Two dimensional general covariance from three dimensions

TL;DR

This paper introduces a 3D generally covariant field theory that effectively behaves as a 2D theory, exhibiting infinite conserved charges, classical integrability, and quantum states linked to 2D graphs.

Contribution

It presents a novel 3D covariant theory with properties akin to a 2D theory, including conserved charges, integrability, and quantum states, expanding understanding of lower-dimensional gravity models.

Findings

The theory has an infinite number of conserved charges for boundaryless 2-manifolds.

The theory is classically integrable with explicit solutions.

Quantum states are associated with 2D graphs in the quantized theory.

Abstract

A 3d generally covariant field theory having some unusual properties is described. The theory has a degenerate 3-metric which effectively makes it a 2d field theory in disguise. For 2-manifolds without boundary, it has an infinite number of conserved charges that are associated with graphs in two dimensions and the Poisson algebra of the charges is closed. For 2-manifolds with boundary there are additional observables that have a Kac-Moody Poisson algebra. It is further shown that the theory is classically integrable and the general solution of the equations of motion is given. The quantum theory is described using Dirac quantization, and it is shown that there are quantum states associated with graphs in two dimensions.