Large Diffeomorphisms in (2+1)-Quantum Gravity on the Torus

TL;DR

This paper investigates the impact of large diffeomorphisms, specifically modular transformations, on (2+1)-dimensional quantum gravity on a torus, revealing fundamental obstructions to finite-dimensional unitary representations.

Contribution

It demonstrates that modular transformations prevent finite-dimensional unitary representations in the quantum gravity Hilbert space, challenging existing approaches.

Findings

Finite-dimensional unitary representations are ruled out by modular transformations.

Piecewise continuity assumptions also exclude finite-dimensional representations.

The behavior of modular transformations complicates the representation theory in quantum gravity models.

Abstract

The issue of how to deal with the modular transformations -- large diffeomorphisms -- in (2+1)-quantum gravity on the torus is discussed. I study the Chern-Simons/connection representation and show that the behavior of the modular transformations on the reduced configuration space is so bad that it is possible to rule out all finite dimensional unitary representations of the modular group on the Hilbert space of $L^2$-functions on the reduced configuration space. Furthermore, by assuming piecewise continuity for a dense subset of the vectors in any Hilbert space based on the space of complex valued functions on the reduced configuration space, it is shown that finite dimensional representations are excluded no matter what inner-product we define in this vector space. A brief discussion of the loop- and ADM-representations is also included.