⇤Math diary
May 24, 2023: Index theorem as infinite-dimensional equivariant localization
Today I read some of Localization Formulas, Superconnections, and the Index Theorem for Families by J.M Bismut. It gave a new perspective on the heat kernel proof of the index theorem, which I discussed here. In particular, it is a manfiestation of the duestermaat-heckman formula on loop space. This is informally explained in a very readible article by Atiyah.
Duestermaat-Heckman via gaussians
The deustermaat-heckman formula is a localization formula for equivariant cohomology. In its usual statement, it says that the equivariant cohomology for an $S^1$ action on a manifold can be computed entirely with local information around the fixed point set of the $S^1$ action. The deustermaat-heckman formula is the explict expression of equivariant cohomology classes as charecteristic classes of vector bundles over the fixed point set. Specifically, let $\omega$ is the symplectic form on a manifold $M$ of dimension $2n$, and $H$ is the hamiltonian function generating the $S^1$ function. If $X$ is the fixed point set of the $S^1$ action, then
\(\int_M e^{-tH} \frac{\omega^n}{n!} = \int_X e^{-tH}\frac{e^\omega}{\Pi (t m_i - i \alpha_i)}\) where $m_i$ are the charecters of the $S^1$ action on the notmal bundle to $X$, and the $\alpha_i$ are charecteristic classes of the normal bundle to $X$.
The paper above first supplies a new proof of the localization theorem, using guassians which limit to the fixed point set. We start with an equivariantly closed form $\mu$, which is a differential form closed under a deformation of the exterior derivative that accounts for the $S^1$ action. Then, we compute the integral of the form over the whole manifold via an integral fixed point set (this is the localization formula). To derive this formula, we introduce a one form $\beta$ dual to the generating vector field $X$ of the $S^1$ action. We can take this to be an equivariantly closed form. Then, we deform
\[\mu \to e^{s(\mathrm{d} + \iota_X)\beta}\mu\]and show that the integreal remains constant over the manifold. Next, since $\beta$ is dual to $X$, $\iota_X(\beta)$ is equal to $\vert X\vert^2$. around the zero set of $X$ (the fixed point set of the $S^1$ action), this looks locally quadratic. As we limit $s\to 0$, this quadradic part is dominant, and the mass of the deformed $\mu$ concentrates around the limit set. The other part involves $\mathrm{d} \beta$, which contributes some first order terms in the limit $s \to 0$ which relate to curvature. Computing the gaussian integral in the $s\to 0$ limit results in an integral around the zero set of $X$, involving the curvature on the normal bundle. Careful evaluation gives a formula, which agrees with that of Duestermaat-Heckman.
To summarize, we have a very renormalization-colored proof of localization. We concentrated the equivariant form around the fixed point set, in a way that preserves the total integral. Thus, we can compute cohomology from local data. This is very reminiscent of Witten’s approch to morse theory
Loop spaces and index theory
Now to relate it to the index theorem. The Mckean-Singer theorem tells us that the index of the dirac operator can be computed from the supertrace of the heat kernel: \(\text{index} D_+ = \int_M\text{sTr}( P_t(x,x)) \quad \text{where } -D^2P_t(x, \cdot) = \partial_t P_t(x, \cdot)\) In particular, the heat kernel term asks, if we start with a spinor at $x$ and let it diffuse for time $t$, what will the spinor remaining at $x$ be?
We can recontextualize the heat operator as diffusion. To find the heat kernel $P(x,y)$, we integrate over all possible paths (using the proper Weiner measure), and evaluate what percentage of them who start at $x$ end at $y$. To compute $P(x,x)$, we need to look at paths starting and ending at the same point, or loops. The intergal of $P(x,x)$ over all $M$ becomes an integral of the weiner measure over loop space $LM = \text{Maps}(S^1,M)$ (A path integral, in physics parlance). Specifically, we wish to evaluate
\[\int_{LM} e^{i E(\phi)} \mathrm{D}\phi\]Where $\mathrm{D}\phi$ is the weiner measure on the loop space, and $E(\phi)$ is the energy of a loop. We wish to evaluate this integral using a formal application of the Duesteramaat-heckman formula. First, we note that $LM$ carries a natural almost symplectic structure. Let $X_E$ be the vector field generating rotation around the loops of $LM$. Using the riemannian mertic structure, we can turn this to a one form $\theta$, and define the symplectic form via $\omega = \text{d} \theta$. Next, we note that the energy function $E(\phi) = \int_{S^1} \vert \text{d}\phi \vert ^2$ is actually the Hamiltonian generating the rotation around a loop! That is, its vector field is $X_E$. The integral we want is exactly the left side of the Deustermaat-heckman formula, on an infintie dimensional manifold.
Before we apply localization, we need to switch the masure $\cal{D}\phi$ to the louiville measure using $\omega$. The difference is the square root of the determinant of $\omega$, written as a matrix. This is undefined in infinite dimensions, but using zeta function regularization, we can make sense of something. This pffafian turns out to be the index of the dirac operator on $M$.
Now, apply localization. The fixed point set for the “rotate around each loop” consists of the constant loops, which are just points in $M$. Thus, the set consits of $M\subset LM$,and our localization theorem transforms the path integral to an integral over $M$. The right hand side rearranges to \(\int_M \Pi \frac{\alpha_i /2}{\sinh \alpha_i/2}\) Which is a combination of charecteristic classes on $M$. This numer is called the $\hat{A}$-genus of $M$. Applying Duestermaat-Heckman localization formula to the loop space $LM$ proves that the index of the Dirac operator is the $\hat{A}$ genus!
In fact, this method can be applied to all elliptic operators, to prove the general index theorem. And here’s the kicker: The heat kernel proof of the index theorem is exactly the finite dimensional proof of the localization form, summarized above, carried out in infinite dimenions. A very cool perspective on index theory!