The Dirac Index on Manifolds and Loop Spaces: Part I

Edward Witten*
December 10, 1997

I Introduction
One mathematical approach to the proof of the Hirzebruch index theorem is through the heat
equation. One considers, for each t > 0, the operator e t
acting on the dierential forms on
a closed, oriented riemannian manifold Y . A direct spectral argument shows that the trace of
this operator (or more precisely the dierence of its trace on the self-dual forms and on the
antiself-dual forms) is independent of t and limits as t 7! 1 to the index of the manifold. On
the other hand, the limit as t 7! 0 of this operator can be computed in terms of local invariants,
curvature, of the manifold. One can even explicitly evaluate these local expressions, and in
this manner recover the result that the signature of the manifold is given by L-polynomials of
Hirzebruch. In a related context, that of equivariant signatures when there is a group action,
one sees a localization taking place in the Atiyah-Hirzebruch formula for the equivariant signa-
ture in terms of local data (characteristic classes) at the xed points. These localizations are
reminiscent of localizations in quantum eld theory, e.g., perturbation expansions around the
classical solutions of Lagrange's equations, and raise the question of whether there are quantum
eld theory arguments establishing these and similar results.
The answer to this question is yes, and it is our purpose here to show how to establish results
of a similar nature (for Dirac operators on a spin manifold) by doing a perturbation expansion of
an appropriate path integral or Hamiltonian eigenstates. The explanation of the quantum eld
* Lecture notes by John Morgan

1
theories that do indeed lead to index theorems for the Dirac operator on a spin manifold (an its
equivariant analogue) are the rst subject of these lectures. Once these classical cases are been
covered, we turn to more sophisticated applications of the same techniques. This involves the
index of Dirac operators on the loop spaces of spin manifolds. While the results on loop spaces
were rst discovered by the path integral approach that we take here, eventually, mathematical
proofs were also found for these results [Bott-Taubes].
Before we get into the details of the computations, one general comment is in order. So far the
perturbative expansions we have encountered have been around an isolated, non-degenerate crit-
ical point. That is to say the space of classical solutions (the space of minima of the Lagrangian)
is a single point and the Hessian (the quadratic term in the Lagrangian) is positive denite at
that point. The expansions that we encounter here will be of a more general type. The set of
classical solutions will be a compact, nite dimensional smooth manifold (or super-manifold)
and the quadratic term in the Lagrangian restricted to the normal bundle to the critical set is
positive denite. The same techniques will work for oscillatory integrals as long as the Hessian in
the normal directions is non-degenerate (the Chern-Simons functional in three dimensions gives
examples of this type). Of course, one could meet situations where the critical set is singular
and the Hessian has null-space which is not of constant rank. Such situations require techniques
that are not discussed here.
In the situation where the critical set is a super-manifold M and at each point of the critical
set the Hessian has null space equal to the tangent space of the critical set, the basic approach
is to apply the usual perturbative expansion using Feynman diagrams and regularization in the
normal direction where there is an isolated, non-degenerate critical point to obtain a result, and
then integrate this result over the critical set. Of course, it may well happen that the individual
expansions over the normal spaces have no intrinsic meaning and depend on choices but we
expect the resulting integral over M to be well-dened. We will also be computing only the
constant term in the expansions. These of course will depend only on the quadratic term in the
Lagrangian and will be written in terms of the determinant of the propagator for the Bosons
and Pfaan of the propagator of the fermions integrated over the normal directions.

2
II The Dirac Operator on a Spin Manifold
In this section we shall give a path integral derivation of the index theorem for the Dirac operator
on a spin manifold. Fix for this section Y a compact, oriented, 2n-dimensional spin manifold.

II.1 The Lagranian Formulation

Let R1;1 denote the super Euclidean space with one even variable and one odd variable; i.e.,
C 1(R1;1) = C 1 (R)
^ (R). We denote by t;  2 C 1 (R1;1) the natural even and odd variables,
respectively. We consider a quantum theory of maps
X: R1;1 ! Y;

where Y is a compact spin manifold of dimension 2n. The Lagrangian we take is

Z Z
1 1 g (X) dX DX J = 1 hX;
I
L= dtd I;J _ DX i;
R1;1 2 dt R1;1 2
where as usual gI;J denotes the metric tensor on Y and D is the vector eld
@ @
D = @ @t
on R1;1.
Let us recall some of the material from Problem 2 in Problem Set One. We can view X as a
map x from R1 to Y together with a section of the odd vector bundle (x TY ) over R1. We
write X = x(t) +  (t) with this understanding as to the meanings of x and . Then we can
re-write the Lagrangian as
Z
L= 1 hx(t)
_ + rt( (t)); (t) x(t)
_ idtd 1 : (1)
R1;1 2
If one prefers an expression in local coordinates, doing the -integration yields:
Z
L = 12 hx(t);
_ x(t)_ i hrt (t); (t)idt:
R1
The classical solutions to this Lagrangian form a supermanifold of maps X: R1;1 ! Y satis-
fying
rtDX = 0:
3
We can also write this equation as a pair of equations:

rt = 0
_ ) = rt(x);
F( ; x)( _
where F is the curvature of Y . Fixing a time t0 and evaluating at this time identies this
supermanifold with space of \initial conditions" at time t = t0. The initial conditions are simply
a pair { a tangent vector x_ 0 at some point x0 of Y and an element 0 in a copy of TYx0 considered
as an odd vector bundle. That is to say, the space of initial conditions is identied with the
odd vector bundle p (TY ) over TY , where p: TY ! Y is the natural projection mapping. The
symplectic structure on this space of classical solutions is given by (see Problem 2 of Problem
Set One)
! = !T  Y + 21 (r ; r ) 12 (F( ); ) ;
where the rst term is the canonical symplectic form on the cotangent bundle which is identied
with the tangent bundle using the riemannian metric on Y .

