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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.