Gravity with torsion as deformed $BF$ theory

TL;DR

This paper introduces a deformed $BF$ theory called quadratically extended General Relativity (qeGR), which is classically equivalent to gravity models with dynamical torsion, and explores its formal properties within the Batalin--Vilkovisky framework.

Contribution

It formulates a new deformation of $BF$ theory that captures gravity with torsion and establishes its equivalence to known gravity models in a rigorous formalism.

Findings

qeGR is classically equivalent to gravity with dynamical torsion.

Standard gravity models are recovered from qeGR via BV formalism.

The formalism extends to non topological deformations of $BF$ theory.

Abstract

We study a family of (possibly non topological) deformations of $BF$ theory for the Lie algebra obtained by quadratic extension of $\mathfrak{so}(3,1)$ by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin--Vilkovisky formalism. In particular, both Palatini--Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin--Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields.