 ∈ H 4 (BSO, Z) becomes divisible precisely by 2 when pulled back to
BSpin. The corresponding preimage under multiplication by two, denoted 12 p1 , is a generator of the group
H 4 (BSpin, Z) ∼
= Z.
As in the Spinc case discussed above, this means that at the level of cocycles a certain homotopy exists.
Here it is this homotopy which is the representative of the B-field to which the string couples. In detail, write
1
3
2 p1 : BSpin → B U (1) for a representative of the universal first fractional Pontrjagin class, and similarly
3
c2 : BSU → B U (1) for a representative of the universal second Chern class, where now B 3 U (1) ≃ K(Z, 4) is
equivalent to the Eilenberg-MacLane space that classifies degree-4 integral cohomology. Then if T X : X →
BSpin is a classifying map of the Spin-bundle and E : X → BSU is one of the gauge bundles, the above
anomaly cancellation condition says that there is a homotopy, denoted H3 , in the diagram
(2.11)
E
/ BSU
✂✂
✂
✂
✂✂✂✂
.
c2
TX
✂✂✂✂
✂
✂
|
BSpin 1 / B 3 U (1)
X
H3
2 p1
Notice that if both 12 p1 (T X) as well as c2 (E) happen to be trivial, such a homotopy is equivalently a map
H3 : X → ΩB 3 U (1) ≃ B 2 U (1). So in this special case the B-field in the background of the heterotic string is
a U (1)-gerbe, or a circle 2-bundle, as in the previous case of the type II string in section 2.1. Generally, the
homotopy H3 in the above diagram exhibits the B-field as a twisted gerbe, whose twist is the difference class
1
2 p1 (T X) − c2 (E). This is essentially the perspective adopted in [22] (in a somewhat different language).
For the general discussion of interest here it is useful to slightly shift the perspective on the twist. To
that end, first consider, by analogy with (2.2), the following definition.
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
9
Definition 2.4. The String group is the loop space of the homotopy fiber BString of the universal class
1
2 p1 from Proposition 2.3:
BString
/∗
⌋⌋
.
BSpin
1
2 p1
/ B 3 U (1)
For more details on this and a collection of references see section 4.1 of [49]. Accordingly, one says that a
String structure on the Spin bundle T X : X → BSpin is a homotopy filling the outer square of the diagram
X❍
(2.12)
❍
❍
❍
❍#
BString
,
/# ∗
TX
$
BSpin
1
2 p1
/ B 3 U (1)
or, equivalently– by the universal property of homotopy pullbacks– a choice of dashed morphism filling
the interior of this square, as indicated. 2 Therefore, now by analogy with (2.8), we say that a c2 (E)twisted String structure is a choice of homotopy H3 filling the diagram (2.11). This notion of twisted String
structures was originally suggested in [61]. For it to be useful, we need to say what homotopies of twisted
String structures are, homotopies between these, etc. Hence we need to say what the space of twisted String
structures is. This is what the following definition provides, analogously to Definition 2.1.
Definition 2.5. For X a manifold, and for [c] ∈ H 4 (X, Z) a degree-4 cohomology class, we say that the
space of c-twisted String structures on X is the homotopy pullback 12 p1 Struc[c] (X) in the diagram
1
2 p1 Struc[c] (X) ⌋⌋
/∗
(2.13)
c
Maps(X, BSpin)
Maps(X, 21 p1 )
,
/ Maps(X, B 3 U (1))
where the right vertical morphism picks a representative c of [c].
In terms of this then, we find
Observation 2.6. The anomaly cancellation condition (2.10) is, for a fixed gauge bundle E, precisely the
condition that ensures a lift of the given Spin structure to a c2 (E)-twisted String structure on X, through
the left vertical morphism of def. 2.5.
Of course the full background field content involves more than just this topological data; it also consists
of local differential form data, such as a 1-form connection on the bundles E and on T X and a connection
2-form on the 2-bundle whose curvature is H3 . Below, in section 3, we identify this differential anomaly-free
field content with a differential twisted String structure.
2 Originally, this condition was considered in its weaker incarnation after transgression to loop space [33].
10
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
The 10-dimensional string theory backgrounds discussed so far are supposed to be part of a bigger picture,
which schematically looks as follows
UV-complete theory
low energy approximation /
effective field theory
(2.14)
ℓ1
M-theory
/ D = 11 Supergravity
∂2
∂1
Heterotic String
ℓ2
/ D = 10 Supergravity SYM .
Here the bottom and right structures and maps are fairly well-defined. The right vertical map denotes
the restriction of 11-dimensional supergravity on a manifold with boundary to 10-dimensional heterotic
supergravity on the boundary. The bottom left morphism indicates that heterotic supergravity is the effective
low energy field theory whose UV-completion is heterotic string theory. The top left entry denotes M-theory,
which is what meaningfully completes this schematic diagram. Accordingly, the 2-brane and 5-brane solutions
of 11-dimensional supergravity have incarnations as fundamental objects in M-theory, the M2-brane and the
M5-brane.
We discuss the topological anomalies for the M2-brane, first in the bulk theory, then after restriction to
the boundary, where it becomes the heterotic string of section 2.2.1. At the level of fields, this is the C-field
in the bulk becoming the B-field on the boundary.
2.2.2. M-theory in the bulk. The bosonic field content of 11-dimensional supergravity on a Spin 11-manifold
Y consists of a Spin-bundle T Y (with connection), as well as of the C-field, which has underlying it a 2-gerbe
– or circle 3-bundle – with class [G4 ] ∈ H 4 (Y, Z). The M2-brane that couples to these background fields has
an anomaly that vanishes if [63]
(2.15)
2[G4 ] =
1
p1 (T Y ) − 2a(E) ∈ H 4 (Y, Z) ,
2
where E → Y is an auxiliary E8 -principal bundle, whose class a(E) is defined by this condition. This is also
called the quantization condition for the C-field.
In the absence of smooth or differential structure, since E8 is 15-coskeletal, one could therefore replace
the E8 -bundle here by a U(1)-2-gerbe, hence by a B 2 U (1)-principal bundle, and replace condition (2.15) by
(2.16)
2[G4 ] =
1
p1 (T Y ) − 2DD2 ,
2
where DD2 is the canonical 4-class of this 2-gerbe (the “second Dixmier-Douady class”, hence the notation).
While topologically this condition is equivalent, over an 11-dimensional X, to (2.15), the spaces of solutions
of smooth refinements of these two conditions will differ, because the space of smooth gauge transformations
between E8 bundles is quite different from that of smooth gauge transformations between circle 2-bundles. In
the Hořava-Witten reduction [29] of the 11-dimensional theory down to the heterotic string in 10 dimensions,
this difference is supposed to be relevant, since the heterotic string in 10 dimensions sees the smooth E8 bundle with connection.
In either case, we can understand the situation as a refinement of that described by (twisted) Stringstructures, as above, via a higher analogue of the passage from Spin-structures to Spinc -structures, as in
section 2.1. To that end, notice the following fact, which provides an alternative perspective on (2.2), and
which uses the point of view on twisted structures advocated recently in [44] [45].
Proposition 2.7. The classifying space BSpinc is the homotopy fiber product of a representative of the
universal second Stiefel-Whitney class w2 ∈ H 2 (BSO, Z2 ) with a representative of the mod 2-reduction of
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
11
the universal first Chern class c1 ∈ H 2 (BU (1), Z)
BSpinc
/ BU (1)
⌋⌋
c1 mod 2
BSO
.
w2
/ B 2 Z2
This remains true after refinement to smooth and differential structures, as we discuss below in 3. Due
to the universal property of the homotopy pullback, this says, in particular, that a lift from an orientation
structure to a Spinc -structure is a cancelling by a Chern class of the class obstructing a Spin-structure.
In this way, lifts from orientation structures to Spinc -structures are analogous to the divisibility condition
(2.15), since in both cases the obstruction to a further lift through the Whitehead tower of the orthogonal
group is absorbed by a universal “unitary” class.
In higher analogy to this situation, and in the spirit of [44] [45], we therefore have the following definition.
Definition 2.8. For G some topological group, and α : BG → K(Z, 4) a universal 4-class, we say that
Stringα is the loop group of the homotopy pullback 3
BStringα
/ BG
⌋⌋
α
BSpin
1
2 p1
/ B 3 U (1)
of α along a representative of the first fractional Pontrjagin class 12 p1 ∈ H 4 (BSpin, Z).
Of relevance for the present purpose are the following cases. Notice that on the space B 2 U (1) ≃ K(Z, 3),
which classifies circle 2-bundles / U (1)-bundle gerbes, the canonical 3-class is traditionally called the DixmierDouady class, denoted DD. Accordingly, it makes sense to speak of the canonical 4-class on B 3 U (1) ≃
K(Z, 4), which classifies circle 3-bundle / U (1)-bundle 2-gerbes as the second Dixmier-Douady class DD2 .
We are interested in the case when the first Spin characteristic class λ = 21 p1 is divisible by 2, so that we can
‘divide by 2’ in equation (2.16). For α = DD2 we have that a Spin-structure lifts to a String2DD2 -structure
precisely if 21 p1 is further divisible by 2. Equivalently, a Spin-principal bundle P lifts to a String2DD2 principal bundle precisely if the corresponding Chern-Simons 2-gerbe 21 p1 (P ) is twice some other 2-gerbe.
Equivalently this says that, the fourth Stiefel-Whitney class vanishes, w4 = 0, because, as explained in [43]
[44], w4 is the mod 2 reduction of the integral first Spin characteristic class λ = 21 p1 . In summary, we have
for a given Spin-structure Y → BSpin the following diagram
1
4 p1 (X)
(2.17)
❴❘ ❴ ❴ ❴ ❴ ❴ ❴/ BString2DD2
Y ❘
❘❘❘
⌋⌋
❘❘❘
❘❘❘
❘❘❘
❘❘❘
)
(BO)h4i = BSpin
1
4 p1
+
/ K(Z, 4)
❘❘❘
❘❘❘ 0
❘❘❘
❘❘❘
❘(
/ K(Z, 4)
/ K(Z2 , 4)
×2
1
2 p1
=
(BO)h2i = BSO
w4
/ K(Z2 , 4),
where the dashed arrow is a lift of the given Spin-structure.
