RIEMANN-ROCH, HEAT KERNELS, AND SUPERSYMMETRY

RIEANN-ROCH, HEAT KERNELS, AND SUPERSYETRY SEAN POHORENCE Introduction This talk will serve as a microscopic introduction to the heat kernel approach to index theory. Instead of diving into this deep subject head first, we will seek to understand a more basic result, the Riemann-Roch theorem for Riemann surfaces, using the theory of heat kernels. Our approach will highlight contributions to this subject from physics, particularly the theory of supersymmetry. 1 In order to cover the material in a couple of one hour sessions, the style of the talk will be more admiring from a distance, but most of the material can be found in [2] which goes into considerably more ail. 1. Hermitian vector bundles and Dirac operators We first describe our objects of interest. Let be a compact, complex manifold of (complex dimension m. Recall that a Kähler structure on consists of a Hermitian metric g on T C := T R C for which the complex structure is covariant constant. That is, if g is the Levi-Civita connection associated to g and J is the complex structure on, then g J = 0. Exercise 1.0.1. For any Hermitian manifold, the curvature tensor is an End(T C -valued two form. Prove that for Kähler manifolds 1 the endomorphism part of the curvature tensor respects the decomposition T C = T 1,0 T 0,1, and 2 the form part of the curvature tensor lies in A 1,1 (. This exercise tells us that the curvature tensor of a Kähler manifold lies in A 1,1 (, End(T 1,0 End(T 0,1. Using this decomposition, we may write the curvature tensor as R = R + R. We will also want to consider more general Hermitian vector bundles. Let E be a Hermitian vector bundle with Hermitian metric h. In a previous talk, we saw that a holomorphic structure on E is given by a choice of partial connection E which behaves like the (0, 1-part of a covariant derivative and squares to zero. These two pieces of data uniquely ermine a connection h on E satisfying the following two conditions: ( h 0,1 = E, dh(s 1, s 2 = h( h s 1, s 2 + h(s 1, h s 2 where s 1, s 2 Γ(, E. The connection h is called the Chern connection of E. Exercise 1.0.2. Prove that such a h exists and is unique. We will denote the curvature of h by F E. This is an endomorphism valued 2-form. One can show that the de-rham cohomology class of tr(f E is independent of the choice of Hermitian metric h, somewhat justifying the notation. ore complicated (inhomogeneous differential forms built from F E will also define cohomology classes which do not depend on h, as we will see later. Date: 8/17. 1 Sadly, we will neglect the equally fascinating ties to probability and stochastic calculus. 1

2 SEAN POHORENCE By extending E as a derivation on forms, we can construct the Dolbeault complex 0 A 0,0 (, E E A 0,1 (, E E E A 0,m (, E 0. The cohomology of (A 0, (, E, E is called the Dolbeault cohomology of E. We define the Euler characteristic of E, denoted χ(e as the alternating sum of the dimensions of the Dolbeault cohomology spaces m χ(e := ( 1 i dim(h i (A 0, (, E, E. We now discuss a different viewpoint on the Dolbeault complex which will serve as our entry point into index theory. Define the holomorphic vector bundle E := (T 0,1 E where m (T 0,1 = i (T 0,1. We see that sections of E are objects we already know, Γ(, E = A 0, (, E. The new feature we now point out is that E has a natural Z/2Z grading coming from differential form degree (mod 2. Specifically, E = E + E where E + := i (T 0,1 E, i even E := i (T 0,1 E. i odd We call the summands E + and E the even and odd parts of E, respectively. This gives Γ(, E the structure of a Z/2Z-graded vector space, or super vector space. With respect to this grading, E defines an odd endomorphism of Γ(, E E : Γ(, E ± Γ(, E. Note that E inherits a Hermitian metric built from the ones on and E. Using this Hermitian metric, we can define an conjugate-linear pairing on sections of E via integration: (s 1, s 2 := s 1 (x, s 2 (x x dx where, x denotes the Hermitian metric on E x and dx is a volume form on which we have chosen. Using this pairing, we can define the formal adjoint 2 of E by ( Es 1, s 2 = (s 1, E s 2 for any sections s 1, s 2 Γ(, E. We see that E is also an odd endomorphism of Γ(, E. Exercise 1.0.3. Show that ( E 2 = 0. We will mainly be intersted in the composite operator D := 2( E + E. This too is an odd endomorphism of Γ(, E. Using the direct sum decomposition of E, we may write D in matrix form as ( 0 D D = D + 0 2 We will not pay close attention to the functional analytic ails of this procedure.

