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BV description of $N = 1$, $D = 4$ Supergravity in the first order formalism

Alberto S. Cattaneo, Filippo Fila-Robattino

TL;DR

This work develops a BV/BFV framework for $N=1$, $D=4$ supergravity in the Palatini–Cartan first-order formalism, aiming for off-shell supersymmetry within a boundary-compatible setting. Starting from the naive BV action, the authors show that the supersymmetry algebra closes only on-shell ($Q_0^2\approx0$), which motivates a rank-2 extension of the BV action and targeted field redefinitions. They construct the explicit rank-2 BV action, demonstrate how quadratic antifield terms restore off-shell closure, and provide a detailed verification that the full data satisfy the Classical Master Equation, outlining implications for boundary reductions and quantization. The results lay a robust groundwork for BV/BFV quantization of supergravity on manifolds with boundary and point toward future directions in reduced phase spaces and AKSZ-inspired formalisms.

Abstract

This note examines the BV formulation of $N=1$, $D=4$ supergravity in the first-order Palatini--Cartan framework. Challenges in achieving an off-shell formulation are addressed by introducing corrections to the rank 2 BV action, offering in addition a solid foundation for the study of the theory on manifolds with boundary.

BV description of $N = 1$, $D = 4$ Supergravity in the first order formalism

TL;DR

This work develops a BV/BFV framework for , supergravity in the Palatini–Cartan first-order formalism, aiming for off-shell supersymmetry within a boundary-compatible setting. Starting from the naive BV action, the authors show that the supersymmetry algebra closes only on-shell (), which motivates a rank-2 extension of the BV action and targeted field redefinitions. They construct the explicit rank-2 BV action, demonstrate how quadratic antifield terms restore off-shell closure, and provide a detailed verification that the full data satisfy the Classical Master Equation, outlining implications for boundary reductions and quantization. The results lay a robust groundwork for BV/BFV quantization of supergravity on manifolds with boundary and point toward future directions in reduced phase spaces and AKSZ-inspired formalisms.

Abstract

This note examines the BV formulation of , supergravity in the first-order Palatini--Cartan framework. Challenges in achieving an off-shell formulation are addressed by introducing corrections to the rank 2 BV action, offering in addition a solid foundation for the study of the theory on manifolds with boundary.

Paper Structure

This paper contains 16 sections, 8 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Review of the formalism and setting
  3. Palatini-Cartan SUGRA
  4. On-shell vs off-shell supersymmetry invariance
  5. The BV/BFV formalism in field theories
  6. The case of gravity in the PC formalism
  7. The $\mathcal{N}=1$, $D=4$ Supergravity BV action in the first order formalism
  8. The second rank BV action
  9. Tools, lemmata and identities
  10. Results for gamma matrices and Majorana spinors in $D=4$
  11. Majorana flip relations
  12. Fierz identities
  13. Lemmata
  14. Lengthy computations of section \ref{['sec: BV formalism']}
  15. Computation of $Q_0^2$
  16. ...and 1 more sections

Key Result

Theorem 8

The collection $(\mathcal{F}_{PC},\varpi_{PC},Q_{PC},\mathcal{S}_{PC})$ defines a BV structure, where $\mathcal{F}_{PC}:=T^*[-1]F_{PC}$ and The symplectic form is canonically defined as while the BV action reads Lastly, one easily recovers the cohomological vector field acting on the fields and the ghosts as

Theorems & Definitions (21)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 8: CS2017
  • Remark 9
  • Remark 10
  • ...and 11 more