Families index theorem

Remarks

The proof follows the K-theoretic proof of the Atiyah–Singer index theorem, carried out over the base $B$. Both indices are homomorphisms with the same formal properties, and they agree for the basic families on vector bundles over $B$. For $B$ a point it is the Atiyah–Singer index theorem, since $K(\text{point}) = \Z$.

Taking Chern characters gives a cohomological form: $\operatorname{ch}(\operatorname{ind} P)$ is obtained, up to a sign fixed by orientation conventions, by integrating $\operatorname{ch}[\sigma(P)]$ times the Todd class of $T_v Z \otimes \C$ over the fibres. The index bundle carries more information than the numerical index. Its rank is the numerical index of each $P_b$, and its Chern classes record how the kernels and cokernels twist over $B$.

Example: let $Q \colon C^\infty(M; E_0) \to C^\infty(M; F_0)$ be one elliptic operator on $M$ and let $V \to B$ be a complex vector bundle. On $Z = B \times M$ take $P_b = Q \otimes \id_{V_b}$. Then $\ker P_b = \ker Q \otimes V_b$ and $\operatorname{coker} P_b = \operatorname{coker} Q \otimes V_b$, so $\operatorname{ind} P = (\operatorname{ind} Q)\,[V] \in K(B)$, an integer multiple of $[V]$.

Going deeper

For a student

The families index theorem is used to compute determinant line bundles and their curvature, which is important in gauge theory and string theory. It is also the starting point for index theorems with values in the K-theory of C*-algebras, as in the work of Mishchenko and Fomenko on higher signatures. Its real version, with values in real K-theory, gives mod $2$ invariants of families such as those used in the study of spin manifolds.