RIEANN-ROCH, HEAT KERNELS, AND SUPERSYETRY 3 where D ± : Γ(, E ± Γ(, E. Exercise 1.0.4. Show that (at least in the formal sense D is self-adjoint, so that (D + = D. Computing the square of D using the definition, we see that D 2 = 2( E E + E E = 2 E where E is the generalized Laplacian associated to the Dolbeault operator. In particular, D is a Dirac operator (see [2] for ails. Exercise 1.0.5. This exercise is for those who have seen Dirac operators in physics and are wondering what the connection to D as defined above is. Define an action c : A 1 ( Γ(, End(E by where v 1,0 is defined via α 1,0 = g(v 1,0,. Show that c(αs = 2(α 0,1 s ι(v 1,0 s c(αc(βs + c(βc(αs = 2g (α, β s where g is the induced Hermitian metric on T. Thus, c factors through an action map c : C( Γ(, End(E, where C( is the bundle of Clifford algebras over with fibers C( x = Cliff(T x, g x. We say that E inherits the structure of a Clifford module. Now, define a connection on E by = g h. Finally, define an endomorphism of Γ(, E locally by / = c(dx i i. and show that / = D as defined above. For ails and hints, see chapter 3 of [2]. We define the index of D to be ind(d := dim(ker(d + dim(ker(d. For those of you inclined to super-thinking, we can define the index of D as the super-dimension of ker(d. We will now focus on finding ways to compute ind(d. 2. The heat kernel and its supertrace In this section, we introduce the heat kernel associated to D 2 and relate it to the index of D. We will not go through the construction of the heat kernel itself, since this is quite involved (at the very least, it requires understanding the spectrum of D 2. The general idea of the heat kernel is that it allows us to construct solutions to the heat equation ( t + D 2 s = 0 where s is a t-dependent section of E satisfying an initial condition s t=0 = s 0 Γ(, E. If we could construct a section p t Γ(, E E which satisfies the heat equation itself in the first argument, ( t + D 2 xp t (x, = 0, and gave a reproducing operator in the t 0 limit, p t (x, yr(ydy = r(x lim t 0 for any section r, then we could solve the heat equation for any initial condition s 0 Γ(, E by setting s(t, x := p t (x, ys 0 (ydy.

