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Equivariant BV-BFV Formalism

Alberto S. Cattaneo, Nima Moshayedi

TL;DR

This work extends the equivariant BV formalism to manifolds with boundary by embedding it into the BV–BFV framework, showing that boundary data can be consistently incorporated when the symmetry-generated vector fields are tangent to the boundary. It clarifies how the classical master equation and its quantum version are modified by boundary contributions and introduces boundary BFV data that remain compatible with bulk BV data under suitable conditions. The paper develops a systematic approach for equivariant AKSZ theories, provides precise boundary compatibility criteria, and works out an explicit Abelian BF theory example, including a Schrödinger-quantized boundary operator and the associated emQME. Together, these results enable perturbative quantization of gauge theories with boundary symmetries and lay the groundwork for controlled boundary phenomena in equivariant settings.

Abstract

The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example.

Equivariant BV-BFV Formalism

TL;DR

This work extends the equivariant BV formalism to manifolds with boundary by embedding it into the BV–BFV framework, showing that boundary data can be consistently incorporated when the symmetry-generated vector fields are tangent to the boundary. It clarifies how the classical master equation and its quantum version are modified by boundary contributions and introduces boundary BFV data that remain compatible with bulk BV data under suitable conditions. The paper develops a systematic approach for equivariant AKSZ theories, provides precise boundary compatibility criteria, and works out an explicit Abelian BF theory example, including a Schrödinger-quantized boundary operator and the associated emQME. Together, these results enable perturbative quantization of gauge theories with boundary symmetries and lay the groundwork for controlled boundary phenomena in equivariant settings.

Abstract

The recently introduced equivariant BV formalism is extended to the case of manifolds with boundary under appropriate conditions. AKSZ theories are presented as a practical example.
Paper Structure (17 sections, 17 theorems, 113 equations)

This paper contains 17 sections, 17 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. The BV-BFV Formalism
  3. Classical BV Formalism
  4. 2D $BF$ theory
  5. Quantum BV Formalism
  6. Classical BV-BFV Formalism
  7. Quantum BV-BFV Formalism
  8. AKSZ Theories
  9. Equivariant BV-BFV Formalism
  10. Equivariant BV Formalism
  11. Preliminary Computations
  12. Classical Equivariant Setting
  13. Reduction to the Boundary
  14. Summary
  15. Prequantization
  16. ...and 2 more sections

Key Result

Theorem 2.2.1

Consider two half-densities $f,g$ on an odd-symplectic supermanifold $(\mathcal{M},\omega)$. ThenWe assume below that the given half-densities have fast decay at infinity.

Theorems & Definitions (49)

  • Definition 2.1.1: Lagrangian field theory
  • Definition 2.1.2: BV manifold
  • Remark 2.1.3
  • Definition 2.1.4: BV theory
  • Remark 2.1.5
  • Theorem 2.2.1: Batalin--Vilkovisky--SchwarzBV2S
  • Definition 2.3.1: BFV manifold
  • Remark 2.3.2
  • Definition 2.3.3: BV-BFV manifold over exact BFV manifold
  • Remark 3.1.1
  • ...and 39 more