Atiyah-Singer Index Theorem

Prerequisites: Topology, Differential Geometry. Full path (6 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Many problems in physics come down to counting the solutions of a linear differential equation: how many zero-energy states a fermion has in a given background field, or how many independent ways a solution can be deformed. Counting them directly is hard, because the answer depends on every detail of the equation and of the space it lives on. If we change the metric a little, or the background field, individual solutions can appear or disappear.

The index theorem looks at a slightly different number. For an operator we count the solutions of the equation in which the operator gives zero, and subtract the number of solutions of the corresponding "adjoint" equation. When we deform the equation smoothly, solutions can only appear or disappear in pairs, one of each kind, so this difference (the index) cannot change. A number that does not change under smooth deformations can only depend on the global shape of the space and of the fields on it, which is its topology.

The theorem by Michael Atiyah and Isadore Singer says how to compute this number: as an integral over the space of certain curvature expressions. These integrals are integers that measure how the space or the field is "twisted" globally, like the number of holes of a surface or the winding of a gauge field. A difficult analytic question (how many solutions?) becomes a computation that only uses topology. In physics this explains why fermions in an instanton background have a fixed number of zero modes, and why the axial charge changes by a whole number in such a background.

Concrete

Let $M$ be a compact manifold without boundary, and let $D$ be a linear elliptic differential operator that maps sections of a vector bundle $E$ to sections of a vector bundle $F$ (for example, wave functions with values in $\mathbb{C}^n$ to other such wave functions). Ellipticity means that the highest-derivative part of $D$ is invertible for every nonzero momentum, like the Laplacian $-\nabla^2 \to |k|^2$. For such operators the spaces of solutions of $D\psi = 0$ and $D^\dagger \chi = 0$ are finite-dimensional, and the index is $$\operatorname{ind} D = \dim \ker D - \dim \ker D^\dagger .$$

The theorem states that $\operatorname{ind} D$ equals a topological index, given by an integral over $M$ of characteristic classes built from the curvature of $M$ and of the bundles. Two standard examples show what this means.

Euler characteristic. Take a closed oriented manifold of even dimension and the operator $D = d + d^\dagger$ acting from differential forms of even degree to forms of odd degree. Its kernel consists of harmonic forms, which by Hodge theory are in one-to-one correspondence with the de Rham cohomology. The index is $$\operatorname{ind} D = \sum_k (-1)^k b_k = \chi(M),$$ the Euler characteristic, with $b_k$ the Betti numbers. For a closed surface with Gaussian curvature $K$ the index theorem gives $$\chi(M) = \frac{1}{2\pi}\int_M K \, dA ,$$ which is the Gauss-Bonnet theorem. For a sphere, $K = 1/R^2$ and the area is $4\pi R^2$, so $\chi = 2$.

Dirac operator in a gauge field. On a compact four-dimensional spin manifold, let $\mathcal{D} = \gamma^\mu(\partial_\mu + \omega_\mu + A_\mu)$ be the Euclidean Dirac operator coupled to a gauge field $A$ in a representation of dimension $r$. Using $\gamma_5$ we split spinors into chiralities, and $\mathcal{D}$ maps positive chirality to negative chirality, $D_+ : S_+ \to S_-$, with $D_+^\dagger = D_-$. The index $n_+ - n_-$ counts zero modes of positive minus negative chirality. The theorem gives $$n_+ - n_- = \int_M \hat{A}(M)\, \operatorname{ch}(F) ,$$ where, with the gauge field strength $F$ taken anti-Hermitian, $\operatorname{ch}(F) = \operatorname{tr} \exp\left(\frac{iF}{2\pi}\right)$ is the Chern character and $\hat A(M) = 1 - \frac{p_1(M)}{24} + \dots$ is the $\hat A$-genus built from the Riemann curvature. In four dimensions only the 4-form parts contribute: $$n_+ - n_- = -\frac{1}{8\pi^2}\int_M \operatorname{tr}(F\wedge F) \; - \; \frac{r}{24}\int_M p_1(M) .$$ On the flat torus or on the round four-sphere the gravitational term vanishes and the index is given by the instanton number of the gauge field (up to a sign fixed by orientation conventions). For an $SU(2)$ instanton of charge $k$ on $S^4$ and fermions in the fundamental representation, one chirality has $|k|$ zero modes and the other has none. These zero modes are what makes instantons violate axial charge conservation, see anomalies.

