Non-orientability constrains couplings in 2+1 quantum gravity

Posted on by
Jorma Louko

Jorma Louko is an Associate Professor in Applied Mathematics at the School of Mathematical Sciences, University of Nottingham

2+1 gravity is topological even without spacetime orientability, and quantisable for selected couplings

General relativity in four and more spacetime dimensions has local dynamical degrees of  freedom, as manifested for example in gravitational waves. In three spacetime dimensions,  by contrast, Einstein’s equations preclude local dynamics but allow still dynamics in the  global properties. This makes (2+1)-dimensional general relativity a dynamically simple but geometrically interesting arena for quantising gravity. Continue reading →

Quantum gravity on a Klein bottle

Posted on by
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 →

Boundary states in higher-dimensional loop quantum gravity

Posted on by

Higher-dimensional Chern-Simons theory appears in the description of isolated horizon boundaries in higher-dimensional General Relativity.

It is a well-known fact that the presence of boundaries (“edges”) leads to the concept of boundary states, which e.g. ensure gauge invariance for parallel transporters ending on the boundary. Most famously, the quantum Hall effect can be explained using such states. In the context of black hole (quantum) physics, boundary states are important since they are microscopic states associated to the horizon of the black hole. Counting such boundary states in agreement with the macroscopic properties of a black hole is thus a good candidate for a microscopic explanation of the Bekenstein-Hawking entropy. This paradigm has been successfully employed in 3+1 dimension in the context of loop quantum gravity, a canonical quantisation of General Relativity. Continue reading →