The art of using quantum field theory to derive mathematical results often lies in a mysterious transition between infinite dimensional geometry and finite dimensional geometry. In this talk we describe a general mathematical framework to study the quantum geometry of \sigma-models when they are effectively localized to small fluctuations around constant maps. 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. This leads to a theory of “counting constant maps” in a nontrivial way. We explain this program by a concrete example of topological quantum mechanics and show how “counting constant loops” leads to a simple proof of the algebraic index theorem.