Observation 2.9. The class 41 p1 is the obstruction to lifting a String2DD2 bundle, to a String-bundle. Here,
String2DD2 is the loop space of the space BString2DD2 , defined above.
3We are using the notation Stringα to distinguish from the notion of Stringc structure (related to Spinc in the same way
that String is related to Spin) studied in [45]. The latter is a special case of the former, as explained in op. cit.
12
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Proof. The pasting law for homotopy pullback implies that the homotopy fiber of 14 p1 is BString.
BString
/∗
⌋⌋
0
BString2DD2
1
4 p1
⌋⌋
/ K(Z, 4) .
×2
BSpin
/ K(Z, 4)
This observation is analogous to proposition 2 in [47] for the Fivebrane case, where there we were considering
1
p2 to the obstruction to Fivebrane structure, given by 16 p2 . We will discuss this further
the comparison of 48
later in the paper.
Similarly, with a : BE8 → B 3 U (1) the canonical universal 4-class for E8 -bundles and X a manifold of
dimension dimX ≤ 14 we have that a Spin-structure on X lifts to a String2a -structure precisely if 21 p1 is
further divisible by 2.
(2.18)
2a
/ BE8 .
BString
▲▲▲ 1
✈;
▲▲▲4 p1
✈
▲▲▲
2a
✈
✈
▲%
1
✈
2 p1
/ BSpin
/ B 3 U (1)
X
Using this we can now reformulate the anomaly cancellation condition (2.15) as follows.
Definition 2.10. For X a manifold and for [α] ∈ H 4 (X, Z) a cohomology class, the space ( 12 p1 −2a)Struc[α] (X)
of [α]-twisted String2a -structures on X is the homotopy pullback
( 21 p1 −2a)Struc[α] (X)
/∗
⌋⌋
α
Maps(X, B(Spin × E8 ))
1
2 p1 −2a
,
/ Maps(X, B 3 U (1))
where the right vertical map picks a cocycle α representing the class [α].
This can be viewed as a “twist for the twisted String structure”. In terms of this definition, we have
Observation 2.11. Condition (2.15) is precisely the conditon guaranteeing a lift of the given Spin- and the
given E8 -principal bundle to a [G4 ]-twisted String2a -structure through the left vertical map from def. 2.10.
2.2.3. M-theory with boundary: Heterotic M-theory. In [29], Horava and Witten carefully analyzed the map
∂1 in diagram (2.14) and gave arguments on how it must extend to ∂2 . If we denote by Q := ∂Y ֒→ Y the
boundary inclusion, then the condition they find is a boundary condition on the C-field, saying that the
restriction of its 4-class to Q has to vanish,
(2.19)
[G4 ]|Q = 0 .
This implies that over Q the anomaly-cancellation condition (2.15) becomes
(2.20)
1
p1 (T Y )|Q = 2a(E)|Q ∈ H 4 (Q, Z) .
2
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
13
This is equivalent, in direct analogy with (2.8), to the existence of a homotopy of the form
/ BSpin
Q ❖❖
❖❖❖
ss
s
s
s
❖❖❖ u} s
1
.
❖❖
2 p1
2a(E)|Q ❖❖❖'
/ K(Z, 4)
X
(2.21)
2a(E)
Essentially in this form twisted string structures in the context of string theory were proposed in (the first
eprint version of) [61]. We see here a general pattern of twisted structures occuring as relative trivializations
on branes.
Notice that on Q this is the Green-Schwarz anomaly cancellation condition (2.10) of the heterotic string,
but refined by a further cohomological divisibility condition. The following statement says that this may
equivalently be reformulated in terms of String2a structures.
Proposition 2.12. For E → Y a fixed E8 -bundle, we have an equivalence
1
Maps(Y, BString2a )|E ≃ ( p1 )Struc(Y )[2a(E)]
2
between, on the right, the space of 2a(E)-twisted String-structures from def. 2.5, and, on the left, the space of
String2a -structures with fixed class 2a, hence the homotopy pullback Maps(Y, BString2a ) ×Maps(Y,BE8 ) {E}.
Proof. Consider the diagram
Maps(Y, String2a )|E
⌋⌋
/∗
E
Maps(Y, String2a )
/ Maps(Y, BE8 )
⌋⌋
Maps(X,2a)
Maps(Y, BSpin)
Maps(Y, 21 p1 )
/ Maps(Y, B 3 U (1))
The top square is a homotopy pullback by definition. Since Maps(Y, −) preserves homotopy pullbacks (for Y
a manifold, hence a CW-complex), the bottom square is a homotopy pullback by definition 2.8. Therefore,
by the pasting law, also the total rectangle is a homotopy pullback. With def. 2.5 this implies the claim.
Therefore the boundary anomaly cancellation condition for the M2-brane has the following equivalent
formulation.
Observation 2.13. For X a Spin-manifold equipped with a complex vector bundle E → Y , condition (2.20)
precisely guarantees the existence of a lift to a String2a -structure through the left vertical map in the proof
of prop. 2.12.
2.3. Twisted Fivebrane structures and physical applications. We discuss twisted fivebrane structures
and their role in anomaly cancellation in
(1) the NS-5-brane and dual heterotic string theory;
(2) the M5-brane.
2.3.1. The NS-5-brane. The magnetic dual of the (heterotic) string is the NS-5-brane. Where the string is
electrically charged under the B2 -field with class [H3 ] ∈ H 3 (X, Z), the NS-5-brane is electrically charged
under the B6 -field with class [H7 ] ∈ H 7 (X, Z) [14]. As we discuss in detail shortly, in the presence of a
String-structure, hence when 21 p1 (T X) = 0, the anomaly of the 5-brane σ-model vanishes if the background
fields satisfy
1
(2.22)
p2 (T X) = 8 ch4 (E) ∈ H 8 (X, Q) ,
6
14
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
where E is an E8 × E8 - or Spin(32)/Z2 -principal bundle, and ch(E) denotes its Chern character, and where
1
6 p2 denotes the second fractional Pontrjagin class given by the following classical fact (see [9]).
Fact 2.14. The second Pontrjagin class p2 ∈ H 8 (BSO, Z) becomes divisible precisely by 6 when pulled back
to BString. The corresponding preimage under multiplication by six, denoted 16 p2 , is a generator of the group
H 8 (BSpin, Z) ∼
= Z.
It is clear now that a discussion entirely analogous to that of section 2.2.1 applies. For the untwisted case
the following terminology was introduced in [47].
Definition 2.15. Write Fivebrane for the loop group of the homotopy fiber BFivebrane of a representative
1
6 p2 of the universal second fractional Pontrjagin class
BFivebrane
/∗
⌋⌋
.
BString
1
6 p2
/ B 7 U (1)
In direct analogy with def. 2.5 we therefore have the following notion.
Definition 2.16. For X a manifold and [c] ∈ H 8 (X, Z) a class, we say that the space of [c]-twisted Fivebranestructures on X, denoted ( 16 p2 )Struc[α] (X), is the homotopy pullback
( 61 p2 )Struc[c] (X)
/∗
⌋⌋
c
Maps(X, BString)
Maps(X, 61 p2 )
,
/ Maps(X, B 7 U (1))
Explicitly, a [c]-twisted Fivebrane structure on a brane ι : Q → X equipped with String structure
f : Q → BString is a homotopy η in a diagram analogous to (2.21)
(2.23)
/ BString
Q PP
PPP
✇
✇
✇
✇
✇
PPP w ✇η
1
.
PP
6 p2
c|Q PPP'
/ K(Z, 8)
X
c
Two [c]-twisted Fivebrane structures η and η ′ on Q are regarded as equivalent if there is a homotopy between
η and η ′ . In the case that [c] = 0 this reduces to the untwisted Fivebrane structures considered in [47].
In terms of these notions we now have
Observation 2.17. For X a manifold with String-structure and with a background gauge bundle E → X
fixed such that 8ch(E) is integral, condition (2.22) is precisely the condition for the existence of 8 ch(E)twisted Fivebrane-structure on X.
We now consider the above anomaly cancellation condition in more detail.
In [47] the main example of a Fivebrane structure came from the dual formulation [42] [25] of the GreenSchwarz anomaly cancellation mechanism [26], using the dual H-field H7 of [14]. The expression is given
by
1
1
1
2
(2.24)
dH7 = 2π ch4 (FA ) − p1 (Fω )ch2 (FA ) + p1 (Fω ) − p2 (Fω ) ,
48
64
48
where FA and Fω are the curvatures of the connections A and ω on the gauge bundle E and the tangent (or
Spin) bundle of the ten-manifold M , respectively. In order to define a Fivebrane structure, we assume we
already have a String structure, so we require 21 p1 (T M ) = 0. Then the expression (2.24) becomes
1
(2.25)
dH7 = 2π ch4 (FA ) − p2 (Fω ) .
48
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
15
In [47] we had to find ways to get rid of the extra terms to isolate the non-decomposable terms. In the
twisted formalism in this paper we see that the presence of such terms amounts to a piece of the twist and
that it does not matter how many terms we have, as long as they are integral and have the same total degree
and hence provide a map to K(Z, 8). Indeed, if we can define
(2.26)
[β] := −ch4 (E) : M
!
/ K(Z, 8) ,
i.e. require factorization
(2.27)
M
[β]
then we can reinterpret expression (2.25) as
an exact form.
/ K(Q, 8) ,
s9
s
s
s
s
s
s
+ sss
#
K(Z, 8)
1
48 p2 (T M )
+ [β] = 0, since [dH7 ] = 0, the cohomology class of
We discuss the validity of the map in (2.26). The Chern character is in general not an integral expression,
but rather
ch : K 0 (X) → H even(X; Q).
(2.28)
One way out of this is to first define a rational version of the twist, for which the map in (2.26) is replaced
by a map from M to the rational Eilenberg-MacLane space
(2.29)
[β] := −ch4 (E) : M → K(Q, 8),
which gives that indeed ch4 (E) is in general in [M, K(Q, 8)] = H 8 (M, Q). Hence
Definition 2.18. A rational Fivebrane twist on M is a map from M to K(Q, 8), i.e. an element of H 8 (M ; Q).
However, we can also give conditions under which the map in (2.26) is valid. The degree four Chern
character is given by
1 4
c1 − 4c21 c2 + 4c1 c3 + 2c22 − 4c4 .
(2.30)
ch4 =
24
The Chern classes are integral classes and so the Chern character is a priori integral up to a factor of 24.