II.2 The Quantization

We wish to apply canonical quantization to this super-manifold of classical solutions. Were
the space of classical solutions simply TY with the usual symplectic structure, then canonical
quantization would yield the Hilbert space L2 (Y ) of complex-valued L2 -functions on Y . (More
precisely, we would get the space of L2 half-densities.) The functions on Y act by multiplication
and sections of T Y (i.e., vector elds) act by dierentiation. But in our case where the space
of classical solutions is a super-manifold, we must mix this canonical quantization of TY with a
quantization of the vector spaces which are the bers of the odd bundle p (TY ) ! TY .
Let us consider a real, even dimensional super vector space V which is purely odd with
respect to the Z=2Z-grading. Suppose further that V has a positive denite inner product and
an orientation and a spin structure. We quantize this space by taking as the Hilbert superspace
the complex spin bundle S(V ), with the Z=2Z-grading given by the usual decomposition of S(V )
into S + (V )  S (V ). The linear forms on V are quantized to operators on this space which act

4
by Cliord multiplication:
V 
S(V ) ! S(V ):
Now we must mix together the two constructions. We have the odd vector bundle p (TY )
over the symplectic manifold TY . The Hilbert space is the space of L2 (S(Y )), the space of
L2-sections of the spin bundle S(Y ), again tensored by the half-densities on Y . The functions
on Y act as usual by multiplication; the linear forms in TY act by covariant dierentiation, and
the linear forms on T Y act by Cliord multiplication. This space of sections of S(Y ) is a super
Hilbert space { the Z=2Z-grading is induced with the usual decomposition of spinors in even
dimensions S(Y ) = S + (Y )  S (Y ).
The Hamiltonian operator for this quantization is given by quantizing the Hamiltonian func-
tion h associated with the ow @=@t on the theory:
dh = i@=@t!:
In this case it is easy to see (cf, Problem 2 in Problem Set 1) that
h = 21 jx_ j2 :
In quantizing this function there is in general an ambiguity in the resulting second order operator
(adding zero and rst order operators), but one representative for it is the Laplacian r  r on
spinor elds. In our situation the vector eld
Q = @@ + @
@t
on R1;1 (whose square is @=@t) yields a vector eld acting on the theory. The Hamiltonian
function for this vector eld, when quantized, yields a naturally dened rst-order operator
whose square is the Laplacian r  r on spinor elds, up to lower order terms. This rst-order
operator is in fact the Dirac operator @= on the spinor bundle. Thus, it is fairly natural to use
the square of the Dirac operator on the spin bundle as the Hamiltonian operator. (Of course, all
the various operators we could use will have the same symbol and hence give the same answers
in the computations we do below.)
For any  > 0, let us denote by [0; ]1;1 the pre-image in R1;1 under the natural projection
mapping of the interval [0; ]  R. Quantize as above. The Hamiltonian operator associated

5
with innitessimal time evolution is given by
H = @=2
acting on the L2 -sections of spin bundle S(Y ) thought of as a super vector bundle over Y . Let
S 1;1 denote the supercircle of length  obtained by dividing R1;1 by a translation of distance .
The partition function for the resulting supersymmetric sigma model of maps of S 1;1 into Y is
the supertrace of the operator of time evolution from t = 0 to t = ; that is to say with


trs rmexp( H) = trs exp( =@2 ) (2)
acting on the L2 -sections of the spin bundle S(Y ). Of course, by the usual spectral analysis (as
alluded to int the introduction in the case of the heat equation proof of the HirzeCruch signature
theorem), we see that the supertrace of this operator is independent of  > 0 and as  7! 1 it
approaches



dim Ker @=2j (S + (Y )) dim Ker @=2 j (S (Y ))
which is the index of
@=+ : (S + (Y )) ! (S (Y )):
Our strategy is to compute a perturbative expansion for this partition function near  = 0. In
fact, as we mentioned above, we will only compute the constant term in this expansion.

II.3 The Path Integral

Now we wish to compute a power series expansion near  = 0 for the supertrace in Equation 2.
Of course, as was explaned in Fadde'ev's lectures, the basic connection between the Lagrangian
and Hamiltonian formulations is given by the Feynman-Kac formula. For the current application,
the formula must be worked out in the context of spinor elds instead of scalar elds. This was
done in detail in [Bismut]. In any event, the perturbative power series expansion that we shall
compute by path integrals over Feynman diagrams is in fact a power series that is asymptotic to
the above supertrace. We shall ignore all terms which are positive powers of , thus computing
the constant term in this power series. Since we know that the series is asymptotic to a constant
function, this will suce to compute the constant term in the series.

6
 Recall that we are considering the space of maps of the supercircle S 1;1 of length  into Y .
The super Lagrangian is
Z
L(X) = 1;1 dtd 1 21 gI;J (X) dX
I
dt DX J ;
S
The supermanifold of classical solutions is the supermanifold of maps of S 1;1 ! Y , which are
critical points for this Lagrangian. The underlying geometric manifold of this super-manifold
is the space of closed geodesics in Y . The supermanifold M of critical points that minimize
the Lagrangian is identied with TY ! Y , an odd vector bundle over the space of constants
maps from S 1 to Y . We shall do our perturbation expansion around M. Other critical points
give contributions which are exponentially small in . (In fact, one can prove directly that the
non-constant maps contribute zero to the nal answer.)
By Equation 1 we can re-write this path integral as
  I !
Z Z  J 
DxD exp dt gI;J dx dxJ
2 dt dt
ID
Dt :
x2Maps(S 1 ;Y ); 2 (x TY ) 0

Rescale by setting t0 = t and 0 = p1 . Doing this and then rewriting the equation
dropping the primes from the notation yields
Z  Z 1   J 
1
DxD exp  dt 2 dt dt g I;J dxI dxJ I D : (3)
x2Maps(S 1 ;Y ); 2x TY 0 Dt
This change of variables on the circle introduces a change in the measure of integration by a
formally innite constant. The regularization procedure will determine a nite value for this
constant which depends only on the dimension of Y . In fact it is of the form C 2n for some
universal constant. (Recall that the dimension of Y is 2n.) We shall not try to explicitly
evaluate this constant nor others like it which arise later in the argument, though it is possible
to do so. Thus, our nal answer must be taken up to a multiplicative constant of the form
C dimY .
As we have already remarked a minimum for the Lagrangian is a pair (x0 ; 0) where x0 2 Y
is regarded as the constant map of S 1 to Y and 0 (t) is a constant section of T(Y )x0 . We
expand to rst order around this point by taking x = x0 + a where a(t) 2 TYx0 satises
Z 1
a(t) = 0 (4)
0

