Quantizing models of (2+1)-dimensional gravity on non-orientable manifolds

Abstract

Motivated by the relation between the Chern–Simons gauge theory and (2+1)-dimensional gravity, we find a formulation of gauge theories which applies to both orientable and non-orientable manifolds, using orientation bundles and density-valued forms. We show that on a non-orientable manifold, (2+1)D gravity is equivalent to BF theory, which is still topological and can be mapped in turn to Chern–Simons theory on the orientable double cover. By quantizing U(1) BF theory on a non-orientable manifold, we find that non-orientability introduces additional constraints on the quantized BF theory beyond those present for an orientable manifold, such that the coupling constant can only adopt a small number of discrete values. Specifically, for both the Klein bottle of demigenus 2 (N2) and the compact surface of non-orientable genus 3 (Dyck’s surface or N3), we find explicit representations for the holonomy, large gauge, and mapping class groups, as well as the Hilbert space; here the above symmetries along with the non-orientability of the surface constrain the coupling constant k to only take values 1/2, 1, or 2.