Extended matter coupled to BF theory

TL;DR

This paper explores a four-dimensional topological field theory coupled with matter, revealing classical solutions related to cosmic strings and providing a quantum framework with dualities to quantum gravity Feynman diagrams.

Contribution

It introduces a detailed analysis of classical and quantum aspects of matter-coupled BF theory in four dimensions, including solutions and a regularization scheme.

Findings

Classical solutions correspond to Einstein's equations with cosmic strings.

Transition amplitudes are dual to Feynman diagrams in 3D quantum gravity.

The theory's topological invariance allows removing the regularization.

Abstract

Recently, a topological field theory of membrane-matter coupled to BF theory in arbitrary spacetime dimensions was proposed [1]. In this paper, we discuss various aspects of the four-dimensional theory. Firstly, we study classical solutions leading to an interpretation of the theory in terms of strings propagating on a flat spacetime. We also show that the general classical solutions of the theory are in one-to-one correspondence with solutions of Einstein's equations in the presence of distributional matter (cosmic strings). Secondly, we quantize the theory and present, in particular, a prescription to regularize the physical inner product of the canonical theory. We show how the resulting transition amplitudes are dual to evaluations of Feynman diagrams coupled to three-dimensional quantum gravity. Finally, we remove the regulator by proving the topological invariance of the transition amplitudes.