7
and = 0+ where (t) 2 TY jx0 satises
Z 1
dt = 0: (5)
0
These variations describe the normal bundle to M at (x0 ; 0). To project this parametrization
of the normal bundle to a submanifold in the space of maps transverse to M we consider
x(s; t) = expx0 (sa(t))
as a map of S 1  ( ; ) into Y and
(s; t) = P(s;t) ( 0 + (t))
where Ps;t denotes parallel translation in TY from x0 along the path x(s0 ; t) as s0 runs from 0
to s. Thus, (s; t) 2 T Yx(s;t), so that x(s; t) +  (s; t) is a map from S 1;1 to Y .
Since we are at a critical point, the rst-order variation of the Lagranian with respect to a
and  is zero. We are interested in the quadratic term, which is all that is needed to compute the
constant term in the -expansion of the path integral in Equation 3. To compute the quadratic
term with respect to a and  in the Lagrangian we take the second derivative at a minimum
(x0; 0). We nd that the quadratic term is
 
I = 2 1 Z 1 dt ja_ j2 + h; _ i 1 I J R a K a_ L :
0 2 0 0 IJKL
As noted above the higher order terms in a and  in the action will contribute to positive powers
of  in the asymptotic expansion. In this expression the term _ is the ordinary derivative of 
with respect to t, which makes sense because for all t, the image (t) is contained in the vector
space T Yx0 . The term RIJKL is the curvature of Y . Notice that in this integral there is no
mixing in the quadratic terms between the a's and the 's. Thus, I is a sum I1 + I2 where
Z 1
1
I1 (x; ) = 2 ja_ j2 21 ha; 0I 0J RI;J a_ i
0
and
I2 = 2 1 Z 1h; _ i:
0
Hence, the path integral is a product
Z Z
Da exp ( I1 )
D exp ( I2 ) :
a 

8
Also notice that the  integral Z  
Dexp 1 h; _ i
 2
is independent of x0 ; a and , since this integral depends only on TYx0 with its inner product and
the isomorphism class of this nite dimensional positive denite inner product space depends
only on the dimension of Y . This path integral is purely quadratic and is identied with the
Pfaan of dtd acting on the Hilbert space of maps . This is a constant which depends only on
the dimension. In fact it is of the form C 2n for some universal constant C. Notice also that in
the regularization that we are performing, these purely quadratic integrals are independent of .
Thus, ignoring such multiplicative constants, to compute the constant term of the expansion
in , we must compute
Z  Z 1 
Da exp 21 ja_ j2 21 ha; 0I 0J RI;J a_ i
a 0
By integrating by parts from 0 to 1 we can re-write the integrand of the functional integral as
the exponential of
1 Z 1 hD (a); ai
B
2 0
where the propagator DB depends on (x0 ; 0) and is given by
2
DB (x0 ; 0) = dtd 2 12 0I 0J RI;J dtd :
Thus, modulo , the path integral over the normal space to the super manifold M of minima at
the point (x0 ; 0) of Ia gives
p
1
deta DB (x0 ; 0)
where the determinant is computed over the space of a for xed (x0 ; 0) and is regularized using
the -function regularization. The nal formula for the path integral is then
Z Z
DxD exp( Ia I )  C 2n p
1 (mod ): (6)
TY det DB (x; )
Here C is a universal constant (explicitly calculable given enough patience).
It is convenient to realize that this integral over the supermanifold TY can be re-interpreted
as an ordinary integral of a top dimensional form over the underlying manifoldY . To see this note
that, as explaned in Berstein's lectures, the C 1 -functions on TY are simply the C 1 -sections

9
of the bundle ^ (T  Y ) over Y with the usual Z=2Z-grading; that is to say the C 1 -functions
on T Y are exactly the C 1 -dierential forms on Y with the Z=2Z-grading being the degree
modulo two. Let f be such a C 1 -function and let ! be the corresponding dierential form on
Y . We have an orientation on Y . For a function f 2 C 1 (TY ) the integral
Z
f
TY
is simply the integral over the fundamental cycle of Y of the the corresponding form !. The point
is Ber(T Y ) and Ber(T  Y ) are naturally dual line bundles so that their product is canonically
trivial.
Thus, we have the path integral expression for the index of the Dirac operator:
Z
index(=@+ ) = (C 0)n p
1 ;
Y det DB (x; )
where the determinant is computed using the -function regularization, where C 0 is a universal
constant, and, as discussed above, the -dependence turns the integrand into a dierential form
(with compnents of various degrees).

II.4 Evaluation of the Determinant

Our next task is to compute this determinant as a dierential form. Let us x rst a element
g 2 so(T Yx0 ) and consider the operator
2
Dg = dtd 2 1 d
2 g dt
acting on the space of maps a: S 1 ! TYx0 which satisfy the constraint given in Equationi 4. We
can decompose T Yx0 into two-dimensional spaces invariant under g such that on the ith -space g
is given by the matrix !
0 xi
xi 0
with respect to an orthonormal basis. The determinant of Dg will be the product over the blocks
in this decomposition of the determinants associated to the individual two-dimensional pieces
associated with each block. This allows us to reduce to the two-dimensional case. (This step