4 SEAN POHORENCE There are other subtler conditions on the regularity p t which we shall ignore, since for the operator D 2 it turns out that such a p t exists which is smooth in both arguments, as well as in t for t > 0. We define a t-dependent operator 3 e td2 Γ(, End(E associated to the heat kernel by (e td2 s(x = p t (x, ys(ydy. We now state three of the most important properties of p t for our upcoming discussion. The first property is that the operators e td2 for various values of t form a semigroup, that is, e t1d2 e t2d2 = e (t1+t2d2. The second property is that, like D 2, e td2 is essentially self-adjoint. The third property is that the operator e td2 is trace class for all t > 0, with trace given by tr(e td2 = tr(p t (x, xdx. Note that in the equation above, the trace under the integral is the pointwise trace of the endomorphism p t (x, x Γ(, E E = Γ(, End(E where we have used the metric on E to associate E and its dual. Exercise 2.0.6. Use the first and second properties of the heat kernel to prove the trace formula above. We now introduce the notion of a supertrace. Let V = V + V be a super vector space and let O be an endomorphism of V. In matrix notation, we have ( a O = + b where a ± End(V ±, b ± Hom(V ±, V. We define the supertrace as b + a Str(O = tr V +(a + tr V (a. In particular, the supertrace of any odd endomorphism vanishes. The supertrace satisfies most of the usual identities of the trace, the most important being (in finite dimensions Str([O, P ] = 0 where [, ] is the graded commutator. This identity holds for arbitrary vector spaces if one of the operators is nice enough (since e td2 is defined by a smooth kernel, it will usually be nice enough. As our first exercise in working with the supertrace, consider the projection operator onto the kernel of D, denoted P 0. We see that Str(P 0 = dim(ker(d + dim(ker(d = ind(d. Furthermore, since D is self-adjoint, we see that (D 2 s, s = (Ds, Ds = Ds 2. Thus, we see that ker(d 2 ker(d, hence ker(d 2 = ker(d. In summary, we see that P 0 is the projection onto ker(d 2 and that its supertrace is equal to the index of D. Our next goal will be to understand the supertrace of e td2. 2.1. Large t behavior of Str(e td2. In order to understand the behavior of e td2 as t, we will invoke the spectral theorem for essentially self-adjoint operators. Define H and H ± be the spaces of L 2 sections of E and E ±, respectively. For any non-negative real number λ, let n λ be the dimension of the λ-eigenspace of D 2. Furthermore, let n ± λ be the dimension of the intersection of this eigenspace with H±. This allows us to write the supertrace, for t > 0, as Str(e td2 = λ 0(n + λ n λ e tλ. 3 The notation here is meant to be sugestive, but is not meaningful without the following definition.

RIEANN-ROCH, HEAT KERNELS, AND SUPERSYETRY 5 It follows that lim t Str(e td2 = n + 0 n 0 = dim(ker(d+ dim(ker(d = ind(d. This result provides us with our first method for computing ind(d. Before moving on to the analysis of the small t behavior of this supertrace, we use Hodge theory to relate ind(d to an invariant of E. Proposition 2.1.1. The projection P 0 defines a quasi-isomorphism (A 0, (, E, E = (ker(d, 0. Note that this result is a consequence of the Hodge decomposition theorem for E. Since we have already taken the existence of the heat kernel for D 2 for granted, we may use this to construct a simple proof. Proof. Using the heat kernel, we may construct the Green s operator for D 2. That is, an operator G which satisfies D 2 G = 1 P 0. and super-commutes with D. Hence [D, DG] = 1 P 0. Thus, DG provides a chain homotopy between P 0 and the identity. Corollary 2.1.2. ind(d = χ(e. The corollary follows immediately using the definition of χ(e given above. 2.2. Small t behavior of Str(e td2. The behavior of e td2 for small t is more difficult to understand. To see why, let s step back and consider the heat kernel on R for the second derivative operator. This heat kernel is easier to construct explicitly. It is given by the Gaussian function p Euc t (x, y = (4πt 1/2 e (x y2 /4t. We see that as t the exponential factor controls the behavior of p Euc t which decays quickly to zero. However, as t 0 the heat kernel develops a singularity at x = y and can only be treated as a distribution. Indeed, we see that exp( t 2 x tends to the reproducing operator for smooth functions, which is δ(x y. 2 In our more general setting, the heat kernel also develops singularities as t 0. In fact, we have the following result (see chapters 2 and 4 of [2]. Proposition 2.2.1. For D and e td2 where p i (x Γ(, End(E. as defined above, p t (x, x has an small t asymptotic expansion p t (x, x (4πt m t i p i (x In particular, we see that p t (x, x develops singularities for small t if p i (x 0 for i = 0,..., m 1. Here is where taking the supertrace swoops in to save the day. Proposition 2.2.2. Str(p i (x = 0 for all x and i = 0,..., m 1. This is stated in chapter 4 of [2] as a consequence of Theorem 4.1. This shows that Str(e td2 has an asymptotic expansion without t = 0 singularities Str(e td2 = Str(p t (x, xdx (4πt m t i Str(p i (xdx. i=m This tells us that the supertrace Str(e td2 should have well defined limits for both large (which we have seen already and small t. Surprisingly, these two limits turn out to be equal.

