  Into a geometric setting, we import the physical interpretation of index
theorems via semi-classical analysis in topological quantum field theory. We
develop a direct relationship between Fedosov's deformation quantization of a
symplectic manifold X and the BV quantization of a one-dimensional sigma model
with target X. This model is a quantum field theory of AKSZ type and is
quantized rigorously using Costello's homotopic theory of effective
renormalization. We show that Fedosov's Abelian connections on the Weyl bundle
produce solutions to the effective quantum master equation. Moreover, BV
integration produces a natural trace map on the deformation quantized algebra.
This formulation allows us to exploit a (rigorous) localization argument in
quantum field theory to deduce the algebraic index theorem via semi-classical
analysis, i.e., one-loop Feynman diagram computations.
