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Combinatorial quantisation of the Euclidean torus universe

C. Meusburger, K. Noui

TL;DR

This paper develops a Hamiltonian, combinatorial quantisation of the Euclidean torus universe by exploiting the representation theory of the Drinfel'd double DSU(2) and its dual. It constructs a graph algebra for the torus, identifies a unique kinematical representation on DSU(2)^*, and defines a gauge-invariant physical Hilbert space as a regularised, $L^2$-space on the torus, with gauge-invariant observables forming two commuting copies of the Heisenberg algebra. The quantum flatness constraint is implemented via adjoint-invariance under DSU(2), enabling explicit Wilson-loop observables and a unitary representation of the modular group SL(2, Z) on the gauge-invariant sector; however, enforcing full modular invariance at the level of physical states is shown to be problematic. The results establish a clear link between the combinatorial quantisation framework and the classical Fock–Rosly Poisson structure, providing a robust route to analyze spectra, uncertainty relations, and the classical limit in a non-compact setting, with potential generalisations to other groups and topologies.

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.

Combinatorial quantisation of the Euclidean torus universe

TL;DR

This paper develops a Hamiltonian, combinatorial quantisation of the Euclidean torus universe by exploiting the representation theory of the Drinfel'd double DSU(2) and its dual. It constructs a graph algebra for the torus, identifies a unique kinematical representation on DSU(2)^*, and defines a gauge-invariant physical Hilbert space as a regularised, -space on the torus, with gauge-invariant observables forming two commuting copies of the Heisenberg algebra. The quantum flatness constraint is implemented via adjoint-invariance under DSU(2), enabling explicit Wilson-loop observables and a unitary representation of the modular group SL(2, Z) on the gauge-invariant sector; however, enforcing full modular invariance at the level of physical states is shown to be problematic. The results establish a clear link between the combinatorial quantisation framework and the classical Fock–Rosly Poisson structure, providing a robust route to analyze spectra, uncertainty relations, and the classical limit in a non-compact setting, with potential generalisations to other groups and topologies.

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.

Paper Structure

This paper contains 33 sections, 13 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The classical Euclidean torus universe
  3. Geometry of the torus universe
  4. The static torus universe
  5. The evolving torus universe
  6. The description in terms of a Chern-Simons gauge theory
  7. Fock and Rosly's description of the phase space
  8. Quantisation and constraint implementation
  9. Combinatorial quantisation of the Euclidean torus universe
  10. The combinatorial quantisation formalism
  11. Quantum group structures for the Euclidean torus universe
  12. The Hopf algebra $DSU(2)$ and its dual
  13. Irreducible representations of $DSU(2)$
  14. Construction of the graph algebra
  15. Representations of the graph algebra - kinematical states
  16. ...and 18 more sections

Key Result

Lemma 3.2

The Alekseev isomorphism $\Gamma:{\mathcal{L}}(T, ISU(2))\rightarrow H(DSU(2))$ between the graph algebra $\mathcal{L}(T, ISU(2))$ of the torus and the Heisenberg double $H(DSU(2))$ is given by

Figures (2)

  • Figure 1: The static and the evolving torus universe: a) Action of $\pi_1(T)$ on $\mathbb E^3$ for the static torus universe ($\mu_A=\mu_B=0$). b) Action of $\pi_1(T)$ on $\mathbb E^3$ for the generic evolving torus universe with $s_A,s_B\neq 0$. c) Action of $\pi_1(T)$ on $\mathbb E^3$ for the evolving torus universe for $s_B=0$.
  • Figure 2: Ciliated fat graph associated with a set of generators of the fundamental group of the torus. The dotted line corresponds to the cilium. The lines labelled $a$ and $b$ to the $a$- and $b$-cycle of the torus.

Theorems & Definitions (17)

  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Definition 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Definition 5.1
  • Lemma 5.2
  • ...and 7 more