Quantum gravity on a Klein bottle

Figure 1a

(A) Klein bottle, or the non-orientable surface of genus 2; The fundamental polygon representation of the Klein bottle is shown in the inset.

Figure 1b

(B) The orientable double cover of the Klein bottle is the orientable surface of genus 1, or the toroid. Closed loops on the double cover that traverse the non-orientable boundary— red/blue line in (B)— wind around the non-orientable surface in panel (A) twice.

In this work we study a model of quantum gravity on two-dimensional, non-orientable manifolds, for example a Klein bottle. We find that for a simplified version of quantum gravity called U(1) BF theory, a generalization of U(1) Chern-Simons theory, the fact that the manifold is non-orientable induces severe constraints on the values allowed for the coupling constant appearing in the action; in fact it can only take values of ½, 1, or 2. This comes about because the coupling constant appears in the commutation relation (or uncertainty relation) for the fields, and because the fields in the effective gauge theory must be consistent with the discrete symmetry groups for homeomorphisms on manifold. These discrete symmetry groups include the large gauge transformation group, the holonomy group, and the mapping class group. Continue reading