Why the index is stable. The nonzero eigenvalues of $D^\dagger D$ and $D D^\dagger$ are the same, with the same multiplicities: if $D^\dagger D \psi = \lambda \psi$ with $\lambda \neq 0$, then $D\psi$ is an eigenvector of $D D^\dagger$ with the same eigenvalue. Hence $$\operatorname{ind} D = \operatorname{Tr}\, e^{-t D^\dagger D} - \operatorname{Tr}\, e^{-t D D^\dagger} \quad \text{for every } t>0 .$$ Under a smooth deformation, eigenvalues of $D^\dagger D$ and $DD^\dagger$ can only reach or leave zero together, one state of each kind, so the index stays fixed. Taking $t \to 0$, the heat kernel traces become integrals of local curvature expressions, which is one standard route to the index formula.

Abstract

Let $M$ be a closed smooth manifold and $D : \Gamma(E) \to \Gamma(F)$ an elliptic differential operator between smooth vector bundles. Ellipticity means that the principal symbol $\sigma(D)(x,\xi) : E_x \to F_x$ is an isomorphism for every nonzero covector $\xi \in T^*_xM$. Elliptic operators extend to Fredholm operators between Sobolev spaces, so the analytic index $\dim\ker D - \dim\operatorname{coker} D$ is a well-defined integer. It is constant on connected components of the space of Fredholm operators and therefore depends only on the homotopy class of the symbol.

The symbol, being invertible away from the zero section, defines a class $[\sigma(D)] \in K_c(T^*M)$, the compactly supported K-theory of the cotangent bundle. Atiyah and Singer define a topological index $K_c(T^*M) \to \mathbb{Z}$ by embedding $M$ into $\mathbb{R}^N$, pushing the class forward with the Thom isomorphism, and using Bott periodicity $K_c(\mathbb{R}^{2N}) \cong \mathbb{Z}$. The theorem says that the analytic and the topological index agree as homomorphisms on $K_c(T^*M)$.

Evaluated with the Chern character, the topological index becomes an integral of characteristic classes. For the Dirac operator on a closed even-dimensional spin manifold, twisted by a Hermitian bundle $E$ with connection, the formula is $$\operatorname{ind} D_E^+ = \int_M \hat A(TM)\, \operatorname{ch}(E) .$$ The classical theorems appear as special cases: the signature operator gives the Hirzebruch signature theorem $\sigma(M) = \int_M L(TM)$, the Dolbeault operator on a compact complex manifold gives the Hirzebruch-Riemann-Roch theorem $\chi(M,E) = \int_M \operatorname{ch}(E)\operatorname{Td}(TM)$, and the de Rham operator gives the Chern-Gauss-Bonnet theorem.

Several generalisations are used in physics. For a family of elliptic operators parametrised by a space $B$, the kernels and cokernels assemble into an index bundle with class in $K(B)$; when $B$ is the space of gauge fields modulo gauge transformations, its Chern character (through the determinant line bundle) gives the perturbative gauge anomalies of chiral fermions, while global anomalies need finer invariants such as the eta invariant. On manifolds with boundary, the Atiyah-Patodi-Singer theorem adds a boundary correction involving the eta invariant of the boundary operator. The equivariant index theorem, and the related Atiyah-Bott fixed point formula, compute the trace of a symmetry on the kernel minus the cokernel in terms of data at its fixed points. In supersymmetric quantum mechanics the index appears as the Witten index $\operatorname{Tr}(-1)^F e^{-\beta H}$, and evaluating its path integral in the $\beta \to 0$ limit reproduces the local index density.

Why is it interesting?

The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of topology. Topology is that part of mathematics that deals with those aspects of geometrical objects that don't change as one deforms the object (the standard explanatory joke is that 'a topologist is someone who can't tell the difference between a coffee cup and a doughnut'). One important aspect of the index theorem is that it can be proved by relating the differential equation under study to a generalised version of the Dirac equation. Atiyah and Singer rediscovered the Dirac equation for themselves during their work on the theorem. Their theorem says that one can calculate the number of solutions of an equation by finding the number of solutions of the related generalised Dirac equation. It was for these generalised Dirac equations that they found a beautiful topological formula for the number of their solutions. […] Atiyah became interested [in Yang-Mills theory], and they quickly realised that their index theorem could be applied in this case and it allowed a determination of exactly how many solutions the self-duality equations would have.

page 118 in Not Even Wrong by Peter Woit

The next major development in the story of the so-far-only-physicist’s Yang-Mills equations was one by mathematicians! The Atiyah-Singer Index Theorem said something about the dimension of the space of solutions to the Yang-Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics community began to change. It is here that the cooperation and cross-fertilization really began. I don’t know how to describe the tremendous difference this made. The entire position that mathematics held within the scientific community changed. The mathematicians actually had something to say!

http://www.ma.utexas.edu/users/lfredrickson/Attachments/TempleLectures.pdf

• See also: http://physics.stackexchange.com/a/1865/37286

Path to this page

1. Calculus
2. Linear Algebra
3. Manifold
4. Product Spaces
5. Differential Geometry
6. Topology