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.