10
is similar in spirit to the reduction step to riemann surfaces in some other proofs of the same
result.) Fix a two-dimensional subspace V of TYx0 on which g has the form
!
0 x
g= :
x 0
We use the metric and orientation of TYx0 to induce a complex stucture on V . With respect to
this structure g is simply multiplication by ix. We can decompose the complexication of the
space of maps of S 1 into V into two-dimensional spaces of the form exp(2ikt)a for a 2 V . Notice
that by Condition 4, this decomposition is over k 6= 0. It is easy to see that the eigenvalues of
Dg on the two-dimensional subspace indexed by k are
(2k)2  (2k)ix=2:
Thus,
p YYp
detDg = ((2k)2  i(2k)x=2):
 k6=0
Grouping together the four terms for each pair k; k we get
p Y1   x 2
detDg = (2k)4 1 + 4k :
k=1
As in Problem 2 of Problem Set Five, the -function regularization sets the innite factor
1
Y
(2k)4
k=1
equal to C2 for some universal constant C. Thus, the result is that
p 1   x 2 
Y

x=2
 1
detDg = C 2 1 + 4k = C 2 sinh(x=2) :
k=1
This computation was done under the assumption that TY was two-dimensional. In the general
case when T Y is a sum of two-dimensional spaces of the above form we nd
n   1
p
detDg = C 2n
Y xi=2 :
i=1 sinh(xi=2)
This power series can be re-written using the elementary symmetric functions of the xi. When
we do so by the denition of A^ we obtain
(detDg ) ^
1=2 = C 2nA(g):

11
 Now let us return to detDB 1=2 as a function on TY , or as we have seen as a dierential form
P
on Y . Consider the function I;J RI;J (x0) 0I 0J on TY with values in ad(TY ). As indicated
above this function is identied with a two-form on Y with values in ad(TY ). Not suprisingly
under this identication the sum becomes the curvature two-form R(Y ) of Y . Thus,
det(DB ) ^
1=2 = C 2nA(R(Y )):
We have now established that
Z
index (=@+ ) = C 2n ^
A(R(Y ))
Y
where C is a universal constant. Of course, the usual formula of Atiyah-Singer is
Z
index @=+ ) = ^
A(R(Y )):
Y
In this derivation we have been careless with universal constants depending only on dimension
(innite but regularized in some manner). To evaluate the constant C one can either follow
through the regularization procedure or simply evaluate it on one example where the index is
non-trivial.

III The Case of a Circle Action

In this section Y is a compact 2n-dimensional spin manifold with an isometry g: Y ! Y . We
suppose that this automorphism lifts to the bundle of spinors. We dene the g-index of the Dirac
operator to be
TrH+ g TrH g
where H are the plus and minus harmonic spinors. Equivalently, we can compute this as
trs (g exp( H)) ;
where H is the square of the Dirac operator. As before, there is a path integral computation of
a power series expansion in  aysmptotic to this supertrace at  = 0. The constant term in this
series computes this g-index. In fact, this procedure works for any isometry g which lifts to the
spin bundle, (see the rst problem in the nal exam for the Fall Semester). But here we shall

12
suppose that g lies in a connected, compact group of isometries, which is equivalent to assuming
that it lies in a circle of symmetries.
So we consider an action S 1  Y ! Y by isometries. At the expense of doubling the circle,
this action automatically lifts to the bundle of spinors. We take a Killing vector eld V for this
action (the vector eld generating the action), and denote by LV the Lie derivative of this vector
eld on the spinors. We consider
X
F() = trs (exp(iLV )exp( H)) = exp(in)indn@=+ ;
n2Z
where indn@=+ is the index of the Dirac operator from subspace H+n of plus spinors on which
the character of S 1 is the nth -power to the corresponding subspace Hn of the minus spinors. In
what follows we denote indn @=+ by an .
Our goal is to prove the theorem of Atiyah-Hirzebruch [AH];
Theorem III.0.1 If S1  Y ! Y is a non-trivial isometric circle action on a spin manifold Y
then indexn(=@)+ = an = 0 for all n.

III.1 Proof of Atiyah-Hirzebruch Theorem using the Hamiltonian For-

malism
Instead of computing this via path integrals as we did in the case of trivial action above, we
introduce a Hamiltonian approach. We perturb the Dirac operator to an operator depending on
a real parameter t dened by
@=t ( ) = @=( ) + tV

on spinors. (Here, V
is Cliord multiplication.) It is a formally self-adjoint operator. The
Hamiltonian is
Ht = (=@t )2 = (=@)2 + t2jV j2 + t (2rV + div(V ) + dV 
) ;
where V  is the one-form dual under the metric to V and the last term is Cliord multiplication
by dV  . Once again this is a formally self-adjoint operator. Since the circle action perserves the
metric, div(V ) = 0. Thus, we have
Ht = (=@t)2 = (=@)2 + t2jV j2 + t (2rV + dV 
) :

13
We also have a formula for the Lie derivative LV as follows

LV = rV + 41 dV 
( );
so that we can re-write the Hamiltonian as
 
Ht = (=@t )2 = (=@)2 + t2 jV j2 + t 2LV + 12 dV 
( ) :

Since LV generates a circle action on the spin bundle, its eigenvalues are integers. We denote
by Hn the eigenspace for LV of eigenvalue n in the Hilbert space of all L2-sections of the spin
bundle. The vector eld V generates a symmetry of H and in fact of Ht for every t 2 R. Hence,
LV commutes with Ht. The index an we are trying compute is simply the index of the restriction
of the operator dened by H (or Ht) from the invariant space H+n to the invariant subspace Hn ;
or put another way, using self-adjointness of Ht it is the dimension of the kernel of Ht on H+n
minus the dimension of the kernel of Ht on Hn . We shall prove the theorem by showing that if
tn  0 and jtj >> 1 then Ht is positive on Hn and in particular has no kernel on this subspace.
It follows immediately that an = 0.
For large jtj the spectrum of Ht restricted to the subspace Hn can be calculated in an
asymptotic expansion in powers of 1=jtj by expanding near the minima of the potential. These
minima are the zeros of V . We shall work rst in the case when the S 1 -action has isolated xed
points. We consider the more general case brie y at the end of this section. Let us consider an
isolated xed point  2 Y . We decompose the linearization of the S 1 -action at this xed point
into a sum of two-planes:
X
TY = Wj
j
where the restriction of V on Wj is given in orthonormal coordinates (x2j 1; x2j ) by
 
kj x2j 1 @x@ x2j @x @ ; kj 2 Z+ :
2j 2j 1
We use the orientation induced on Wj by these coordinates. The conditition that the S 1 -action
lifts to the spin bundle implies that
X
kj  0 (mod 2): (7)
j

