Diff(SIGMA) and Metrics from Hamiltonian-TQFT's in 2+1 Dimensions

TL;DR

This paper explores the construction of Hamiltonians in 2+1 and 1+1 dimensional topological quantum field theories using $BF$ gauge theories, revealing connections to canonical gravity and metric structures.

Contribution

It introduces a method to derive Hamiltonians from $BF$ theories as anti-commutators of BRST operators, linking TQFTs with canonical gravity constraint algebras.

Findings

Hamiltonians constructed as BRST anti-commutators in TQFTs.

Existence of a homomorphism between TQFT and canonical gravity constraint algebras.

Metrics on hypersurfaces derived from the TQFT framework.

Abstract

The constraints of $BF$ topological gauge theories are used to construct Hamiltonians which are anti-commutators of the BRST and anti-BRST operators. Such Hamiltonians are a signature of Topological Quantum Field Theories (TQFT's). By construction, both classes of topological field theories share the same phase spaces and constraints. We find that, for 2+1 and 1+1 dimensional space-times foliated as $M=\Sigma\times\IR$, a homomorphism exists between the constraint algebras of our TQFT and those of canonical gravity. The metrics on the two-dimensional hypersurfaces are also obtained.