Combinatorial quantisation of the Euclidean torus universe

TL;DR

This paper develops a combinatorial quantisation of the Euclidean torus universe using Chern-Simons theory and Drinfel'd double representations, resulting in a quantum algebra with a Hilbert space of square integrable functions on the torus.

Contribution

It introduces a novel combinatorial quantisation approach for the Euclidean torus universe based on its Chern-Simons formulation and Drinfel'd double DSU(2).

Findings

Quantum algebra consists of two commuting Heisenberg algebras.

Hilbert space is isomorphic to square integrable functions on the torus.

The theory exhibits a unitary representation of the modular group.

Abstract

We quantise the Euclidean torus universe via a combinatorial quantisation formalism based on its formulation as a Chern-Simons gauge theory and on the representation theory of the Drinfel'd double DSU(2). The resulting quantum algebra of observables is given by two commuting copies of the Heisenberg algebra, and the associated Hilbert space can be identified with the space of square integrable functions on the torus. We show that this Hilbert space carries a unitary representation of the modular group and discuss the role of modular invariance in the theory. We derive the classical limit of the theory and relate the quantum observables to the geometry of the torus universe.