14
 To leading approximation in 1=jtj the restriction of Ht near  is given by
!
2n
X @2 + Xn
2 k2(jx
n
2 + jx j2 ) t X k
1 0
Htloc() = 2 t j 2j 1 j 2j j + 2tLV :
j =1 @xj j =1 j =1 0 1
Ignoring the last term and splitting along the two-dimensional subspaces for the monent we
consider the operator
!
@ 2 @ 2 1 0
Htloc;j () = @x2 2 + t kj (jx2j 1j + jx2j j ) tkj
2 2 2 2 ; 1  j  n: (8)
2j 1 @x2j 0 1
With these denitions we have
X n
Ht () = Htloc;j () + 2tLV :
loc
j =1
Let us consider the spectrum of the harmonic oscillator which is the sum of all but the last
term in the expression for Htloc;j ():
@2 @ 2 + t2 k2(jx j2 + jx j2 ):
@x22j @x22j j 2j 1 2j
1
Its ground state
has eigenvalue 2jtjjkjj, and its spectrum is

f(4` + 2)jtjjkjg ` = 0; 1; 2;



with each eigenvalue being of multiplicity two.
The last term in Expansion 8 commutes with the harmonic ocsillator and has eigenvalues
tkj . Thus, it splits each of the two-dimensional eigenspaces into two one-dimensional spaces
with distinct eigenvalues. Thus, the lowest eigenvalue for Htloc;j () is jtjjkj j, and its spectrum is

f(2r + 1)jtjjkjjg; r = 0; 1; 2;


:
Though we do not need it now, it will be important later to understand how LV acts on
these eigenstates. A direct computation in these local coordinates shows that the eigenstate
of Htloc;j () with eigenvalue (2r + 1)kj jtj is an eigenstate for LV with eigenvalue (r + 1=2)kj .
Also notice that if n > 0 then KerHt \ Hn consists of plus spinors and if n < 0 it consists of
minus spinors.

15
 Summing up over the two-planes and using the denition of Hn , we see that the lowest
eigenvalue for Htloc () on Hn is
X
jtjjkj j + 2tn:
j
As jtj 7! 1, any element of norm one in the kernel of the global Hamiltonian operator must
localize around the xed points. In fact it must converge at each xed point an element in the
kernel of Htloc(). But we have already seen that for tn  0 the local operators are positive and
have no kernel. It follows that if t >> 0 and tn  0 there is no kernel for the restriction of Ht
to Hn .
This proves the Atiyah-Hirzebruch theorem stated above at least in the case when the circle
action has isolated xed points. The case of non-isolated xed points is brie y discussed at the
end of this section.
Notice that in proving this result for all n we had to use both t > 0 and t < 0. When we get
to the loop space setting in the next lecture, we will not be able to take t = 0 and hence we can
not pass from t > 0 to t < 0. The consequence of this is that we will only establish that half the
S 1 -indices vanish.

III.2 A local integral for the index of the Dirac operator

Even though we have established the Atiyah-Hirzebruch theorem that the an vanish, it is still
interesting to give a dierent computation of the an in terms of local data around the xed
points of the circle action. This computation is particularly relevant when we get to the case of
twisted Dirac operators where the equivariant indices do not all vanish.
P
The function F() = n an ein is computed as supertrace of the circle action on the kernel of
the operator Ht for any t. The convergence of the operators Ht to the sum of the local operators
Htloc for large jtj implies that in fact we can do the computation by using the kernels of the local
operators at the xed points. Let us x t < 0 and compute the kernel of Htloc (). Note that the
argument in the last subsection did not show that this kernel vanished, only that the restriction
of Ht to the Hn with tn  0 had no kernel.
As we have already seen, the spectrum of the operator Htloc;j () is f(2r + 1)kj jjtjg; r =
0; 1;


, with the eigenspaces of dimension one, generated by r , say. Furthermore, the vector

16
eld V is the innitessimal generator of an S 1 -action on the state =lambdar is given by

e(r+1=2)kj  :

Thus, V is the innitessimal generator of an S 1 -action whose character on the sum of the states
given by
kj i=2
ekj i=2 + e3kj i=2 +


= 1 e e ikj = 2i sin(k1 =2) : (9)
j
Nex we consider the sum of all possible tensor products over j of eigenspaces of the Htloc;j : The
vector eld V is the innitessimal generator of an S 1 -action on this sum of tensor products whose
character is given by n
F() = 2i sin(k1i=2) :
Y
(10)
j =1 j
Notice also that since t < 0, the computation above of the action of LV on the eigenstates of
the Htloc;j immediately implies taht all these tensor product states are in the kernel of our local
operator Htloc () and are plus spinors. It is clear that this accounts for the entire kernel of the
local operator. Notice that even though the kj are not necessarily even, all the exponents that
appear in the above product expansion are of the form

((2r1 + 1)k1 +


+ (2rd + 1)kd) =theta=2;

and by the spin condition (Equation 7) are integral multiples of . Hence, F= alpha() is a
character of an action of the original S 1 , not some covering of it. It is the character of the
action of the circle on the kernel of Htloc (). Thus, F () is the local contribution at  of the
S 1 -equivariant index.
Summing up over the various isolated xed points  gives
 n X n
F() = trs eiLV e H


= i ( 1)
Y 1 :
2  j =1 sin(k ;j =2)

where  ranges over the xed points of the circle action (still assumed to be isolated), and 
compares the orientation on TY coming from the ambient orientation of Y with the orientation
induced from the direct sum of the orientations on the various two-planes V;j . (The k;j ; j =
1; : : :n are the rotation numbers of the circle action at the xed point .)

17
 Of course, in this case we have shown that this sum is in fact zero. This is a non-trivial
statement about the nature of the S 1 -action at the xed points. (Recall that we are assuming
in the discussion so far that the S 1 -action has only isolated xed points.)