We describe this as follows. The Chern character is not integral in BU but it will be integral in some lift,
say BU, of BU . Then we ask: when can we lift to this new space? This is given in terms of the following
diagram
K(Z24 , 7)
(2.31)
f
M
8 BU ⌋⌋
ch4
/ K(Z, 8)
24ch4
/ K(Z, 8)
×24
/ BU
/ K(Z24 , 8) .
The right-most factor K(Z24 , 8) represents the obstruction: there is a class k in H 8 (M ; Z24 ) which measures
this obstruction. The top-most factor K(Z24 , 7) represents the different labeling of lifts f to the new space
BU. If we take connected covers of BU rather than BU itself in the diagram, then we have that the space BU
is isomorphic to another space in which 16 c4 , instead of ch4 , is integral. The relevance of the unitary groups
here is because they provide the adjoint representation for our structure groups and this is the representation
relevant for Yang-Mills theory. For E8 , the adjoint representation is ad : E8 → SU (248), so that the adjoint
representation of G = E8 × E8 is (ad, ad) : E8 × E8 → SU (248) × SU (248) ֒→ SU (496).
16
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Note that the above general discussion can be simplified. For both structure groups E8 × E8 and
1
(c2 (E)2 − 2c4 (E)). We now conSpin(32)/Z2 we have c1 (E) = 0, so that for these groups ch4 (E) = 12
sider the case when, in addition, we have c2 (E) = 0. In this case, the formula for the Chern character ch4 (E)
further simplifies to
1
(2.32)
ch4 (E) = − c4 (E).
6
Here what we have really done is lifted the unitary group to its connected cover BU h8i. Indeed let us
consider the result from [54] where the mod p (p an odd prime) cohomology of the connective cover BU h2ni
was calculated. From that result and the result of Stong [57] for p = 2, the following divisibility result was
deduced for all primes p in [54]. Let ck ∈ H 2k (BU ; Z) be the universal Chern class in BU , then the Chern
class rn∗ (ck ) in BU h2ni where rn : BU h2ni → BU be the canonical projection is divisible by [54]
Y
(2.33)
pq
p
P
with σp (n) = ai the sum of the coefficients in the unique
where q is the least integer part of
decomposition of the integer n as n = a0 + ap + · · · + ak pk , with ai < p. Applying this result for n = 4,
p = 2, 3, and using σ2 (3) = 2, σ3 (3) = 1, we get that r4∗ (c4 ) is divisible by
(n−1)−σp (k−1)
,
p−1
(2.34)
2
2−σ2 (3)
1
·3
3−σ3 (3)
2
=6.
We will give an example where this occurs and where the expression (2.32) is integral.
Example. Consider a complex vector bundle E on the eight-sphere S 8 . For ten-manifold we can simply
take S 8 × R2 for example. The index of the Dirac operator
on S 8 coupled
to the vector bundle E is given by
8
8
8
b
b
b
the evaluation of the twisted A-genus A(S , E) := ch(E) · A(S ) [S ] on the fundamental class [S 8 ] of S 8
b S 8 ) · ch(E) = ch(E)[S 8 ],
(2.35)
IndexDE = A(T
b S 8 ) = 1, since spheres have stably trivial tangent bundles. Since S 8 is a Spin manifold, the index
as A(T
should be an integer. This then gives the requirement
1
(2.36)
ch4 (E)[S 8 ] = − c4 (E)[S 8 ] ∈ Z.
6
2.3.2. The M5-brane and the dual C-field. The magnetic dual of the M2-brane is the M5-brane. Where the
M2-brane is electrically charged under the C3 -field with class [G4 ] ∈ H 4 (X, Z), the M5-brane is electrically
charged under the dual C6 -field with class [G8 ] ∈ H 8 (X, Z). If X admits a String-structure, then, as we
discuss in more detail in a moment, one finds an anomaly cancellation condition on these background fields
analogous to (2.15) which reads
1
(2.37)
8[G8 ] = 4a(E) ∪ a(E) − p2 (T X) ,
6
where a : BE8 → K(Z, 4) is a representative of the canonical degree-4 class on the classifying space of the
exceptional Lie group E8 .
The homotopy-theoretic interpretation of this condition involves the following Fivebrane-analog of Spinc
as it appeared in prop. 2.7, and of Stringα as it was considered in def. 2.8.
Definition 2.19. For G a topological group and [α] ∈ H 8 (BG, Z) a universal 8-class, we say that Fivebraneα
is the loop group of the homotopy pullback
BFivebraneα
⌋⌋
/ BG
α
BString
1
6 p2
/ B 7 U (1)
.
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
17
As before with Stringα , this notion in particular subsumes spaces on which the class 16 p2 is further divisble.
For let G = B 6 U (1) in the above and take α := DD6 to be the canonical degree-8 class on B 7 U (1) ≃ K(Z, 8).
This is the canonical class of circle 7-bundles / U(1)-bundle 6-gerbes. Then Fivebrane8DD6 is the loop group
of the homotopy pullback
(2.38)
BFivebrane8DD6
1
48 p2
⌋⌋
/ K(Z, 8) .
×8
BString
1
6 p2
/ K(Z, 8)
Accordingly, we have that a String-bundle lifts to a Fivebrane8DD6 -bundle precisely if the class of
1
further divisible by 8, hence if 48
p2 exists. In [47] this space was denoted
1
6 p2
is
F h8i := BFivebrane8DD6 .
(2.39)
1
p2 ∈ H 8 (Fivebrane8DD6 ; Z) is the universal obstruction to lifting to a genuine FivebraneMoreover, the class 48
1
bundle. Accordingly, we have a notion of twisted Fivebrane-structures induced by 48
p2 . They form the space
given by the homotopy pullback.
(2.40)
1
48 p2 Struc[β] (X) ⌋⌋
/∗
.
β
Maps(X, BFivebrane)
1
48 p1
/ Maps(X, B 7 U (1))
Such a [β]-twisted Fivebrane-structure on a brane ι : Q ֒→ X is a homotopy η in the diagram
(2.41)
Q
ι
X
ν
tt
tttttt
t
t
t
t
tttttt η
u} tttt
β
/ BFivebrane8DD6 ,
1
48 p2
/ K(Z, 8)
which exists precisely if
(2.42)
1
p2 (X) + ι∗ ([β]) = 0 .
48
We may also regard the situation in analogy with def. 2.10 and consider the following.
Definition 2.20. For X a manifold and for [c] ∈ H 8 (X; Z) a degree 8 cohomology class, the space
( 61 p2 − 2a ∪ 2a)Struc[c] (X) of [c]-twisted Fivebrane2a∪2a -structures on X is the homotopy pullback
( 61 p2 − 2a ∪ 2a)Struc[c] (X)
/∗
c
Maps(X, BString × E8 )
1
6 p2 −2a∪2a
,
/ Maps(X, B 7 U (1))
where the right vertical map picks a cocycle c representing the class [c].
In terms of these notions we thus see that
Observation 2.21. Over a manifold X with String-structure and with a fixed gauge bundle E, condition
(2.37) is precisely the condition that guarantees existence of a lift to [8G8 ]-twisted Fivebrane2a∪2a -structure
through the left vertical morphism in def. 2.20.
18
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
We discuss now how (2.37) arises in more detail. Locally the C3 -field is given by a 3-form, which is
traditionally denoted by the same symbol. The equation of motion for C3 is obtained from varying the
action
Z
1
(2.43)
S(C3 ) =
G4 ∧ ∗G4 + G4 ∧ G4 ∧ C3 − I8 ∧ C3
6
Y
on an eleven-dimesional Spin manifold Y to obtain
1
d ∗ G4 = − G4 ∧ G4 + I8 .
2
Here I8 is the one-loop polynomial [59] [19] given in terms of the Pontrjagin classes of the tangent bundle
T Y to Y
p2 (T Y ) − 12 ( 12 p1 (T Y ))2
,
(2.45)
I8 =
48
and ∗ is the Hodge duality operator for a given metric in eleven dimensions.
(2.44)
The integral lift of (2.44) leads to a class defined in [16]
1 2
[G8 ] =
G − I8
2 4
7λ2 − p2
1
(2.46)
a(a − λ) +
,
=
2
48
where λ = 12 p1 , and a is the degree four class of an E8 bundle coming from Witten’s shifted quantization
condition for G4 [63]
1
1
(2.47)
[G4 ] = a − λ = a − p1 .
2
4
In [66] Witten interpreted the vanishing of a certain torsion class θ on the M-fivebrane worldvolume as
a necessary condition for the decoupling of the fivebrane from the ambient space ( “the bulk”). Hence the
vanishing of θ meant that the fivebrane can have a well-defined partition function. Consider the embedding
ι : W ֒→ Y of the fivebrane with six-dimensional worldvolume W into eleven-dimensional spacetime Y .
Consider the ten-dimensional unit sphere bundle π : X → W of W with fiber S 4 associated to the normal
bundle N → W of the embedding ι. Then it was shown in [16] that the integration of G8 over the fiber of
X gives exactly the torsion class θ on the fivebrane worldvolume
(2.48)
θ = π∗ (G8 ) ∈ H 4 (W ; Z).
Therefore, the vanishing of G8 is a necessary condition for the existence of a non-zero partition function [16].
We now proceed with the interpretation. Since we have Fivebrane structures in mind, we assume that
Y already admits a String structure, i.e. that 12 p1 (Y ) = 0. Then, from (2.46) we see that the class G8 (Y )
simplifies to
1
1
(2.49)
G8 (Y ) = a2 − p2 (Y ).
2
48
The class a is an integral class of an E8 bundle and hence defines a map to K(Z, 4). Then the square of a
1
p2 , then we have a
defines a map to K(Z, 8), and hence defines a twist for us. As we also have the class 48
twist for the modified Fivebrane structure.
Necessity of the Fivebrane condition? The Fivebrane condition is stronger than simply the requirement
that the one-loop term I8 to vanish. For the former we require the obstructions 21 p1 and 61 p2 vanish separately,
whereas for the latter we only require the combination to vanish. This has been studied in [31] [30] [62]. For
instance, following [62], a Riemannian 8-dimensional spin manifold M 8 is said to be doubly supersymmetric if
and only if the tangent bundle T M 8 and the spinor bundles ∆+ M 8 and ∆− M 8 are associated with a principal
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
19
G-fiber bundle such that there exist G-invariant isomorphisms between any two of the three bundles , i.e.
