The Atiyah–Singer Index Theorem

The index theorem equates an analytical quantity — the number of solutions of a differential equation — with a topological one — a characteristic number of a bundle. It is one of the great unifications of 20th-century mathematics, and in physics it is the chiral anomaly: the imbalance of left- and right-handed zero modes of the Dirac operator equals an instanton number. This page is statement-level, emphasizing the physics; it builds on characteristic-classes.md and chern-weil.md.

Analytical vs. topological index

Let $D:\Gamma (E)\to \Gamma (F)$ be an elliptic differential operator between sections of vector bundles over a closed manifold. Its analytical index is the finite integer $$\mathrm{index}(D)=\dim \ker D-\dim \mathrm{coker}D=(\#\text{solutions of }D\psi =0)-(\#\text{solutions of }{D}^{†}\varphi =0).$$ This counts zero modes and is manifestly analytical. The remarkable fact is that it cannot change under continuous deformations — it is a topological invariant.

Atiyah–Singer index theorem. For an elliptic operator on a closed manifold, $$\mathrm{index}(D)={\int }_M[\text{characteristic classes of }E,F,TM],$$ the topological index — an integral of curvature-built characteristic classes (characteristic-classes.md).

The analytical side (hard PDE data) equals the topological side (Chern/Pontryagin numbers). Special cases recover Gauss–Bonnet (Euler operator → $\chi$), the Riemann–Roch theorem, and the signature theorem.

The Dirac operator and the chiral anomaly

The case physics needs is the Dirac operator $/D={\gamma }^{\mu }{\nabla }_{\mu }$ coupled to a gauge field. Chirality splits spinors into left/right (${\gamma }^5=\pm 1$), and $/D$ maps one chirality to the other. Its index counts the chiral zero-mode imbalance: $$\mathrm{index}(/D)=n_{+}-n_{-}={\int }_M\hat{A}(M)\wedge \mathrm{ch}(F),$$ where $n_{\pm }$ are the numbers of positive/negative-chirality zero modes, $\hat{A}$ is the A-roof genus (Pontryagin classes of $TM$), and $\mathrm{ch}(F)$ the Chern character of the gauge bundle. In flat 4D spacetime this reduces to the instanton number: $$n_{+}-n_{-}=\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)=\nu .$$

Why this is the anomaly

The chiral (ABJ) anomaly of anomalies/chiral-anomaly.md is the statement that the chiral current is not conserved in a gauge background: $${\partial }_{\mu }{j}^{\mu 5}=\frac{1}{16{\pi }^2}{ϵ}^{\mu \nu \rho \sigma }\mathrm{tr}(F_{\mu \nu }{F}_{\rho \sigma }).$$ Integrating over spacetime, the total change of chiral charge equals $2\nu$ — and by the index theorem this is $2(n_{+}-n_{-})$, the zero-mode imbalance. So:

The anomaly is the index theorem. The non-conservation of chiral charge is the analytical index of the Dirac operator, and the right-hand side $\mathrm{tr}(F\wedge F)$ is exactly the topological index. The "one-loop triangle diagram" and the "instanton zero modes" are two computations of the same integer.

This is why the anomaly is (i) one-loop exact (an integer cannot receive continuous corrections) and (ii) tied to topology (it counts a characteristic number). Anomaly cancellation (anomalies/anomaly-cancellation.md) is then the condition that the representation content makes the total topological index vanish — a constraint the Standard Model precisely satisfies.

References

• Nakahara, Geometry, Topology and Physics, Ch. 12–13 — the physics-facing account.
• Atiyah & Singer, The Index of Elliptic Operators (Ann. Math. 1968).
• Nash, Differential Topology and Quantum Field Theory.