BRST-invariant boundary conditions and strong ellipticity

TL;DR

This paper investigates boundary conditions in quantum gravity that preserve BRST symmetry and ensure ellipticity, enabling well-defined effective actions using heat-kernel methods.

Contribution

It introduces new BRST-invariant boundary conditions fixing extrinsic curvature, addressing non-ellipticity issues in quantum gravity boundary value problems.

Findings

New boundary conditions fix extrinsic curvature in quantum gravity.

These conditions restore ellipticity and enable heat-kernel techniques.

Discussion on when boundary gauge-fixing resolves non-ellipticity.

Abstract

The quantisation of gauge theories usually procedes through the introduction of ghost fields and BRST symmetry. In the case of quantum gravity in the presence of boundaries, the BRST-invariant boundary value problem for the gauge field operators is non-elliptic, and consequently the definition of the effective action using heat-kernel techniques becomes problematic. This paper examines general classes of BRST-invariant boundary conditions and presents new boundary conditions for quantum gravity which fix the extrinsic curvature on the boundary and lead to a well-defined effective action. This prompts a discussion of the wider issue of non-ellipticity in BRST-invariant boundary value problems and when the use of gauge-fixing terms on the boundary can resolve the issue.