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 Batalin-Vilkovisky (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.