Worldsheet formulations of gauge theories and gravity

TL;DR

This paper introduces a worldsheet formulation for gauge theories and gravity, representing their evolution as sums over worldsheets, which could be promising for quantum gravity research due to its compatibility with diffeomorphism invariance.

Contribution

It develops a worldsheet framework for gauge theories and discusses its potential application to Euclidean general relativity, linking loop quantum gravity and string-inspired methods.

Findings

Expressed lattice $SU(2)$ BF and Yang-Mills theories as worldsheet theories

Presented formal worldsheet forms of continuum $U(1)$ theories

Argued the framework's suitability for Euclidean GR, though weighting remains unknown

Abstract

The evolution operator for states of gauge theories in the graph representation (closely related to the loop representation of Gambini and Trias, and Rovelli and Smolin) is formulated as a weighted sum over worldsheets interpolating between initial and final graphs. As examples, lattice $SU(2)$ BF and Yang-Mills theories are expressed as worldsheet theories, and (formal) worldsheet forms of several continuum $U(1)$ theories are given. It is argued that the world sheet framework should be ideal for representing GR, at least euclidean GR, in 4 dimensions, because it is adapted to both the 4-diffeomorphism invariance of GR, and the discreteness of 3-geometry found in the loop representation quantization of the theory. However, the weighting of worldsheets in GR has not yet been found.