6 SEAN POHORENCE Theorem 2.2.3. For all t > 0, we have d dt Str(e td2 = 0. In particular, for all t > 0. Str(e td2 = ind(d This result is known as the ckean-singer formula. Combined with the small t asymptotic expansion we have obtained, it shows that ind(d = (4π m Str(p m (xdx, where the left and right hand sides of the equation are the large and small time limits of Str(e td2, respectively. This is the key idea of the heat kernel proof of the Atiyah-Singer index theorem, which generalizes our discussion here. While Theorem 2.2.3 was originally proved by mathematicians, physicists were able to provide intuition behind the result (see [3], [1]. In particular, the somewhat miraculous looking cancelation that must occur in order for the supertrace to only depend on the contribution from the zero modes of D 2 can be explained using supersymmetry. We will discuss these ideas in section 3. Before doing so, we take a moment to better understand the supertrace of the coefficient p m (x in the small t asymptotic expansion. There are a number of ways to compute Str(p m (x. For instance, in [1] this is done using standard perturbation theory techniques for computing path integrals in quantum field theory. Here, we will state the result which is proved in chapter 4 of [2] and try to give some evidence of where this answer comes from. Theorem 2.2.4. The 2m-form Str(p m (xdx is equal to the 2m-form component of the inhomogeneous form give by the expression ( R ( ( 2i m + tr ( exp( F E. e R+ 1 Some explanation is needed. In general, if we are given a formal power series a(z = a i z i then we can define the expression a(f, where F is the curvature associated to a vector bundle over equipped with a connection, by the same power series a(f := a i F i. Since F is a 2-form, sufficiently high powers of F will vanish and the sum above is finite. This defines an inhomogeneous endomorphism-valued differential form. We can then use the well known power series expressions for the functions e z z and e z 1 to define the factors in formula ( as tr ( exp( F E ( (F E i = tr, i! ( ( R + ( 1 i B i = (R + i, e R+ 1 i! where B i is the i-th Bernoulli number. In particular, if is a Riemann surface and E is a line bundle, then R + and F E are just (1, 1-forms on and the two series above terminate after the linear terms. In this case, we have ( R + e R+ 1 (exp( F E = (1 R+ (1 F E 2

RIEANN-ROCH, HEAT KERNELS, AND SUPERSYETRY 7 so our formula for the index of D becomes ind(d = (4π 1 Str(p m (xdx = 1 ( R + 2πi 2 + F E. We can recognize both summands in the integral as topological invariants of and E. The (normalized integral of the trace of the curvature of E gives the first Chern number, or degree, of E: ( F E deg(e =. 2πi Additionally, realizing that in the case of a Riemann surface, the full curvature tensor R + is equal (up to a factor of i to the Ricci form ir + = Ric(g and recalling the Gauss-Bonnet theorem, we see that 1 ir + = 1 2π 2 2π Ric(g 2 = g 1 where g is the genus of. Combining these gives ind(d = (4π 1 Str(p m (xdx = deg(e + 1 g. Thus, a corollary of Theorem 2.2.3 for this special case is the classical Riemann-Roch theorem: χ(l = deg(l + 1 g where L Σ is a holomorphic line bundle over a compact Riemann surface of genus g. We now take a brief moment to give a clue as to where the result of Theorem 2.2.4 comes from. We begin with an exercise. ( Exercise 2.2.5. Show that R + e R+ 1 ( = 1/2 R where R = R + R is the curvature tensor of. e R 1 Using this, we may observe the following identity, ( R (4π m + tr ( exp( F E ( = (4π m R + e R+ 1 e R+ /2 e R+ /2 = (4π m 1/2 ( R/2 sinh(r/2 tr ( exp( R + F E tr ( exp( R + F E. This manipulation is beneficial because, upon recalling the ehler kernel, ( 1/2 rt/2 ( m t (x, y = exp r ( coth(rt/2(x 2 + y 2 2cosech(rt/2xy tf, 4πt sinh(rt/2 8t which is the heat kernel for the harmonic oscillator H = d2 dx 2 + 1 16 r2 x 2 + f on R 2m, we may observe that the expression we obtained above resembles m 1 (0, 0 if we replace r with R and f with R + + F E. Indeed, the approach used in [2] to obtain the small t asymptotics of p t (x, x is to use the Bochner-Kodaira formula to obtain a local description of D 2, and then transfer the problem to Euclidean space using the exponential map. When the dust settles, D 2 takes a form similar to the harmonic oscillator H described above, so its heat kernel is given by a modified ehler kernel. A ailed proof of Theorem 2.2.4 is provided in chapter 4 of [2].