T M 8 = ∆+ M 8 = ∆− M 8 . If M 8 is doubly supersymmetric,
(2.50)
w1 = w2 = 0,
e = 0,
4p2 = p21 ,
where e is the Euler class. Then this implies for the signature sgn(M 8 ) = 16Â[M 8 ]. In particular, sgn(M 8 ) ≡
0 mod 16. One example is P SU (3)-structure for which
(2.51)
(2.52)
wi
=
e =
0 (i 6= 4),
w42 = 0
0,
p21 = 4p2 .
In particular, all Stiefel-Whitney numbers vanish.
A second example is a differentiable 8-fold M 8 with an odd topological generalized Spin(7)-structure (in
the sense of [62]) for which
(2.53)
χ(M 8 ) = 0,
p1 (M 8 )2 − 4p2 (M 8 ) = 0.
The 7-sphere admits a Spin structure and therefore admits a generalized G2 -structure. The tangent bundle of
the 8-sphere is stably trivial and therefore all the Pontrjagin classes vanish. Since the Euler class is non-trivial,
there exists no generalized Spin(7)-structure on an 8-sphere. However, equation (2.53) is automatically
satisfied for manifolds of the form M 8 = S 1 × N 7 with N 7 Spin.
3. Twisted differential structures in String theory
We now indicate a theory of nonabelian differential cohomology in which the topological structures considered in section 2 have smooth and differential refinements. A full account of this theory is given in [49].
The complete differential refinements are based on a smooth refinement of topological spaces by structures
called higher smooth stacks or higher smooth groupoids. The full constructions of these, for the cases considered here, are discussed in [21]. As shown there, underlying any such differential cohomological structure
is, locally, explicit differential geometric data, given by differential forms with values in L∞ -algebras. This
L∞ -algebra valued connection data by itself was discussed in [46], a brief collection of relevant L∞ -algebraic
notions is in the appendix section A.
Here, after a brief survey of the general theory, we concentrate on a discussion of this local differential
form data for the differential refinements of the twisted String- and Fivebrane-structures from section 2. We
show that these reproduce the equations on differential forms that are traditional in the physics literature
on the Green-Schwarz anomaly cancellation as well as its magnetic dual.
The key fact is the existence of definition 3.2 below, which gives smooth and differential refinements of
the homotopy pullbacks that defined twisted String- and Fivebrane-structures in def. 2.5 and def. 2.16,
respectively.
3.1. Differential twisted cohomology. Differential twisted cohomology is the pairing of the notions of
twisted cohomology with differential cohomology (see [28]). We have that
• a cocycle in differential cohomology is to the underlying bare cocycle as a connection on an ∞-bundle
is to the underlying principal ∞-bundle;
• a cocycle in twisted cohomology is to an ordinary cocycle as a twisted bundle is to a principal bundle.
We indicate now briefly a formal definition of such objects as described in full detail in [49], showing how they
connect via the constructions from [21] the cohomological discussion that we had so far to the L∞ -algebraic
differential form data of [46] in terms of which we shall obtain the relevant twisted Bianchi identities.
The basic mechanism is to refine the homotopy theory by passing from the ∞-topos [37] Top ≃ ∞Grpd
of topological spaces – equivalently: geometrically discrete ∞-groupoids – to that of smooth ∞-groupoids,
defined as the ∞-category of ∞-stacks over the site of smooth manifolds.
It is a familiar fact that Lie groupoids, such as for instance orbifolds, are naturally to be thought of as
stacks on the site of smooth manifolds, often called differentiable stacks [7]. Such differentiable stacks form
a common generalization of smooth manifolds and a small fragment of homotopy theory. For instance, for
20
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
each Lie group G there is a Lie groupoid, denoted ∗//G or BG, with a single object and G as its space
of morphisms, such that for X any smooth manifold, regarded as a Lie groupoid, the collection H(X, BG)
of stack morphisms X → BG forms the groupoid of smooth G-principal bundles on X and smooth gauge
transformations between them. So the Lie groupoid BG serves as a smooth refinement of the topological
classifying space BG.
But in the context of ordinary stacks, not all the classifying spaces that we considered in the previous
section, such as B n U (1) ≃ K(Z, n + 1) for higher n, have a smooth refinement to stacks on manifolds. But
there is a natural generalization of the theory of stacks to a general theory of higher stacks, often called
∞-stacks. The collection of all of these over a given site is an ∞-topos [37] and this notion serves as a
complete joint unification of the geometry over the given site with homotopy theory.
For instance, over the trivial site an ∞-stack is the same as an ∞-groupoid (a Kan complex), which in
turn is equivalently – in the sense of homotopy theory – a topological space. We say that the collection of
all of these
Top ≃ ∞Grpd
is the canonical base ∞-topos. Next, in generalization to the relation between Lie groupoids and differentiable
stacks, we speak of the ∞-stacks on the site of smooth manifolds as being smooth ∞-groupoids. We write
H := Smooth∞Grpd for the ∞-topos of smooth ∞-groupoids. For emphasis, we will say that an object
in ∞Grpd is a bare or discrete ∞-groupoid (not equipped with nontrivial smooth structure). Notice that
therefore also topological spaces are, in the sense of homotopy theory, discrete ∞-groupoids.
It turns out that the ∞-topos Smooth∞Grpd sits over that of bare ∞-groupoids by an adjoint quadruple
of ∞-functors
Π
Smooth∞Grp
o
o
Disc
Γ
/
/ ∞Grpd
coDisc
and this controls the notion of smooth refinement of bare cohomology. The geometric interpretation is this:
• “Disc” produces smooth ∞-groupoids with discrete smooth structure;
• “Γ” forgets the smooth structure on a smooth ∞-groupoid;
• “Π” sends a smooth ∞-groupoid X to its fundamental path ∞-groupoid ; combined with the equivalence ∞Grpd ≃ Top this is the operation of geometric realization.
For any object or diagram in ∞Grpd by a smooth lift of it we mean a lift through Π to an object or diagram,
respectively, in Smooth∞Grpd.
A smooth ∞-groupoid with a single object we write BG, where G is the smooth ∞-group of automorphisms
of that single object. The boldface B denotes delooping in Smooth∞Grpd as opposed to in ∞Grpd. This
completely defines both structures: pointed connected smooth ∞-groupoids are equivalent to smooth ∞groups. For instance there is for each n ∈ N a smooth ∞-groupoid Bn U (1), defined this way inductively
from the ordinary smooth circle group U (1) = B0 U (1). This is such that ΠBn U (1) ≃ B n+1 Z ≃ K(Z, n + 1)
is the Eilenberg-MacLane space that classifies integral cohomology in degree n. Hence Bn U (1) is a smooth
refinement of the classifying space B n U (1) ≃ K(Z, n + 1).
A morphism X → Bn U (1) of smooth ∞-groupoids classifies a circle n-bundle on X. For n = 1 and X an
ordinary manifold this are ordinary circle bundle, for n = 2 this are bundle gerbes, for n = 3 this are bundle
2-gerbes, etc. Generally, a morphism X → BG classifies a G-principal ∞-bundle.
Note that X here can be much more general than a smooth manifold. Notably, we can have X = BG for
G a Lie group or more general smooth ∞-group. A morphisms c : BG → Bn U (1) then defines a cocycle in
generalized (Segal-Brylinski-)smooth group cohomology on G with coefficients in U (1) in degree n. If G is
a Lie group or an ∞-group presented by a simplicial Lie group, then ΠBG ≃ BG is the ordinary classifying
space of G. Therefore such a cocycle maps under Π to an ordinary integral cocycle
(3.1)
c
c
[Π(BG → Bn U (1))] ≃ [BG → K(Z, n + 1)] ∈ H n+1 (BG, Z)
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
21
on the classifying space. So the cocycle c is a smooth refinement of the topological characteristic map c. We
n
say that c lives in the smooth cohomology Hsmooth
(BG, U (1)).
Given any such cocycle c, we say that the c-twisted cohomology on X is the connected components of the
homotopy pullback
cStructw (X)
tw
⌋⌋
/ Hn
smooth (X, U (1))
,
(3.2)
H(X, BG)
c
/ H(X, Bn U (1))
where the right vertical morphism is the canonical effective epimorphism that picks one cocycle in each
cohomology class. The map tw here sends twisted cocycles to their twist. For instance for c : BP U (n) →
B2 U (1) the cocycle that classifies the extension BU (1) → BU (n) → BP U (n) we have that cStructw (X) is
the groupoid of twisted complex vector bundles of rank n on X, those that appear in the geometric model
for twisted K-theory in degree 0.
The reader may be more familiar with twisted cohomology formulated in terms of sections of certain
bundles. We briefly indicate how this is equivalently another perspective on the above setup.
Consider first the example of a Lie group G acting on a vector space V . The weak quotient of this action
is a Lie groupoid V //G whose objects form the space V and where there is precisely one morphism for every
ordered pair of points related by the group action. This Lie groupoid is equipped with a cononical projection
ρ : V //G → BG. We may think of this as the smooth incarnation of the vector bundle that is associated via
ρ
ρ to the universal G-bundle over BG. In fact we have a fiber sequence V → V //G → BG of Lie groupoids,
and this may equivalently be taken to define the action ρ of G on V .
Now consider a morphism g : X → BG classifying a G-principal bundle P → X, as above. By inspection
one finds that a lift σ of this morphism along this projection
(3.3)
V //G
②<
②
ρ
②②
②②
②② g
/ BG
X
σ
is precisely a section of the vector bundle P ×ρ V that is associated to P by the given representation. On
the other hand, in terms of the above notion of twisted cohomology, such a lift is also precisley an element
in the ρ-twisted cohomology of X with coefficients in V //G, where the twist is the class of P : we have an
equivalence
(3.4)
ρStruc[P ] (X) ≃ ΓX (P ×ρ V )
of cocycles in ρ-twisted cohomology with sections of the ρ-associated vector bundle.
