Batalin-Vilkovisky quantization and the algebraic index

Abstract

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.

V2: Significant re-write, 51 pages, 9 figures

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Cited by in corpus (18)

• Vertex algebras and quantum master equation
• Chiral differential operators via Batalin-Vilkovisky quantization
• Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary
• AKSZ constructions for topological membranes on $G_2$-manifolds
• On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism
• Homotopy Algebras in Higher Spin Theory
• Quantization of Kähler manifolds
• On Globalized Traces for the Poisson Sigma Model
• Kapranov's $L_{\infty }$ structures, Fedosov's star products, and one-loop exact BV quantizations on Kähler manifolds
• Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index
• Quantizable functions on Kähler manifolds and non-formal quantization
• Lie algebroids as $L_{\infty }$ spaces
• Quantizing Derived Mapping Stacks
• The standard cohomology of regular Courant algebroids
• Formal Global Perturbative Quantization of the Rozansky-Witten Model in the BV-BFV Formalism
• Deformation Quantization for Supermanifolds via Gelfand-Kazhdan Descent
• Algebras over not too little discs
• Holomorphic field theories and higher algebra