Using 3D Stringy Gravity to Understand the Thurston Conjecture

TL;DR

This paper introduces a string-inspired 3D Euclidean field theory with additional fields to analyze the Thurston conjecture through a modified Ricci flow, linking topology to field theory parameters.

Contribution

It develops a novel 3D field theory framework that encodes Thurston geometries via modified Ricci flow equations derived from string-inspired models.

Findings

Nine classes of solutions correspond to Thurston geometries and a squashed sphere.

The theory obeys a Birkhoff theorem for constant dilaton.

Topology is encoded in the parameters of the underlying field theory.

Abstract

We present a string inspired 3D Euclidean field theory as the starting point for a modified Ricci flow analysis of the Thurston conjecture. In addition to the metric, the theory contains a dilaton, an antisymmetric tensor field and a Maxwell-Chern Simons field. For constant dilaton, the theory appears to obey a Birkhoff theorem which allows only nine possible classes of solutions, depending on the signs of the parameters in the action. Eight of these correspond to the eight Thurston geometries, while the ninth describes the metric of a squashed three sphere. It therefore appears that one can construct modified Ricci flow equations in which the topology of the geometry is encoded in the parameters of an underlying field theory.