III.3 Dirac Operators with Values in a Bundle

If we pass from the case considered above of the pure Dirac operator to the case of a Dirac
operator twisted by a bundle with a connection, then the vanishing theorem of Atiyah-Hirzebruch
does not generalize. Nevertheless, the second argument computing the equivariant index in terms
of local data at the xed points does generalize. It produces the Atiyah-Bott formula [AB].
Let  ! Y be a vector bundle over a compact, even dimensional spin manifold. We consider
the Dirac operator twisted by this bundle. The local analysis around the xed points  involves
the two-plane bundles W;j at  as well as the restriction  j. The formula is (assuming that
the S 1 -action has only isolated xed points)
n
X
F () = indS 1 (=@
) = ( 1)
YY 1 Tr e iLV
:
 j 
 j j =1 sin(k;j =2)
If  is a trivial bundle then we are back to the case of the pure Dirac operator and we know
that the S 1 -index F () is identically 0. There is an important generalization of this result for a
certain class of spin bundles  ! Y . Let us consider the case when  = S(Y ), the spin bundle of
Y . While it is no longer true that F () = 0, it is true is that F () is a constant function. The
reason is that S(Y )
S(Y ) can be identied with the dierential forms and hence the kernel
spaces are identied with the harmonic forms. Of course, LV acts trivially on these spaces. One
could ask a more general question here: Let PY ! Y be the principal Spin(2n)-bundle of Y and
let R be (nite dimensional) representation of Spin(2n). We dene
 = P Spin(2n) R
and consider the character-valued index function F () as above. The question is: For which
W constructed in this manner is the function F () equal to a constant? We shall see in the
next lecture that there is an innite series of such representations R with this property. By
constructing enough examples Landweber and Stong [LS] showed that the innite series that we
construct includes all the representations with this property.

18
III.4 The Case of Non-isolated Fixed Points
Let us nish this section with a brief discussion of the case when the S 1 -action does not have
`
isolated xed points. Suppose that the xed point set is r Mr , with the Mr being connected.
Again we consider the deformation

@=t ( ) = @=( ) + tV
( ):

Exactly as before, taking jtj 7! 1 localizes the low eigenstates of Ht around the xed components
Mr of the circle action. The local Hamiltonian Htloc;Mr at the xed component Mr has harmonic
ocsillators in the normal direction, analogous to the one encountered when the xed points were
isolated, and is the square of the Dirac operator along Mr . The decomposition of TY into
two-dimensional spaces V;j , which occurred in the case of isolated xed points, is replaced in
this case by a decomposition of the normal bundle Vr to Mr as a direct sum of sub-bundles
Vr;j ! Mr of dimension 2dj , say, on which the circle action is given by the character which is
raising to the kjth -power. We orient the Vr;j so that kj > 0, hence inducing an orientation on Vr
and consequently on Mr . (If Mr is a point, then this orientation diers from the natural one on
the point by the sign ( 1) encountered before.) The argument given above generalizes easily
to this case to show that for tn  0 there is no kernel for Htloc;Mr on the subspace Hn. Thus, we
see in this case as well, that Ht has no kernel on Hn for jtj >> 1 and tn  0, establishing the
Atiyah-Hirzebruch theorem in the case of general circle actions.
One can also generalize the second argument to produce a formula in terms of xed point
data for the equivariant index. Once again it suces to consider jtj >> 1 and to nd the kernel
of Htloc;Mr . Let us suppose rst, for the purposes of illustration, that the component Mr is
of codimension two and is a spin submanifold. Let k be the character of the circle action in
the normal direction. Of course, our spin hypotheses imply that k  0 (mod 2). For t < 0
we nd, exactly as before, one state in the kernel of the normal operator restricted to Hn for
n = (2s+1)k=2 ; s = 0; 1; 2; : : :. Each of these states varies with the point of Mr and hence forms
a complex bundle over Mr . We then have line bundles L1 ; L3; : : : with L2s+1 having S 1 -character
(2s + 1)k=2. Let us relate the L2s+1 to the normal bundle Vr (which is a complex line bundle).
Since the eigenstate 's with character (2s + 1)k=2 is homogeneous of degree 2s + 1 in the spin

19
 p
bundle associated to Vr , it follows that L2s+1 = Vr
(2s+1) , the 2s + 1 tensor power of the
square root of the normal bundle of Mr . (The fact that Mr is spin implies that Vr has a square
root and even picks out one, so that these bundles are well-dened.) Thus, the spaces of vacua
in the normal direction t together to form a sum of S 1 -equivariant complex line bundles. The
local contribution around this component of the xed point set is given by a sum like the one
given by Equation 9 with ik=2 replaced by (x + ik)=2 where x = c1 (Vr ). Of course, along the
xed point set Mr , the operator Htloc;Mr is the square of the Dirac operator. The usual index
theorem for the Dirac operator then computes the local contribution at this component Mr to
the equivariant index to be Z
^ r) 1
A(M 1
Mr 2 sinh((x + ik)=2) :
If Mr is codimension-two but not spin, then two things breakdown in the above argument:
There are no spin bundles on Mr and hence no Dirac operator on Mr , and we cannot form the
square root line bundle to Vr . Nevertheless, the description of the kernel states in the normal
direction is still valid locally. What happens is that the obstruction to dening the spin bundles
and Dirac operator globally on Mr is exactly cancelled by the obstruction to forming the square
root line bundle to Vr . Thus, the local descriptions t together to dene an operator which is
locally described as a Dirac operator with values in a line bundle but is globally twisted. The
index theorem still applies in this twisted situation and gives the same formula. (Though of
course, x=2 is only a rational cohomology class.)
More generally, when the normal bundle Vr to Mr splits as a sum of complex line bundles
invariant under the S 1 -action, the space of vacua is a sum of terms which are tensor products
of the above type, and the normal contribution is a product of terms of the above type:
Y e(xj +ikj )=2 Y 1
1 exj +ikj  = 2sinh((xj + ikj )=2) ;
j j

where xj is the rst Chern class of the j th -line bundle and kj is the weight of the S 1 -action
on this line bundle. In the more general case when Vr does not split as a sum of complex
line bundles invariant under the S 1 -action case we get the expression for normal contribution is
obtained using the splitting principal. Suppose that the normal bundle Vr to Mr splits as j Vr;j
as described before. Recall that dj = dimC (Vr;j ). The resulting S 1 -equivariant Chern character

