Polynomial $BF$-type action for general relativity and anti-self-dual gravity

TL;DR

This paper introduces a polynomial BF-type action for gravity that unifies general relativity and anti-self-dual gravity, revealing a family of models with the same degrees of freedom and a closed constraint algebra.

Contribution

It presents a novel polynomial action principle for gravity depending on two parameters, unifying different gravity theories and generalizing the scalar constraint in the Hamiltonian formulation.

Findings

The action reproduces complex general relativity with a cosmological constant.

A special parameter choice yields anti-self-dual gravity.

An infinite family of models with closed constraint algebra is constructed.

Abstract

We report a gravitational $BF$-type action principle propagating two (complex) degrees of freedom that, besides the gauge connection and the $B$ field, only employs an additional Lagrange multiplier. The action depends on two parameters and remarkably is polynomial in the $B$ field. For a particular choice of the involved parameters the action provides an alternative description of (complex) general relativity with a nonvanishing cosmological constant, whereas another choice corresponds to anti-self-dual gravity. Generic values of the parameters produce "close neighbors" of general relativity, although there is a peculiar choice of the parameters that leads to a Hamiltonian theory with two scalar constraints. Given the nontrivial form of the resulting scalar constraint for these models, we consider a more general setting where the scalar constraint is replaced with an arbitrary analytic function of some fundamental variables and show that the Poisson algebra involving this constraint together with the Gauss and vector constraints of the Ashtekar formalism closes, thus generating an infinite family of gravitational models that propagate the same number of degrees of freedom as general relativity.