Str(e td2 = n + 0 n 0 = ind(d. 8 SEAN POHORENCE 3. Supersymmetry In this section, we give a proof of Theorem 2.2.3. The crucial idea comes from physics, specifically from supersymmetry. The relation of supersymmetry to geometry was perhaps first noted by Witten in [3]. We first give an introduction to what is meant by supersymmetry and then see how to apply this to our current situation. Our setup will be as follows. Let F be a separable Hilbert space and let H be a non-negative, (essentially self-adjoint operator acting on F. Physically, F is the space of states of some quantum system. In the case of quantum mechanics, this space is usually L 2 (X, dx where X is the physical space we are working over. In this case, elements ψ(x of F are called wavefunctions, and the densities ψ(x 2, once normalized, give the probability distribution we would obtain by sampling the location of a particle in the state ψ over points in X. The operator H is called the Hamiltonian. If v F is an eigenvector for H, then we call its eigenvalue the energy of the state v. Physically, we are interested in time-dependent states whose evolution satisfy ( t + ihv(t = 0. We can write a solution to this formally as e ith v(0. In particular, we see that the time evolution operator (once we have defined it! should be unitary. We can say this rigorously for an eigenstate v of H, whose time dependence is given by e itλ v where Hv = λv. In order to introduce supersymmetry, we must assume that F is a super Hilbert space. That is, F = F + F. We think of the even and odd summands of F as the spaces of two species of states; physicists would call the even states bosonic and the odd states fermionic. We say that the system defined by the state space F and Hamiltonian H exhibit supersymmetry if there is an odd operator Q on F such that H = Q 2. The following result describes in what sense Q is a symmetry. Proposition 3.0.6. Let F, H, and Q be as above and let λ > 0 be a real number. Define F λ to be the λ-eigenspace and F ± λ := F λ F ±. Then F + λ = F λ. The content of this theorem is that the even definite energy states and odd definite energy states are in one to one correspondence for each positive energy level. In a more physical language, we say that every massive boson has a superpartner which is a fermion with equal mass. The proof of the proposition is elementary. Proof. We claim that Q F + provides the isomorphism required, with inverse given by λ 1 Q λ F. The only λ thing we need to check is that Q(F + λ = F λ. This follows from the fact that [H, Q] = [Q2, Q] = 0. Surprisingly, this seemingly simple result provides the exact sort of cancellation we need to prove the ckean-singer formula. Proof of Theorem 2.2.3. Recall that, using the spectral theorem for essentially self-adjoint operators, we were able to write Str(e td2 = (n + λ n λ e tλ λ 0 for all t > 0. However, applying the previous proposition to the current situation (here, D corresponds to the operator Q above we see that n + λ n λ = 0 for all λ > 0. Thus, for t > 0, References [1] Luis Alvarez-Gaumé. Supersymmetry and the atiyah-singer index theorem. Comm. ath. Phys., 90(2:161 173, 1983. [2] N. Berline, E. Getzler, and. Vergne. Heat Kernels and Dirac Ooperators. Springer-Verlag Berlin Heidelberg, 1992. [3] Edward Witten. Supersymmetry and morse theory. J. Differential Geom., 17(4:661 692, 1982.