Diffeomorphism invariant subspaces in Witten's 2+1 quantum gravity on ${\bf R} \times T^2$

TL;DR

This paper investigates the structure of the Hilbert space in Witten's 2+1 quantum gravity on ${\bf R} \times T^2$, showing that large diffeomorphisms act reducibly and constructing explicit invariant subspaces.

Contribution

It demonstrates the reducibility of the large diffeomorphism representation and constructs explicit infinite-dimensional invariant subspaces in the spacelike sector.

Findings

No nontrivial finite-dimensional invariant subspaces under large diffeomorphisms.

Explicit construction of infinite-dimensional invariant subspaces.

Comparison with higher genus surface case.

Abstract

We address the role of large diffeomorphisms in Witten's 2+1 gravity on the manifold ${\bf R} \times T^2$. In a ``spacelike sector" quantum theory that treats the large diffeomorphisms as a symmetry, rather than as gauge, the Hilbert space is shown to contain no nontrivial finite dimensional subspaces that are invariant under the large diffeomorphisms. Infinite dimensional closed invariant subspaces are explicitly constructed, and the representation of the large diffeomorphisms is thus shown to be reducible. Comparison is made to Witten's theory on ${\bf R} \times \Sigma$, where $\Sigma$ is a higher genus surface.