User:Silly rabbit/Sandbox/Atiyah-Singer index theorem

Heat equation

Another method of proof is to use the heat equation for an elliptic differential operator. From this perspective, the index theorem is a statement of the following form:

$\mathrm {index} (P)=\int \alpha _{P}$

where P is an elliptic differential operator on a vector bundle, and α is some particular "invariant" differential form constructed out of P in a manner similar to the construction of the Euler class from the exterior derivative and divergence operators. The essential feature of the index theorem is that it provides a local expression of the index: a particular m-form whose integral is the analytical index. The approach is to construct such forms by examining the asymptotic expansion of the heat kernel associated to the operator P, and at the same time to develop an expression for the analytical index of P by using eigenfunction expansions and L² traces. The remaining part of the proof, although still quite involved, is then to show that a form obtained in this manner is cohomologous to the class of the topological index.

The heat equation for a self-adjoint operator P with positive definite leading symbol is a partial differential equation of the form

$f_{t}(x,t)+Pf(x,t)=0,\quad t\geq 0,\,\,f(0,x)=f(x).\,$

For each choice of boundary condition f(x), the solution is written formally as

$e^{-tP}f(x):=\int _{M}K(t,x,y)f(y)\,\mathrm {dvol} (y)\,$

where K is the heat kernel for P. Using eigenfunction expansions in L², one defines the L² trace of the operator e^-tP by

$\mathrm {Tr} _{L^{2}}e^{-tP}=\sum e^{-t\lambda _{n}}=\int _{M}\mathrm {Tr} K(t,x,x)\,\mathrm {dvol} (x).$

If Q is an elliptic differential operator, then it is not difficult to show that

$\mathrm {index} (Q)=\mathrm {Tr} _{L^{2}}e^{-tQ^{*}Q}-\mathrm {Tr} _{L^{2}}e^{-tQQ^{*}}.$

Here QQ^* and Q^*Q are the Laplacians for the elliptic complex determined by Q. In particular, they are elliptic self-adjoint operators with positive-definite leading symbols.

For the other part of the construction, the heat kernel K for an elliptic operator P of order d with positive definite leading symbol (e.g., the Laplacians of an elliptic complex) has an asymptotic expansion of the form

$K(t,x,x)\sim \sum _{n=0}^{\infty }t^{\frac {n-m}{d}}e_{n}(x,P).\,$

The scalars formed by taking an ordinary fibre-trace of the e_n(x, P) are then invariants naturally associated to P: a_n(x, P) = Tr e_n(x, P). Since the index is independent of t, it follows that

$\mathrm {index} (Q)=\int _{M}\left(a_{m}(x,Q^{*}Q)-a_{m}(x,QQ^{*})\right)\,\mathrm {dvol} (x)$

Thus the local index of an operator is identified with a particular coefficient in the asymptotic expansion of its heat kernel.