(2+1)-Dimensional Chern-Simons Gravity as a Dirac Square Root

TL;DR

This paper demonstrates that in (2+1)-dimensional Chern-Simons gravity on a torus, wave functions are modular forms of weight 1/2 and their evolution follows a Dirac equation, linking quantum gravity to spinor structures.

Contribution

It explicitly constructs the transformation between Chern-Simons and ADM quantum gravity formulations and identifies the wave functions as spinors on the moduli space.

Findings

Chern-Simons wave functions are modular forms of weight 1/2.

Wave function evolution is governed by a Dirac equation.

Establishes a link between quantum gravity and spinor structures on moduli space.

Abstract

For (2+1)-dimensional spacetimes with the spatial topology of a torus, the transformation between the Chern-Simons and ADM versions of quantum gravity is constructed explicitly, and the wave functions are compared. It is shown that Chern-Simons wave functions correspond to modular forms of weight 1/2, that is, spinors on the ADM moduli space, and that their evolution (in York's ``extrinsic time'' variable) is described by a Dirac equation. (This version replaces paper 9109006, which was garbled by my mailer.)