  In this paper we propose a general framework to study the quantum geometry of
$\sigma$-models when they are effectively localized to small quantum
fluctuations around constant maps. Such effective theories have surprising
exact descriptions at all loops in terms of target geometry and can be
rigorously formulated. We illustrate how to turn the physics idea of exact
semi-classical approximation into a geometric set-up in this framework, using
Gauss-Manin connection. As an application, we carry out this program in details
by the example of topological quantum mechanics, and explain how to implement
the idea of exact semi-classical approximation into a proof of the algebraic
index theorem. The proof resembles much of the physics derivation of
Atiyah-Singer index theorem and clarifies the geometric face of many other
mathematical constructions.