20
is dj
A^ (Vr ) = 2 dim(Vr )=2 Y Y 1
j `=1 sinh((x `;j + ikj )=2)
with the x`;j being the formal roots whose elementary symmetric functions are the Chern classes
of Vr;j . (Of course, as always, this expression must be re-written in terms of the elementary
symmetric functions in order to nd the expression in terms of the Chern classes of Vr;j .) This
describes the bundle of vacua over Mr for the operator Htloc;Mr . Of course, so far we have
been describing this operator only in the normal directions to Mr . Along Mr it is the square
of the usual Dirac operator. Thus, the index of Htloc;Mr is simply that of the Dirac operator
with values in the bundle of vacua in the normal direction. (As before, if Mr is not a spin
submanifold, then neither the spin bundles along Mr nor the spin bundle in the normal direction
can be independently dened, but the index formula works exactly as if they could. The usual
index theorem now evaluates the S 1 -index of this operator to be
Z
FMr () = ^ r )A^ (Vr ):
A(M
Mr
Adding up the above expression for the local S 1 -indices over all the components of the xed
point set yields the following formula for the S 1 -index of H:
XZ
F() = ^ r )A^ (Vr ):
A(M (11)
r Mr
This formula together with the Atityah-Hirzebruch theorem that the equivariant signature
vanishes can then be viewed as giving non-trivial information about the nature of the xed points
submanifolds of the S 1 -action.
More generally, if we are considering a Dirac operator coupled to a connection on an S 1 -vector
bundle  ! Y then we replace Vr in Equation 11 by


(j (Vr;j )
b eibr;b
where
 jMr = b r;b
and the character of the circle on r;b is the bth -power. With this substitution for Vr , the same
equation gives the index of the coupled Dirac operator as a function on S 1 .

21
IV  -models in 1 + 1 dimensions
Let us study in a rough qualitative way the case of maps of a two-dimensional supermanifold
into a compact manifold Y of dimension 2n. We shall consider several contexts, but always with
an action given by the same formula. First, let us consider the super-manifold
Maps(R2;2; Y );
with the action given by
Z
L= dudv(d+ ) 1 (d ) 1 gI;J D+ X I D X J
R2;2
where the even coordinates on R2;2 are (u; v), the odd coordinates are + ;  , and
D+ = @@ + @u @
+

D = @@  @v @

are vector elds on R2;2. Notice that a map of X: R2;2 ! Y can be written as
x+ + + +  + F+ 
wher x is a map x: R2 ! Y , + ; are sections of the pullback x TY of the tangent bundle
of Y , considered as an odd bundle and a section F of xTY considered as an even bundle. For
any superriemann surface  (supermanifold of type (2; 2)) we have this action, but invariantly
the  are sections of S  ()
x TY . With this identication, we can re-write the action as
Z
 I J K L + g F IF J  :
L= dudv gI;J @u X I @v X J + i + @v ++i @u + RI;J;K;L + + I;J
R 2

Since the eld F enters the Lagrangian purely quadratically and only algebraically (no derivatives
of F appear), it is easy to integrate out the F dependence. When we do this gaussian integration
in F we are left with the constraint F = 0 and the same action (without the jF j2-term to integrate
over the space of elds x;  .
We shall study to some degree this Lagranian formulation of this supersymmetric -model,
but we shall also need a Hamiltonian formulation. In the Hamiltonian formulation we study a
quantum theory of maps of the super cylinder R1=S  R1;2 into Y where S is translation by a

22
xed amount. Equivalently, we consider the space of maps of R1;2 into the loop space LY of Y .
Letting t be the geometric variable in R1;2 and letting s be the loop variable in LY , we identify
t = u + v and s = u v. Under this identication the action above becomes a ow equation in
this context: it is not the usual analogue of the geodesic ow equation for R1;2 in LY since there
are extra potential terms. Nevertheless, the space of classical solutions is given by the initial
conditions at t = 0, where as before F = 0 is a constraint. That is to say, the space of classical
solutions is identied with the odd vector bundle T LY  T LY over T LY which is two copies
of the tangent bundle to the loop space. Quantizing this bundle yields the super-Hilbert space
of sections of Sym T(LY ) which should be viewed as the dierential forms on LY .
There is one important feature we wish to discuss. There is a Z=2Z  Z=2Z-symmetry of the
Lagrangian given by ! !
+ +

7!
and ! !
+ +

7! :
We wish to discuss whether these symmetries lift to the quantum eld theory.
Before addressing this problem directly, let us consider the analogous question for the space of
maps of R1;1 into a nite, even dimensional spin manifold Z. Quantizing the odd vector bundle
T Z over an even dimensional spin manifold Z yields the Hilbert space of L2 -sections of the spin
bundle S(Z), and the eld (analogous to say + above) acts by Cliord multiplication. The
analogue of the symmetry  above is then implemented on this Hilbert space by the involution
 whose 1-eigenspaces are the sections of S  (Z). By this we mean
Cl( + ) =   Cl( + )  ;

where Cl() denotes Cliord multiplication by . Now in the case of maps of R1;2 ! Z one can
implement both of the symmetries  and  using the two copies of the spin bundle associated to
the elds  . Thus, it is clear that in this nite dimensional analogue the symmetries  and 
lift to the quantum level when Z is an even dimensional spin manifold.
Let consider the case of mapping R1;2 into an nite, even dimensional manifold Z, no longer
assumed to be spin. Even though the manifold is not spin, as we indicated above, we can

23
quantize the direct sum of the two copies of TZ resulting in the dierential forms on Z with
their mod 2 grading. (Of course, if Z is spin we got the space of L2 -sections of the tensor product
of S(Z)
S(Z), but this is naturally identied with the space of L2 dierential forms on Z.)
With this quantization, it turns out that, provided that Z is oriented, it is still the case that the
symmetries  and  lift to the realization. They are given by

