[Paper Review] Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory

Pavel Mnëv|arXiv (Cornell University)|Jul 25, 2017

Black Holes and Theoretical Physics30 references19 citations

TL;DR

This paper provides a comprehensive, mathematically rigorous introduction to the Batalin-Vilkovisky (BV) formalism for perturbative path integrals in topological quantum field theory (TQFT), focusing on gauge theories like Chern-Simons theory. It demonstrates how the BV formalism—via AKSZ construction and fiber BV integration—explains the algebraic origin of Feynman diagram weights and ensures gauge independence, with key results including the perturbative Chern-Simons partition function's framing dependence and the diagrammatic cancellation of boundary contributions via Jacobi identity.

ABSTRACT

Lecture notes for the course "Batalin-Vilkovisky formalism and applications in topological quantum field theory" given at the University of Notre Dame in the Fall 2016 for a mathematical audience. In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.

Motivation & Objective

• To provide a self-contained, mathematically oriented introduction to the Batalin-Vilkovisky formalism for graduate students and researchers with no physics background.
• To explain the perturbative path integral quantization of gauge theories, especially topological field theories, using finite-dimensional models as a proxy for infinite-dimensional field theories.
• To clarify the algebraic origin of Feynman diagram weights in Chern-Simons theory and their gauge-invariant structure via the BV formalism.
• To establish the role of the BV-Stokes theorem and canonical transformations in ensuring independence of the perturbative partition function from gauge-fixing choices.
• To demonstrate how the AKSZ construction unifies and explains the compact form of perturbative amplitudes in Chern-Simons and related theories.

Proposed method

• Adopting a finite-dimensional model approach to path integrals, using supermanifolds and graded differential geometry to formalize the BV framework.
• Employing the Faddeev-Popov method as a precursor to introduce ghost fields and BRST symmetry before introducing the full BV formalism.
• Utilizing differential graded manifolds (Q-manifolds) and odd-symplectic structures to define the classical and quantum master equations.
• Applying fiber BV integration and BV Laplacian techniques to handle gauge-fixing and derive effective actions on moduli spaces.
• Using the AKSZ construction to derive the Chern-Simons action and its perturbative expansion from a higher categorical and geometric perspective.
• Applying Stokes’ theorem for BV integrals to prove gauge independence and quantum master equation satisfaction in perturbative expansions.

Experimental results

Research questions

• RQ1 How can the perturbative path integral for gauge theories be rigorously defined in the absence of a background metric, particularly in topological field theories?
• RQ2 What is the algebraic mechanism behind the cancellation of gauge-fixing dependence in Feynman diagram amplitudes?
• RQ3 Why does the perturbative Chern-Simons partition function depend on the framing of the 3-manifold, and how is this related to infinite-dimensional boundary contributions?
• RQ4 How does the BV formalism explain the compact, symmetric form of Feynman weights in Chern-Simons theory, especially when decomposing the propagator into (p,q)-forms?
• RQ5 What is the role of the AKSZ construction in unifying and simplifying the perturbative expansion of topological field theories?

Key findings

• The perturbative Chern-Simons partition function is given by a sum over trivalent Feynman graphs with weights determined by the propagator’s (p,q)-type decomposition, yielding a compact expression that explains the Faddeev-Popov miracle.
• The partition function exhibits framing dependence due to hidden boundary strata in the configuration space, a phenomenon absent in finite-dimensional BV models but present in field theory.
• The BV-Stokes theorem ensures gauge independence of the perturbative path integral, with diagrammatic cancellations arising from the Jacobi identity and graph combinatorics.
• The effective action on cohomology satisfies the quantum master equation, proven via Stokes’ theorem on compactified configuration spaces, generalizing to theories with observables on submanifolds.
• The AKSZ construction provides a geometric origin for the Chern-Simons action and its perturbative amplitudes, explaining the origin of the propagator’s decomposition into (2,0), (1,1), and (0,2) components.
• The finite-dimensional BV integral model for Chern-Simons theory, based on a cyclic dgLa, yields a gauge-independent result due to BV-Stokes theorem, with no analogs of infinite-dimensional boundary effects.