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The Reduced Phase Space of $N=1, D=4$ Supergravity in the BV-BFV formalism

Alberto S. Cattaneo, Filippo Fila-Robattino

TL;DR

This work addresses the challenge of describing the reduced phase space for $N=1$, $D=4$ supergravity in the fully off-shell Palatini–Cartan formalism and ensuring compatibility between bulk BV and boundary BFV formalisms. The authors deploy the KT construction to identify boundary first-class constraints and generate a BFV boundary theory, then apply the BV pushforward on a cylinder to eliminate obstructing degrees of freedom, obtaining a reduced BV theory that is BFV-extendible. A two-step symplectomorphism, implemented via a 1D AKSZ framework, relates the reduced bulk BV data to a consistent boundary BFV description, establishing an explicit BV–BFV correspondence for the reduced theory. The results provide a robust route toward perturbative quantization of supergravity with boundaries, including gluing, and pave the way for rigorous BV-BFV treatment of gravity and supergravity in the presence of boundaries.

Abstract

This paper describes the reduced phase space of $N=1$, $D=4$ supergravity in the fully off-shell Palatini--Cartan formalism. This is achieved through the KT construction, allowing an explicit description of first-class constraints on the boundary. The corresponding BFV description is obtained, and its relation with the BV one in the bulk is described by employing the BV pushforward in the particular example of a cylindrical spacetime.

The Reduced Phase Space of $N=1, D=4$ Supergravity in the BV-BFV formalism

TL;DR

This work addresses the challenge of describing the reduced phase space for , supergravity in the fully off-shell Palatini–Cartan formalism and ensuring compatibility between bulk BV and boundary BFV formalisms. The authors deploy the KT construction to identify boundary first-class constraints and generate a BFV boundary theory, then apply the BV pushforward on a cylinder to eliminate obstructing degrees of freedom, obtaining a reduced BV theory that is BFV-extendible. A two-step symplectomorphism, implemented via a 1D AKSZ framework, relates the reduced bulk BV data to a consistent boundary BFV description, establishing an explicit BV–BFV correspondence for the reduced theory. The results provide a robust route toward perturbative quantization of supergravity with boundaries, including gluing, and pave the way for rigorous BV-BFV treatment of gravity and supergravity in the presence of boundaries.

Abstract

This paper describes the reduced phase space of , supergravity in the fully off-shell Palatini--Cartan formalism. This is achieved through the KT construction, allowing an explicit description of first-class constraints on the boundary. The corresponding BFV description is obtained, and its relation with the BV one in the bulk is described by employing the BV pushforward in the particular example of a cylindrical spacetime.
Paper Structure (33 sections, 32 theorems, 239 equations)

This paper contains 33 sections, 32 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Review of previous results
  3. The BV action of $N=1,D=4$ supergravity
  4. The BV--BFV formalism on manifolds with boundary
  5. The reduced phase space of $\mathcal{N}=1$, $D=4$ SUGRA
  6. The Kijowski--Tulczijew construction
  7. BFV data from the reduced phase space
  8. Constraint analysis of $\mathcal{N}=1$, $D=4$ SUGRA
  9. First reduction and structural constraint
  10. Constraints and the first-class condition
  11. Towards a BFV description
  12. Introducing new variables
  13. The BV pushforward of $\mathcal{N}=1, D=4$ supergravity
  14. Obstructions in the free theory
  15. The BV-Pushforward in Field Theories
  16. ...and 18 more sections

Key Result

Theorem 5

CMR2012b With the above assumptions, the cohomological vector field $Q$ induces a cohomological vector field $Q_\Sigma$ on $\mathcal{F}_\Sigma$, which is the Hamiltonian vector field for the functional $\mathcal{S}_\Sigma$.

Theorems & Definitions (84)

  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Theorem 5
  • Remark 6
  • Theorem 7: BV3Stasheff1997Scht08
  • Remark 8
  • Definition 9
  • Theorem 10
  • ...and 74 more