( 1)deg = 

and
 = ;
where  is the Hodge -operator multiplied by a factor of ideg in order to make it an involution.
Thus, in this nite dimensional analogue, the Z=2Z  Z=2Z-symmetry group lifts to the quantum
theory of maps of R1;2 into Z as long as Z is orientable.
We claim that the answer in the innite dimensional case is the same, with a fairly natural
denition of the notion of LY being orientable. We shall examine this question via path integrals
considering the theory of all maps of super riemann surfaces  (supermanifolds of type (2; 2))
into Y . Since  and  are symmetries of the Lagrangian, the question of whether they lift is
the same as asking whether the symmetries are symmetries of the formal measure of integration
DX D + D . It turns out that just as in the nite dimensional case  always lifts to a
symmetry of the quantum theory. Let us consider . First, we claim that  is a symmetry
of the measure of integration if and only if dim KerD+  0 (mod 2) for all maps x:  ! Y
for any riemann surface . The reason is that  clearly preserves the DX D and  will be a
symmetry of D+ provided that the space of sections of the pullback bundle S + ()
x TY is
even dimensional. The idea is that  is acting by 1 on these elds and thus is preserving the
orthogonal form. The issue is whether formally  preserves the orientation. (In nite dimensions
1 is a symmetry of the integration measure if and only if it is orientation-preserving which is
equivalent to the statement that the space is even-dimenional.) Of course, in our context the
space is innite dimensiona, so in order to determine its dimension we have to regularize it. The
way this is done is the following. We have a skew-symmetric pairing on this bundle of sections

24
given by Z
h1 ; 2i = h1 ; D+ (2)i:

Here D+ is the dirac operator on S + () coupled with covariant dierentiation on the pullback

x T Y . This pairing has a nite dimensional null space which is exactly the kernel of the Dirac
operator D+ . Given a skew-pairing A on a nite dimensional vector space V we know that
dim(V )  dim(Ker(A)) (mod 2). In the innite dimensional theory that we are studying here,
this is true for the pairing given above. Thus, we see that for the dimension of the space of
sections of S + ()
x (TY ) modulo two we use the dimension of the kernel of D+ . Since TY
real this coupled Dirac operator is skew-adjoint. Its spectrum is purely imaginary and symmetric
about zero. Thus, its algebraic spectral ow is zero, and the dimension of its kernel modulo two
is a topological invariant (unchanged under deformation of the operators through ellipic skew-
adjoint operators). In fact, there is a mod two index theorem for the dimension of the kernel.
Since T Y is even dimensional, the mod two index theorem identies the dimension of the kernel
with w2 (x T Y ). Thus, we see that  will lift to an involution on the quantum eld theory for
all maps of all superriemann surfaces  into Y if and only if w2(TY ) = 0.
At least in the case when 1(Y ) = f1g, it is natural to view the condition that w2(Y ) = 0
as the statement that the loop space L(Y ) is orientable. (Of course, in analogy with nite
dimensions, we do not expect to implement the Hodge star operator unless the manifold in
question is orientable. Or put another way, we are attemping to use this theory of maps of R1;2
into LY to compute equivariant signatures of the tautological circle action on LY . In order
for these signatures to be dened, we must assume that LY is orientable.) Let us explain how
w2(Y ) = 0 should be viewed as equivalent to the orientability of LY . The tangent space of L(Y )
at a map x: S 1 ! Y is the space of sections of x (TY ). This vector bundle has a natural skew
pairing on it given by Z 1
!x (; ) = h;  i = (; r )d
0
where the inner product on the right-hand-side is induced from the metric on Y . It is clear
that the radical of this pairing is the space of covariantly constant sections of x(TY ). In
particular, the radical of this pairing is always nite-dimensional. As before with the coupled
Dirac operator, it is easy to see that the spectrum of the skew-adjoint operator associated to this

25
pairing is discrete with all eigenvalues of nite multiplicity. Furthermore, the eigenvalues are
purely imaginary and symmetric with repect to the origin. Given a closed path xt in L(Y ), then
a local orientation of L(Y ) at the initial point x0 will be preserved around the loop if and only
if the spectral ow of the family Txt from negative to positive is even. (By skew symmetry the
full spectral ow is algebraically zero: here we are looking at half of it { the half from negative
to positive.) This spectral ow modulo two is easily identied with w2 (Y ) evaluated on map of
the torus S 1  S 1 into Y given by (t; s) 7! xt(s). Thus, provided that Y is simply connected,
we have that w2 (Y ) = 0 if and only if LY is orientable if and only if the kernel of D+ is even
dimensional for every map of a superriemann surface into Y .
If w2(Y ) 6= 0 this doesn't mean that we can not perform path integrals for maps X with

X w2 (Y ) 6= 0. It simply means that the results of such path integrals will be odd under ,
as the measure is. To get something non-zero we need an operator which is a product of local
operators O1 (x1)


On(xn ); xi 2 , which is  invariant but  anti-invariant. Then we can
calculate the expectation value
hO1 (x1 )


On (xn)i:
The symmetry  tells us that we get zero contribution at the minima (constant maps) and the
non-zero contributions come only from maps x:  ! Y with the property that x (w2(Y )) 6= 0.
Fix a homotopy class of maps with this property. Of course, since the maps must be homotopi-
cally non-trivial, we see that Z
jdxj2

is bounded away from zero. Furthermore, the maps minimizing this integral are exactly the
harmonic maps in the given homotopy class. That is to say we need to nd minimal area
surfaces S  Y and then take holomorphic maps  ! S  Y . We consider  = CP 1 , or better
delete a point and consider  = R2. Thus we are considering maps X: R2 ! Y which converge
at innity to a constant map. The expectation value

hO1 (x1)


On(xn )i

will be of the order exp( tM) where M is the energy of the harmonic map. (Notice that we
have not normalized this expectation value by dividing by the partition function Z. If we were

26
to do that we must take the partition function which is the sum over all maps x:  ! Y not
just those that pullback w2 non-trivially. The leading order term in this partition function will
come from the constant maps.) The critical points of the Lagrangian will not be isolated: Even
with the boundary condition at innity of R2 there are four obvious parameters for such maps:
rotation, translation and scaling of R2 . There may well be others in addition to these universal
ones. Thus, we will be computing an expression of the form
Z
(perturbation expansion)
M
where M is the instanton moduli space, i.e., the space of harmonic maps in the given homotopy
class. This time the propagator is more complicated and the linearization around the instanton
moduli space gives only an asymptotic expansion.

27