This perspective generalizes verbatim to all twisting cocycles c on all ∞-groups. For those of the form
c : BG → Bn U (1) considered before, we may think of their homotopy fiber BĜ in the fiber sequence
(3.5)
c
BĜ → BG → Bn U (1)
as the delooping of the shifted central extension Ĝ of G classified by the cocycle c, or equivalently think of
BG as a universal c-associated BĜ-bundle over Bn U (1). Then for P → X classified by g : X → Bn U (1) a
given circle n-bundle on X, the ∞-groupoid cStruc[g] may be thought of equivalently as the space of sections
of the associated BĜ-bundle. If P is trivial, such sections are just maps from X to BĜ.
In order to equip such structures of smooth and twisted cohomology with connections, we reflect Π back
to smooth ∞-groupoids by defining Π := Disc Π : Smooth∞Grpd → Smooth∞Grpd. For X a smooth ∞groupoid we say that Π(X) is its smooth path ∞-groupoid. The adjunction unit gives a canonical morphism
X → Π(X) which includes X as the constant paths in X. A morphism Π(X) → BG therefore has
an underlying G-principal ∞-bundle X → Π(X) → BG but also assigns equivalences between fibers over
22
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
endpoints of paths, that are equivalent around disks, etc. This is the parallel transport of a flat ∞-connection
on the smooth G-principal bundle.
From this in turn derives a notion of G-valued flat differential forms: these are the flat G-connections
whose underlying ∞-bundle is trivial. We write
a
(3.6)
ΠdR X := ΠX
∗
X
for the canonical homotopy pushout and speak of the de Rham homotopy type of X. A morphism ΠdR (X) →
BG is a closed G-valued form on X. In particular a morphism ΠdR (X) → Bn U (1) is a closed n-form.
A key observation now is that there is a canonically induced morphism of cocycle ∞-groupoids
(3.7)
curv : H(X, Bn U (1)) → H(ΠdR (X), Bn+1 U (1))
that sends each circle n-bundle to a curvature characteristic form. We define ordinary differential cohon
mology Hdiff (X, Bn U (1)) of X to be the ∞-pullback of the canonical effective epimorphism HdR
(X) →
n+1
Smooth∞Grpd(ΠdR (X), B
U (1)) (that which picks one cocycle in each cohomology class) along this
curvature characteristic map. Cocyles in this homotopy pullback
Hdiff,Bn U(1)
⌋⌋
/ H n+1 (X)
dR
(3.8)
H(X, Bn U (1))
/ H(ΠdR X, Bn+1 U (1))
are smooth circle n-bundles with connection.
For X an ordinary manifold, this reproduces the ordinary notions in differential cohomology. More
precisely, in this case the n-groupoid Hdiff (X, Bn U (1)) turns out to be that whose objects are cocycles in
Deligne-Beilinson hypercohomology, whose morphisms are smooth gauge transformations of these, whose
2-morphisms are higher gauge transformations of those, and so on. But as before, we can apply this over
any smooth ∞-groupoid. In particular we may consider differential cohomology over moduli stacks BG of
G-principal ∞-bundles that differentially refine smooth lifts c : BG → Bn+1 U (1) of characteristic map.
These are Chern-Simons n-gerbes with connection.
This finally gives rise to the notion of ∞-connections on general nonabelian G-principal ∞-bundles: these
are structures that lift universal curvature classes from de Rham cohomology to differential cohomology.
With all these concepts thus abstractly given, we can look for explicit constructions of these. By standard
theory [37] every smooth ∞-groupoid A is presented a simplicial presheaf on the category of Cartesian spaces
and smooth maps: a functor
(3.9)
A : (U = Rn , [k]) 7→ Ak (U ) ∈ Set
that we read as assigning to each test space U and each k ∈ N the set of k-morphisms of the ∞-groupoid of
possible ways of probing A with U , or equivalently the set of U -parameterized smooth families of k-morphisms
in A.
We consider the construction of such simplicial presheaves from infinitesimal data. For g an an L∞ algebra (see [46] and the appendix for a review or relevant notions) we can define such a presheaf by setting,
in evident generalization of the construction in [27],
Avert
CE(g) ,
(3.10)
exp(g) : (U, [k]) 7→ Ω• (U × ∆k )vert o
where on the right we have the set of dg-algebra homomorphisms from the Chevalley-Eilenberg algebra of g to
the de Rham complex of vertical forms on the trivial simplex bundle U × ∆k → U . The Chevalley-Eilenberg
dg-algebra CE(g) is the free graded-commutative algebra on the the degreewise dual of the graded vector
space underlying g and equipped with the differential obtained by dualizing all the brackets on g. For g an
ordinary Lie algebra it reduces to the ordinary Chevalley-Eilenberg algebra, hence its name.
This simplicial presheaf exp(g) presents the smooth ∞-groupoid BG for G the “universal ∞-connected”
Lie integration of g. Smooth Postnikov truncations τn of this object yield smooth n-groups integrating g.
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
23
For instance for g an ordinary Lie algebra we have that τ1 exp(g) ≃ BG for G the ordinary simply connected
Lie group integrating g.
The crucial point for our discussion here is that one obtains from this also a model for the smooth ∞groupoid that classifies G-principal ∞-connections for such Lie integrated G [21]. If we write W(g) for the
Weil algebra of g, the unique free dg-algebra on the dual graded vector space of g such that the canonical
projection is a dg-algebra homomorphism to CE(g), and inv(g) for the subalgebra of closed elements formed
out of shifted generators – the invariant polynomials on g –, then this is given by the simplicial presheaf
defined by
Avert
•
k
CE(g)
Ωvert (U × ∆ ) o
O
O
(A,FA )
•
k o
(3.11)
exp(g)conn : (U, [k]) 7→
Ω (U × ∆ )
W(g) .
O
O
hF
i
A
•
o
Ω (U )
inv(g)
On the right we have the set of horizontal dg-algebra homomorphisms that makes the diagram commute, see
around def. A.7 in the appendix for more details. This is the L∞ -algebraic differential form data discussed in
detail in [46], here parameterized over all test spaces U and simplices ∆k as discussed in [21]. The horizontal
morphism in the middle constitutes g-valued differential form data on U × ∆k and the fact that it sits in
this commuting diagram encodes the ∞-analogs of the two conditions of an ordinary Cartan-Ehresmann
connection: the top square says that the vertical part of A is flat, and the bottom square says that the
curvature forms “transform covariantly” and make all invariant polynomials descent down to U .
For X a smooth manifold a morphism X → exp(g)conn is equivalently
(1) a choice of good open cover {Ui → X};
(2) on each patch Ui , differential form data with values in g;
(3) on each double intersection, a choice of 1-parameter gauge transformation between the corresponding
differential form data;
(4) on each triple intersection, a choice of 2-parameter gauge-of-gauge transformation;
(5) and so on for higher intersections.
Given then a differential refinement of a cocycle c : BG → Bn U (1) to the corresponding moduli stacks of
bundles with connection
(3.12)
ĉ : BGconn → Bn U (1)conn ,
we can consider the differential refinement of the twisted smooth cohomology discussed above: the cocycle
∞-groupoid of ĉ-twisted differential cohomology is the homotopy pullback
ĉStructw (X)
tw
⌋⌋
/ H n+1 (X)
diff
,
(3.13)
H(X, BGconn )
ĉ
/ H(X, Bn U (1)conn )
where again the right vertical morphism is the canonical effective epimorphism that picks one cocycle in
each cohomology class. Up to some slight technicalities which are discussed in [21], this homotopy pullback
is modeled by the corresponding pullback of the double square diagrams of L∞ -algebra data from above,
which are discussed in detail in the last part of [46]. This we will later unwind in sections 3.2 and 3.3 for
the case of twisted differential String- and Fivebrane structures.
24
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Facts 3.1.
(1) There is, up to equivalence, a unique smooth refinement of the first fractional Pontrjagin
class (from fact 2.3)
1
p1 : BSpin → B3 U (1) .
2
to the moduli stack of smooth Spin-principal bundles with values in the moduli 3-stack of circle
3-bundle.
(2) The smooth string 2-group, whose smooth delooping we denote BString, is the homotopy fiber of
1
2 p1 , sitting in a fiber sequence
(3.14)
1
(3.15)
p1
2
B3 U (1) .
BString → BSpin −→
(3) The models for the smooth string 2-group given in [27] and in [4] are indeed presentations of the
abstract definition (3.15).
(4) There is a smooth refinement 61 p2 : BString → B7 U (1) of the second fractional Pontrjagin class
from fact 2.14 to the moduli 2-stack of String-principal 2-bundles with values in the moduli 7-stack
of circle 7-bundles.
(5) The homotopy fiber of 21 p2 is the smooth delooping of the smooth fivebrane 6-group
1
p2
6
BFivebrane → BString −→
B7 U (1) .
(6) Under geometric realization these smooth lifts indeed reproduce the first steps in the Whitehead of O:
|BString| ≃ BString
and
|BFivebrane| ≃ BFivebrane .
This is shown in [21] and also discussed in section 4.1 of [49].
In summary all this means that we obtain the following canonical refinement of def. 2.5 and def. 2.16 to
a notion of twisted differential string- and fivebrane structures.
Definition 3.2. For X a smooth manifold, the 2-groupoid of twisted differential String-structures 21 p̂1 Structw (X)
on X is the homotopy pullback
1
2 p̂1 Structw (X) ⌋⌋
tw
/ H 4 (X)
diff
1
2 p̂1
/ H(X, B3 U (1)conn )
.
(3.16)
H(X, BSpinconn )
Analogously, the 6-groupoid of twisted differential fivebrane-structures 61 p̂2 Structw (X) on X is the homotopy
pullback
1
6 p̂2 Structw (X) ⌋⌋
tw
/ H 8 (X)
diff
1
6 p̂2
/ H(X, B7 U (1)conn)
.
(3.17)
H(X, BStringconn )
Notice that since these constructions had been announced in [46] the article [60] has appeared which
defines a presentation of twisted differental String-structures in terms of bundle 2-gerbes for the case that
the underlying topological twist vanishes.
We now describe the local L∞ -algebraic differential form data of such twisted differential structures,
following [46]. Details of the following fact are in [21].
Theorem 3.3. (i) The local differential form data of a twisted String(n)-bundle with connection is that
known from the Green-Schwarz mechanism (section 2.2.1).
(ii) The local differential form data of a twisted Fivebrane(n)-bundle with connection is that of the dual
Green-Schwarz mechanism (section 2.3.1).
