A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions

TL;DR

This paper develops a topological quantum gravity model in 2+1 dimensions using a Chern-Simons formulation based on a doubly extended Galilei group, revealing novel topological and quantum features of Galilean gravity.

Contribution

It introduces a Chern-Simons approach to Galilean gravity in 2+1 dimensions using a two-fold central extension of the Galilei group, and analyzes its quantum structure and topological interactions.

Findings

Defined Galilean gravity as a Chern-Simons theory with a doubly extended Galilei group.

Quantized the theory revealing the role of the quantum double and braid group representations.

Explored the implications for non-commutative Galilean spacetime.

Abstract

We define and discuss classical and quantum gravity in 2+1 dimensions in the Galilean limit. Although there are no Newtonian forces between massive objects in (2+1)-dimensional gravity, the Galilean limit is not trivial. Depending on the topology of spacetime there are typically finitely many topological degrees of freedom as well as topological interactions of Aharonov-Bohm type between massive objects. In order to capture these topological aspects we consider a two-fold central extension of the Galilei group whose Lie algebra possesses an invariant and non-degenerate inner product. Using this inner product we define Galilean gravity as a Chern-Simons theory of the doubly-extended Galilei group. The particular extension of the Galilei group we consider is the classical double of a much studied group, the extended homogeneous Galilei group, which is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure of the doubly extended Galilei group, and quantise the Chern-Simons theory using a Hamiltonian approach. Many aspects of the quantum theory are determined by the quantum double of the extended homogenous Galilei group, or Galilei double for short. We study the representation theory of the Galilei double, explain how associated braid group representations account for the topological interactions in the theory, and briefly comment on an associated non-commutative Galilean spacetime.