[Paper Review] Quantisation of Super Teichmüller Theory

Nezhla Aghaei, Michal Pawelkiewicz+1 more|arXiv (Cornell University)|Dec 8, 2015

Black Holes and Theoretical Physics19 references3 citations

TL;DR

This paper presents a quantization of super Teichm"uller theory using Kashaev-type coordinates on ideal triangulations of super Riemann surfaces, with explicit dependence on spin structures encoded via Kasteleyn orientations. It constructs a projective unitary representation of the super Ptolemy groupoid, proving that the quantum theory is independent of triangulation choice, thereby establishing a consistent quantum framework for super Riemann surfaces with applications to super-Liouville theory and N=2 supersymmetric gauge theories.

ABSTRACT

We construct a quantisation of the Teichmueller spaces of super Riemann surfaces using coordinates associated to ideal triangulations of super Riemann surfaces. A new feature is the non-trivial dependence on the choice of a spin structure which can be encoded combinatorially in a certain refinement of the ideal triangulation. By constructing a projective unitary representation of the groupoid of changes of refined ideal triangulations we demonstrate that the dependence of the resulting quantum theory on the choice of a triangulation is inessential.

Motivation & Objective

• To develop a consistent quantization of super Teichm"uller spaces for super Riemann surfaces.
• To incorporate spin structure dependence combinatorially via Kasteleyn orientations on refined triangulations.
• To demonstrate that the resulting quantum theory is independent of the choice of triangulation.
• To construct a projective unitary representation of the super Ptolemy groupoid governing changes of refined triangulations.
• To lay the foundation for quantum invariants of 3-manifolds and connections to super-Liouville conformal field theory.

Proposed method

• Utilizes Kashaev-type coordinates on ideal triangulations of super Riemann surfaces to parametrize super Teichm"uller space.
• Encodes spin structures via Kasteleyn orientations, refining the triangulation data to include fermionic degrees of freedom.
• Introduces a quantum algebra based on the supersymmetric non-compact quantum dilogarithm to define Fock coordinates.
• Constructs unitary flip operators for edge flips (superflips) that implement changes of triangulation in the quantum theory.
• Derives the quantum super Ptolemy groupoid relations, including the superpentagon equation, to ensure consistency across triangulations.
• Uses projective unitary representations to show that different triangulation choices yield unitarily equivalent quantum theories.

Experimental results

Research questions

• RQ1 How can super Teichm"uller theory be consistently quantized using triangulation-based coordinates while accounting for spin structure dependence?
• RQ2 What is the algebraic structure of the quantum observables in super Teichm"uller theory, and how do they transform under changes of triangulation?
• RQ3 How is the dependence on spin structure encoded combinatorially in the quantum framework?
• RQ4 Can a projective unitary representation of the super Ptolemy groupoid be constructed to ensure triangulation independence?
• RQ5 What is the role of the supersymmetric non-compact quantum dilogarithm in realizing the quantum algebra of observables?

Key findings

• The quantum theory of super Teichm"uller space is independent of the choice of triangulation due to the existence of unitary flip operators that relate different triangulation Hilbert spaces.
• Spin structure dependence is fully encoded in the combinatorics of Kasteleyn-structured triangulations, providing a discrete and finite description of fermionic degrees of freedom.
• The quantum algebra of Fock coordinates satisfies a supersymmetric version of the pentagon identity, known as the superpentagon equation, ensuring consistency of the quantum theory.
• The transformation rules for quantum Fock variables under superflips are explicitly computed using the supersymmetric non-compact quantum dilogarithm.
• The projective unitary representation of the super Ptolemy groupoid ensures that all quantum theories associated to different refined triangulations are physically equivalent.
• The construction provides a concrete realization of quantum super Teichm"uller theory that is compatible with the expectations from super-Liouville theory and N=2 supersymmetric gauge theories.