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
25
3.2. Twisted string(n) 2-connections. One finds that the smooth first fractional Pontryagin map
1
p1 : BSpin(n) → B3 (U (1))
2
(3.18)
from the moduli stack of Spin-principal bndles to the moduli 3-stack of circle 3-bundles may be modeled in
terms of simplicial presheaves by the span of smooth 2-groupoids of the form
≃
BSpin ← B(BU (1) → String) → B3 U (1)
(3.19)
given by a span of smooth crossed complexes of the form
(3.20)
(1
/1
1
≃
/ Spin(n) ) ←−
( U (1)
/ Ω̂Spin(n)
2 p1
/ P Spin(n) ) −→
( U (1)
/1
/ 1 ),
where Ω̂Spin denotes the Kac-Moody central extension of the loop group of Spin.
The corresponding span of L∞ -algebras is of the form
≃
so(n) ←− (bu(1) → string(n)) → b2 u(1) ,
(3.21)
where in the middle we we have the mapping cone of the inclusion of the line Lie 2-algebra, example A.2,
into the string Lie 2-algebra, example A.5.
≃
Therefore for C({Ui }) → X the Čech nerve projection out of a sufficiently good open cover of X and for
≃
tw
X ← C({Ui }) → B3 U (1) a cocycle for the twisting circle 3-bundle, a corresponding twisted String-2-bundle
is a lift ĝ in the diagram of simplicial presheaves
.
B(BU (1) → String)
6
◗◗◗
♠♠♠
◗◗◗
♠
♠
◗◗◗
♠♠
◗◗◗
♠♠♠
(
♠♠♠
tw
/ B3 U (1)
C({Ui })
ĝ
≃
X
After passing to the differential refinement of this situation, the corresponding Cartan-Ehresmann L∞ connection is given on (k + 1)-fold intersections of the cover by compatible diagrams
(3.22)
Ω•vert (U × ∆k ) o
O
Ω• (U × ∆k ) o
O
Ω• (U ) o
Avert
(A,FA )
P (FA )
CE(bu(1) → string(n))
O
W(bu(1) → string(n))
O
inv(bu(1) → string(n))
such that the corresponding twist is the prescribed one. If we write (C3 ) for the local differential 3-form
data of the prescribed twisting circle 3-bundle, then this means that over (k + 1)-fold intersections the
L∞ -algebraic data of a connection on the twisted String-bundle is given by a diagram of dg-algebras of the
26
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
form
CE(bu(1) ֒→ gµ )
,
g❖❖❖
OO
♥
❖❖❖
♥
❖❖❖
♥ ♥
❖❖❖
{Avert ,Bvert ,(C3 )vert }
❖❖❖
♥
♥
❖❖❖
❖4 T
v♥ ♥
(C3 )vert
Ω•vert (U × ∆k ) o
CE(b2 u(1))
OO
OO
W(bu(1) ֒→ gµ )
g❖❖❖
OO
❖❖❖
♥ ♥
♥
❖❖❖
♥
❖❖❖
{A,FA ,B,∇B,C3 ,G4 }
❖❖❖
♥
♥
❖❖❖
❖4 T
v♥ ♥
(C3 ,G4 )
Ω• (U × ∆k ) o
W(b2 u(1))
O
O
inv(inn(bu(1)) ֒→ csP (g))
g❖❖❖
❖❖❖
♥ ♥
♥
❖❖❖
♥
❖❖❖
{H3 ,G4 ,P (FA )}
❖❖❖
♥
♥
❖❖❖
♥
♥
❖4 T ?
?
v♥
G4
Ω• (U ) o
inv(b2 u(1))
p∗
where, for short, we write gµ for string. It may be helpful to think of this as forming sections of a higher
associated bundle, as in def. A.10 in the appendix. In terms of this we may read this diagram as indicated
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
27
in the following
Chevalley-Eilenberg algebra of
structure L∞ -algebra
for twisted String 2-bundle
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
OO
tt
flat nonabelian differential forms
on fibers of total space
or equivalently
section of
2-gerbe / line 3-bundle
t
zt t
vertical differential forms
on total space
o
flat abelian differential forms
on fibers
OO
Chevalley-Eilenberg algebra of
structure L∞ -algebra of
2-gerbe/line 3-bundle
OO
Weil algebra of
structure L∞ -algebra
for twisted String 2-bundle
✉✉
connection and curvature on
twisted String 2-bundle
or equivalently
section with covariant derivative
of 2-gerbe / line 3-bundle
z✉ ✉
differential forms
on total space
O
✉
o
OO
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
connection and curvature on
2-gerbe / line 3-bundle
Weil algebra of
structure L∞ -algebra of
2-gerbe / line 3-bundle
O
invariant polynomials on
structure L∞ -algebra
of twisted String 2-bundle
p∗
✉
✉
✉
✉
✉
characteristic forms of
twisted String 2-bundle
✉
?
z✉
differential forms
on base space
✉
✉
o
✉
✉
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
characteristic forms on
2-gerbe / line 3-bundle
?
invariant polynomials on
structure L∞ -algebra of
2-gerbe / line 3-bundle
28
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Chasing the generators of the graded-commutative algebras through this diagram and recording the condition
imposed by the respect of the morphisms of dg-algebras for differentials, one finds that in components the
commutativity of this diagram encodes the following differential form data and the following relations on
that.
a
Aavert
t 7→
b 7→ Bvert
k
7→ (C3 )vert
❧
v❧ ❧
FAavert = 0
dBvert = µAvert − (C3 )vert
d(C3 )vert = 0
O
o
❧✱
❧ ❧
dta = − 12 C a bc tb ∧ tc
db = µ − k
dk = 0
f▼▼▼
O
▼▼▼
▼
k 7→ k▼
▼▼▼
▼▼▼
▼▼▼
▼▼
k 7→ (C3 )vert
dk = 0
O
❴
dt = − 21 C a bc tb ∧ tc
dra = −C a bc tb ∧ rc
a
db = cs + c − k
dc = l − P
dk = l
O
i∗
ta 7→ Aa
✳
ra 7→ FAa ♥ ♥
b 7→ B
c
7→ ∇B
♥ ♥ k 7→ C
♥
3
♥
v♥ ♥
❴
l 7→ G4
H3 := ∇B = dB + C3 − CS(A, FA )
k 7→ C3
o
dH3 = G4 − hFA ∧ FA i
l 7→ G4
dG4 = 0
O
p∗
c 7→ ∇B := H3
l 7→ G4
P 7→ hFA ∧ FA i
❦
❦
u❦ ❦
❴
dH3 = G4 − hFA ∧ FA i
dG4 = 0
o
✰
❦ ❦
❦
❦ ❦
❴
dc = l − P
dl = 0
dP = 0
l 7→ G4
+ ra
k 7→ k
l 7→ 0
e❑❑
k 7→ k
l 7→ l ❑
❑❑
❑❑
❑❑
❑❑
❑☛
❴
✤ dk = l
dl = 0
O
l 7→ l
g◆◆◆
◆◆◆
◆◆◆
l 7→ l ◆
◆◆◆
◆◆◆
◆◆◆
◆◆
❴
✍
✤ dl = 0
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
29
Here, P ∈ W (g) denotes the invariant polynomial on g in transgression with with the cocycle µ ∈ CE(g).
With {ta } a fixed chosen basis of g∗ in degree 1 and {ra } the corresponding basis in degree 2, we have
P = Pab ra ∧ rb and µ = µabc ta ∧ tb ∧ tc and cs = Pab tb ∧ ra + 16 µabc ta ∧ tb ∧ tc . We have
curvature
Bianchi identity
H3 := dB + C3 − CS(A, FA )
dH3 = G4 − hFA ∧ FA i
In [46] this situation was considered from a different perspective for the special case B = 0 and ∇B = 0.
There the dashed morphism was obtained as a twisted lift of a g-connection to a gµ -connection and the
b2 u(1)-connection appeared as the corresponding obstruction. Here now the perspective is switched: the
b2 u(1)-connection is prescribed and the choice of dashed morphisms is a choice of twisted gµ -connections
with prescribed twist G4 .
The covariant derivative 3-form ∇B of the twisted gµ -connection, which we denote by H3 , measures the
difference between the prescribed b2 u(1)-connection and the twist of the chosen twisted gµ -connection. The
Bianchi identity
(3.23)
dH3 = G4 − P (FA )
which appears in the middle on the left says that this difference has to vanish in cohomology, as one expects.
Indeed, this is the structure of the differential forms in the Green-Schwarz mechanism, constituting the
differential refinement of the integral cohomology relation (2.10).
3.3. Twisted fivebrane(n) 6-connections. The discussion of twisted differential Fivebrane structures proceeds in direct analogy to the above discussion of twisted differential string structures. One finds that the
smooth second fractional Pontryagin class 16 p2 : BString → B7 U (1) from the moduli 2-stack of Stringprincipal 2-bundles to the moduli 7-stack of circle 7-bundles is presented by a span of simplicial presheaves
of the form
≃
BString ← B(B5 U (1) → String) → B7 U (1)
whose infinitesimal version is a span of L∞ -algebras of the form
≃
string ← (b5 u(1) → fivebrane) → b6 u(1) ,
where fivebrane = (soµ3 )µ7 denotes the Lie 6-algebra from example A.5, and where the middle piece is the
mapping cone of the defining extension
b5 u(1) → fibrane → string .
≃
Therfore for tw : X ← C({Ui }) → B7 U (1) a Čech cocycle for a twisting circle 7-bundle, the corresponding
twisted Fivebrane 6-bundles are given by lifts ĝ in the disgram of simplicial presheaves
B(B5 U (1) → Fivebrane)
.
❘❘❘
❧❧5
❘❘❘
ĝ ❧❧❧❧
❘❘❘
❧❧
❘❘❘
❧❧❧
❘❘)
❧❧❧
tw
/ B7 U (1)
C({Ui })
≃
X
As before, we consider now the analogous diagrams of local L∞ -algebra valued forms in order to deduce
the local differental form data of twisted differential Fivebrane structures. For transparency of the following
diagrams we indicate both the twist of the differential String-structure with local differential forms (C3 )
as before, as well as the new twist of the differential Fivebrane structure by local differential forms (C7 ).
The former has to be taken to vanish, but it is still instructive to display its differential incarnation, for
comparison with the previous case.
30
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Then the local differential cocycle data of a differential refinement of the above twisted Fivebrane structure
ĝ is given over k-fold intersections by a diagram of dg-algebras of the form.
