Batalin-Vilkovisky quantization and the algebraic index

Ryan E. Grady aMontana State University Qin Li bSouthern University of Science and Technology Si Li YMSC, Tsinghua University

Mathematical Physics mathscidoc:1806.22001

Adv.Math., 317, 575-639, 2017
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.
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  title={Batalin-Vilkovisky quantization and the algebraic index},
  author={Ryan E. Grady, Qin Li, and Si Li},
Ryan E. Grady, Qin Li, and Si Li. Batalin-Vilkovisky quantization and the algebraic index. 2017. Vol. 317. In Adv.Math.. pp.575-639.
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