  We show that ${\rm Tr}(-1)^F F e^{-\beta H}$ is an index for $N$=2
supersymmetric theories in two dimensions, in the sense that it is independent
of almost all deformations of the theory. This index is related to the geometry
of the vacua (Berry's curvature) and satisfies an exact differential equation
as a function of $\beta$. For integrable theories we can also compute the index
thermodynamically, using the exact $S$-matrix. The equivalence of these two
results implies a highly non-trivial equivalence of a set of coupled integral
equations with these differential equations, among them Painleve III and the
affine Toda equations.