.
CE((bu(1) ⊕ b5 u(1)) ֒→ (so(n)µ3 )µ7 )
OO
j❚❚❚❚
❥
❥
❚
❚❚❚❚
❥
❚❚❚❚
❥ ❥
❚❚❚❚
{Avert ,(B2 )vert ,(B
)
,(C
)
,(C
)
}
6
vert
3
vert
7
vert
❥
❚❚❚❚
❥
❚❚❚❚
❥
❥
❚7 W
u❥
•
k o ((C )
,(C
)
)
Ωvert (U × ∆ )
CE(b2 u(1) ⊕ b6 u(1))
3 vert
7 vert
OO
OO
W((bu(1) ⊕ b5 u(1)) ֒→ (so(n)µ3 )µ7 )
j❚❚❚❚
OO
❥
❚❚❚❚
❥ ❥
❚❚❚❚
❥
❥
❚❚❚❚
{A,FA ,B2 ,B6 ,∇B
2 ,∇B6 ,C3 ,C7 ,G4 ,G8 }
❚❚❚❚
❥
❥
❚❚❚❚
❥
❥
❚❚7 W
u❥
•
k o
(C3 ,C7 ,G4 ,G8 )
Ω (U × ∆ )
W(b2 u(1) ⊕ b6 u(1))
O
O
inv(inn(bu(1) ⊕ b5 u(1)) ֒→ csP4 +P8 (so(n)))
j❚❚❚❚
❥
❚❚❚❚
❥❥
❥
❚❚❚❚
❥❥
❚❚❚❚
{H3 ,H7 ,G4 ,G
,P
(F
),P
(F
)}
8
4
A
8
A
❚❚❚❚
❥ ❥
❚❚❚❚
❥
❥
❚❚7 W
❥
?
?
❥
u
(G4 ,G8 )
Ω• (U ) o
inv(b2 u(1) ⊕ b6 u(1))
p∗
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
31
Here is again the meaning in words of the constituents of this diagram:
Chevalley-Eilenberg algebra of
structure L∞ -algebra
for twisted Fivebrane 6-bundle
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
OO
tt
flat nonabelian differential forms
on the fibers
or equivalently
section of
7-bundle
zt t
t
vertical differential forms
on the total space
o
flat abelian differential forms
on the fibers
OO
Chevalley-Eilenberg algebra of
structure L∞ -algebra of
line 7-bundle
OO
Weil algebra of
structure L∞ -algebra
for twisted Fivebrane 6-bundle
tt
connection and curvature on
twisted Fivebrane 6-bundle
or equivalently
section with covariant derivative
7-bundle
zt t
differential forms
on the total space
O
t
o
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
connection and curvature on
7-bundle
Weil algebra of
structure L∞ -algebra of
7-bundle
O
invariant polynomials on
structure L∞ -algebra
of twisted Fivebrane 6-bundle
p∗
t
t
t
t
t
characteristic forms of
twisted Fivebrane 6-bundle
t
t
t
zt
forms on base space
?
OO
t
t
o
t
e❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑
❑❑ S
3
characteristic forms on
7-bundle
?
invariant polynomials on
structure L∞ -algebra of
7-bundle
32
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
By again chasing elements through the diagram one finds the following data:
✲
♠♠
ta 7→ Aavert
♠♠
b2 7→ (B2 )vert
b6 7→ (B6 )vert
v k3 7→ (C3 )vert
k7 7→ (C7 )vert
FAavert = 0
d(B2 )vert = µ3 (Avert ) − (C3 )vert
d(B6 )vert = µ7 (Avert ) − (C7 )vert
d(C3 )vert = 0
d(C7 )vert = 0
O
dta = − 21 C a bc tb ∧ tc
db2 = µ3 − k3
db6 = µ7 − k7
dk3 = 0
dk7 = 0
O
e▲▲▲
k3 7→ (C3 )vert
k7 7→ (C7 )vert
o
dk = 0
O
❴
dta = − 21 C a bc tb ∧ tc + ra
dra = −C a bc tb ∧ rc
db2 = cs3 + c3 − k3
db6 = cs7 + c7 − k7
ta 7→ Aa
i∗
dc3 = l4 − P4
ra 7→ FAa
dc7 = l8 − P8
b2 7→ B2
dk3 = l4
✵
b6 7→ B6 ♣ ♣ dk7 = l8
c3 7→ ∇B2
O
c7 7→ ∇B6
♣
♣ k3 7→ C3
x♣
❴
k7 7→ C7
H3 := ∇B2 = dB2 + C3 − CS3 (A, FA )
l4 7→ G4
H7 := ∇B6 = dB6 + C7 − CS7 (A, FA )
l8 7→ G8 k3 7→ C3
k7 7→ C7
dH3 = G4 − hFA ∧ FA i
o
l4 7→ G4
dH7 = G8 − hFA ∧ FA ∧ FA ∧ FA i
l8 7→ G8
dG4 = 0
dG8 = 0
O
p∗
c3 7→ ∇B2 := H3
✲
c7 7→ ∇B6 := H7
l4 7→ G4
l8 7→ G8
P4 7→ hFA ∧ FA i
♠ P8 7→ hFA ∧ FA ∧ FA ∧ FA i
v
♠
❴
dH3 = G4 − hFA ∧ FA i
dH7 = G8 − hFA ∧ FA ∧ FA ∧ FA i
dG4 = 0
dG8 = 0
o
k3 7→ k3
k7 7→ k7 ▲
▲▲▲
▲▲▲
▲▲▲
▲
❴
dc3 = l4 − P4
dc7 = l8 − P8
dl4 = 0
dl8 = 0
dP4 = 0
dP8 = 0
l4 7→ G4
l8 7→ G8
k3 7→ k3
k7 7→ k7
l4 7→ 0
l8 7→ 0
c k3 7→ k3
k7 7→ k7
l4 7→ l4
l8 7→ l8 ●●●●
●●
❴
●●
●
✞ dk = l
3
4
✤ dk7 = l8
dl4 = 0
dl8 = 0
O
l4 7→ l4
l8 7→ l8
e▲▲▲
▲▲
▲
l4 →
7 l4
l8 →
7 l8 ▲
▲▲
▲▲
▲▲
▲▲
❴
☞
✤ dl4 = 0
dl8 = 0
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
33
Here, P4 , P8 ∈ W (g) are the invariant polynomials on g in transgression with with the cocycles µ3 , µ7 ∈
CE(g). The covariant derivative 7-form ∇B6 of the twisted (so(n)µ3 )µ7 -connection which we denote by
H7 measures the difference between the prescribed b6 u(1)-connection and the twist of the chosen twisted
(so(n)µ3 )µ7 -connection. The Bianchi identity
(3.24)
dH7 = G8 − P8 (FA )
which appears in the middle on the left says that this difference has to vanish in cohomology, being the local
incarnation of the differential refinement of the anomaly cancellation condition discussed in section 2.3.
Appendix A. L∞ -algebraic notions
We collect here some L∞ -algebraic definitions and constructions that are referred to in section 3. Most
of the following can be found in more detail in [46], the main point here being the notion of representations
of L∞ -algebroids and the emphasis of the interpretation of twists as sections, as explained for smooth ∞groupoids in section 3.1. For a more conceptual account of L∞ -algebra in the context of smooth ∞-groupoids
see [49].
A.1. L∞ -Algebras and L∞ -Algebroids. In direct generalization of how Lie algebras are infinitesimal
approximations to Lie groups, L∞ -algebras are infinitesimal approximations to smooth ∞-groups. More
generally, L∞ -algebroids are infinitesimal approximations to smooth ∞-groupoids.
Definition A.1. An L∞ -algebroid a of finite type over a smooth manifold X is a non-positively graded
A := C ∞ (X)-module degreewise of finite rank, together with a degree +1 derivation
(A.1)
d : ∧•A a∗ → ∧•A a∗ ,
linear over the ground field (not necessarily over A) on the free (over A) graded-symmetric algebra generated from the N-graded dual a∗ (over A), such that d2 = 0. The quasi-free (over A) differential gradedcommutative algebra
(A.2)
CE(a) := (∧•A a∗ , d)
defined this way we call the Chevalley-Eilenberg algebra of the L∞ -algebroid a.
We say the category L∞ Algd of L∞ -algebroids is the opposite of the full subcategory of dg-algebras on
those of the above form.
It is useful to distinguish the following special cases of this definition.
• For X = pt and a concentrated in degree 0 on a vector space g we have CE(a) = (∧• g, dg ) where ∧• g
is the Grassmann algebra on g∗ da is the Chevalley-Eilenberg differential uniquely corresponding to
the structure of a Lie algebra on g. L∞ -algebroids arising this we we write a = bg.
• For X = pt and a concentrated in arbitrary (non-positive) degree the above definition is that of an
L∞ -algebra structure (of finite type). For g any L∞ -algebra, we write a = bg for the L∞ -algebroid
corresponding to it.
• For X = pt and d at most co-binary (sending generators to wedge products of at most word length
2 in the generators) we have a dg-Lie algebra.
• For arbitrary X and a concentrated in degree 0 (being finitely generated and projective as a module
over C ∞ (X)) this is equivalent to the usual definition of Lie algebroids as vector bundles E → X with
anchor map [38] ρ : E → T X: we have g = Γ(E) and the anchor is encoded as dg |C ∞ (X) : f 7→ ρ(·)(g).
• If a is concentrated in degrees 0 through −(n − 1), then we speak of a Lie n-algebroid (Lie n-algebra
if X = ∗).
Example A.2 (line Lie n-algebra). For n ∈ N let bn R or equivalently bn u(1) be the Lie n-algebra defined
by the fact that its corresponding Lie n-algebroid bn+1 R has a Chevalley-Eilenberg algebra coming from a
single generator in degree (n + 1) with vanishing differential. We call this the line Lie n-algebra.
Example A.3 (tangent Lie algebroid). For X a smooth manifold, the de Rham complex Ω• (X) is the
Chevalley-Eilenberg algebra of a Lie 1-algebroid over X, called the tangent Lie algebroid T X, CE(T X) =
(Ω• (X), ddR ) .
34
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
For g an L∞ -algebra, we say that a closed element µ ∈ CE(g) of degree n + 1 is an (n + 1)-cocycle on
g. Cocycles are equivalently morphisms of L∞ -algebras µ : g → bn R or, equivalently, morphisms of L∞ algebroids µ : bg → bn+1 R.
