General Relativity and the AKSZ Construction

G. Canepa,A. S. Cattaneo,M. Schiavina

doi:10.1007/s00220-021-04127-6

G. Canepa, A. S. Cattaneo + Show 1 more

Open Access

https://doi.org/10.1007/s00220-021-04127-6

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Abstract

In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein–Hilbert and of the Palatini–Cartan theories in every space-time dimension greater than two. In the former case one obtains a BV theory for the first-order formulation of Einstein–Hilbert theory, in the latter a BV theory for Palatini–Cartan theory with a partial implementation of the torsion-free condition already on the space of fields. All theories described here are BV versions of the same classical system on cylinders. The AKSZ implementations we present have the advantage of yielding a compatible BV–BFV description, which is the required starting point for a quantization in presence of a boundary.

Highlights

A Lagrangian field theory F on a cylinder × I, where I is a “time” interval, can be given a corresponding Hamiltonian description in terms of a symplectic manifold of the possible initial conditions on and a Hamiltonian that describes the time evolution
In this note the AKSZ construction is applied to the BFV description of the reduced phase space of the Einstein–Hilbert and of the Palatini–Cartan theories in every space-time dimension greater than two
The AKSZ implementations we present have the advantage of yielding a compatible Batalin– Vilkovisky (BV)–BFV description, which is the required starting point for a quantization in presence of a boundary

Summary

Introduction

A Lagrangian field theory F on a cylinder × I , where I is a “time” interval, can be given a corresponding Hamiltonian description in terms of a symplectic manifold (the phase space) of the possible initial conditions on and a Hamiltonian that describes the time evolution. Since the outlook of this extended program is that of addressing quantisation of General Relativity (with boundary), we wish to stress that without the observations produced in this preliminary phase, an early attempt at directly quantising PC theory might have been thwarted by the very obstructions highlighted by our investigations In this sense, we believe the correct preparation of a field theory for its perturbative quantisation to be of crucial importance to drive the scientific effort towards sensible questions, and divert it when evidence is presented of a potential roadblock ahead. This should be of particular interest for the scientific community heavily involved with the study of Palatini–Cartan theory as a fundamental building block for a quantum theory of gravity

Background

BFV Theories of Gravity

AKSZ EH

AKSZ PC