SU(2) Flat Connection on Riemann Surface and Twisted Geometry with Cosmological Constant

TL;DR

This paper explores the relationship between SU(2) flat connections on Riemann surfaces and twisted geometries in 3D space, proposing a generalized phase space framework that incorporates the cosmological constant.

Contribution

It introduces a novel connection between flat connections and twisted geometries, extending the phase space of Loop Quantum Gravity to include cosmological constant effects.

Findings

Mapping of flat connection quantities to 3D geometrical quantities

Proposal of a generalized phase space for twisted geometry with cosmological constant

Insight into the geometric interpretation of flat connections on Riemann surfaces

Abstract

SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Riemann surface generalizes the phase space of twisted geometry or Loop Quantum Gravity to include the cosmological constant.