Example A.4 (higher central extensions). Every cocycle µ on an L∞ -algebra g induces a new L∞ -algebra,
to be denoted gµ , which is defined by its CE-algebra being that of g with a single generator b in degree n
adjoined and the differential extended to this generator by the formula dgµ b = µ . This yields a sequence
(A.3)
bn R → bgµ → bg
exhibiting the shifted central extension classified by µ.
Example A.5. For g a semisimple Lie algebra and µ3 = h−, [−, −]i the canonical 3-cocycle, the Lie 2algebra gµ3 is called the corresponding String Lie 2-algebra [27][4]. For g = so there is also the canonical
7-cocycle µ7 ∈ CE(so). This is still a cocycle on soµ3 , too, and so there is a Lie 6-algebra soµ3 ,µ7 , called the
fivebrane Lie 6-algebra in [46].
For brevity we state several constructions only for L∞ -algebras. The generalization to general L∞ algebroids is immediate. In particular for g an L∞ -algebra we shall write CE(g) as shorthand for CE(bg).
Differential form data on a manifold X with values in an L∞ -algebra g is a graded algebra homomorphism
(not necessarily respecting the differentials) from CE(g) into the differential forms on X:
(A.4)
Ω• (X, g) := HomgrAlg (CE(g), Ω• (X)) .
The space of graded algebra homomorphisms is a subspace of the space of linear maps of graded vector
spaces from CE(g) to Ω• (X) and, since CE(g) is freely generated as a graded algebra and of finite type,
this is isomorphic to the space of grading preserving homomorphisms HomVect[Z] (g∗ , Ω• (X)) from the graded
vector space g∗ of dual generators to Ω• (X). By the usual relation in Vect[Z] for g of finite type, this is
isomorphic to the space of elements of total degree 1 in forms tensored with g:
(A.5)
Ω• (X, g) ≃ (Ω• (X) ⊗ g)0 .
(Recall that g is non-positively graded.)
If instead we consider the corresponding homomorphisms of dg-algebras from CE(g) into forms, we find
that respecting the differentials what deserves to be called flatness
Ω•flat (X, g) := HomdgAlg (CE(g), Ω• (X)) .
/ Ω• (X) ⊗ g realizes flat L∞ -algebra valued forms
The inclusion Ω•flat (X, g) = HomdgAlg (CE(g), Ω• (X))
as elements A ∈ Ω• (X) ⊗ g of forms of total degree 0 with the special property that they satisfy a flatness
constraint of the form
(A.6)
(A.7)
dA + ∂A + [A ∧ A] + [A ∧ A ∧ A] + · · · = 0 ,
where d and ∧ are the operations in A ∈ Ω• (X) ⊗ g and where [·, ·, · · · ] are the n-ary brackets in the L∞ algebra and ∂ is the differential in the chain complex g. For g a dg-Lie algebra, only the binary bracket is
present and A is an ordinary Maurer-Cartan element DA + [A ∧ A] = 0, where D = d + ∂. This equation of
course has a long and honorable history in various guises called a Maurer-Cartan equation.
We would like to describe also non-flat L∞ -algebra valued forms by homomorphisms of differential graded
algebras. This is accomplished by passing from the Chevalley-Eilenberg algebra CE(g) to the Weil algebra
W(g).
Definition A.6. For g an L∞ -algebra, let W(g) be the unique dg-algebra free on the underlying graded
vector space g∗ such that the canonical projection morphism W (g) → CE(g) is a dg-homomorphism.
Due to the freeness of W(g) we have an isomorphism
(A.8)
Ω• (X, g) = HomgrAlg (CE(g), Ω• (X)) ≃ HomdgAlg (W(g), Ω• (X)) .
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
35
Definition A.7. The ∞-groupoid of g-valued forms on a smooth manifold X is the Kan complex whose
k-simplices are g-valued forms Ω•si (X × ∆k ) ← W(g) : A on X × ∆k with sitting instants (becoming constant
perpendicularly towards the faces of the k-simplex) and fitting into commutative diagrams of dg-algebras of
the form
Ω•vert (X × ∆k ) o
O
Avert
CE(g)
O
,
(A.9)
•
A
k
Ω (X × ∆ ) o
O
?
Ω• (X) o
W(g)
O
?
inv(g)
where the vertical morphisms are the canonical ones. For more details on this definition see [21]. The
parameterized version of this construction leads to the smooth ∞-groupoid exp(g)conn defined in (3.11).
A.2. L∞ -algebra representations and section. The L∞ -analog of the notion of representations of
smooth ∞-groupoids as in (3.5) is the following.
Definition A.8 (representations of L∞ -algebroids). A representation of an L∞ -algebroid a over X on
a cochain complex V of finite rank (A := C ∞ (X))-modules is an L∞ -algebroid V //ρ a whose ChevalleyEilenberg algebra CE(V //ρ a) is an extension of CE(a) by ∧•SymA V0∗ V ∗
(A.10)
∧•SymA V0∗ V ∗ o o
i
CE(V //ρ a) o
? _ CE(a) .
0
This means that the differential is d|a∗ = da , d|V ∗ = dV ∗ + dρ , where dρ encodes the action of g on V .
Remark A.9. In roughly this latter form, the definition appears in [8], where it is called a superconnection.
Indeed, in cases where the L∞ -algebroid in question is similar to a tangent Lie algebroid of some space, its
representations behave like (flat) connections on that space. In the work of [1], for the special case of 1-Lie
algebroids, such representations are called representations up to homotopy.
For a = bg coming from an L∞ -algebra, the above notion of representation reproduces the notion of
sh-representations of [55] [35].
Example: ordinary adjoint representation. Let g be an ordinary Lie algebra with basis {ta } and
structure constants {C a bc }. Write { |{z}
ta } for the corresponding dual basis elements and { χa } for the
|{z}
deg=+1
deg=0
corresponding basis elements of Vg∗ . Then we have dρ χa = σ −1 (dg ta ) = σ −1 (− 21 C a bc tb ∧ tc ) = C a bc tb χc .
Definition A.10 (sections and covariant derivatives). Let g be an L∞ -algebra, and let V //ρ g be a representation of g, def. A.8.
36
HISHAM SATI, URS SCHREIBER, AND JIM STASHEFF
Then for A ∈ Ω • (X × ∆k , g) a k-morphism of g-valued form data on a smooth manifold X, def. A.7, we
say that a section of the associated V -connection is a choice of the dotted arrows in
(A.11)
(s,Avert )
w
Ω•vert (X × ∆k ) o
OO
CEρ (g, V )
OO e▲▲▲▲
▲▲▲
▲▲▲
3S
Avert
CE(g)
OO
Wρ (g, V )
O e▲▲▲
▲▲▲
(s,∇A s,A,FA )
▲▲▲
▲3 S
w
•
k o
(A,F
)
Ω (X × ∆ )
W(g)
A
O
O
w
?
Ω• (X) o
?
invρ (g, V )
f▲▲▲
▲▲▲
▲▲▲
▲3 S
?
inv(g) .
Here we say that s is the section itself whereas ∇A s is its covariant derivative.
Example A.11. Let g be an ordinary Lie algebra with Lie group G, let V be a vector space (a chain complex
concentrated in degree 0) and ρ an ordinary representation of g on V , let P be a principal G-bundle and
(A, FA ) an ordinary Cartan-Ehresmann connection on P . Then the dotted morphism in
(A.12)
CEρ (g, V )
e❑❑❑
❑❑❑
(s,Avert )
❑❑❑
❑3 S
x
Avert
Ω•vert (P ) o
CE(g)
is dual to a V -valued function on the total space of the bundle (not on base space!) s : P → V , which is
covariantly constant along the fibers in that the covariant derivative
(A.13)
∇A s := ds + (ρ ◦ A)s
vanishes when evaluated on vertical vectors, where (ρ ◦ A)s denotes the action of A on the section s using
the representation ρ. This means that s descends to a section of the associated vector bundle P ×G V . The
covariant derivative 1-form ∇A s of the section s is one component of the extension in the middle part of our
diagram
(A.14)
Wρ (g, V )
.
e❏❏
❏❏
❏❏
(s,∇A s,A,FA )
❏❏
❏2 R
y
•
o
Avert
Ω (P )
W(g)
The equation
(A.15)
∇A ∇A s = (ρ ◦ FA ) ∧ s
is the Bianchi identity for ∇A s. If s is everywhere non-vanishing, this says that the curvature FA of our
bundle is covariantly exact on P . In the case that g = u(1) it follows that FA is an exact 2-form on P and
the choice of the non-vanishing section amounts to a trivialization of the bundle.
TWISTED DIFFERENTIAL STRING AND FIVEBRANE STRUCTURES
37
In sections 3.2 and 3.3 we see twisted differential String-structures and twisted differential Fivebrane
structures as parameterized examples of this notion of L∞ -sections.
Acknowledgements
H. S. and U. S. would like to thank the Hausdorff Institute for Mathematics in Bonn for hospitality and the
organizers of the “Geometry and Physics” Trimester Program at HIM for the inspiring atmosphere during
the initial stages of this project. H.S. thanks Matthew Ando for useful discussions. U. S. also thanks the
Max-Planck institute for Mathematics in Bonn for hospitality later during this work; and the crew of the
nLab, where some of the material presented here was first exposed. This research is supported in parts
by the FQXi mini-grant “QFT and Nonabelian Differential Cohomology” and NSF Grant PHY-1102218.
H. S. thanks the Department of Mathematics at Hamburg University for hospitality during the writing of
this paper. J. S. would like to thank the Department of Mathematics of the University of Pennsylvania
for support of the Deformation Theory Seminar enabling the three authors to have at least one meeting in
person. The authors are indebted to the referee for many useful remarks and suggestions that led to major
improvements of the paper.
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Department of Mathematics, Yale University, New Haven, CT 06511
Current address: Department of Mathematics, University of Maryland, College Park, MD 20742
E-mail address: hsati@math.umd.edu
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, D–20146 Hamburg, Germany
Current address: Department of Mathematics, Utrecht University, 3508 TA Utrecht, The Netherlands
E-mail address: schreiber@math.uni-hamburg.de
Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, Philadelphia, PA 19104-6395
E-mail address: jds@math.upenn.edu