Topological Symmetry Breaking on Einstein Manifolds

TL;DR

This paper investigates topological symmetry breaking in topological gravity on Einstein manifolds, showing how gauge conditions and scalar curvature influence symmetry breaking and connecting it to the Yamabe conjecture, leading to a semiclassical Einstein theory.

Contribution

It demonstrates the conditions under which topological symmetry breaks in topological gravity on Einstein manifolds, linking gauge choices, scalar curvature, and the Yamabe conjecture.

Findings

Topological symmetry is broken when the gauge condition parameter $ eq 0$ and Einstein solutions exist.

A new BRS symmetry can be defined after symmetry breaking, redefining physical states.

The theory reduces to semiclassical Einstein gravity with regularized Gribov zero modes.

Abstract

It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to topological gravity in dimension $n \geq 3$. Our choice of the gauge condition for conformal invariance is $R+{\alpha}=0$ , where $R$ is the Ricci scalar curvature. We find when $\alpha \neq 0$, topological symmetry is not broken, but when $\alpha =0$ and solutions of the Einstein equations exist then topological symmetry is broken. This conditions connect to the Yamabe conjecture. Namely negative constant scalar curvature exist on manifolds of any topology, but existence of nonnegative constant scalar curvature is restricted by topology. This fact is easily seen in this theory. Topological symmetry breaking means that BRS symmetry breaking in cohomological field theory. But it is found that another BRS symmetry can be defined and physical states are redefined. The divergence due to the Gribov zero modes is regularized, and the theory after topological symmetry breaking become semiclassical Einstein gravitational theory under a special definition of observables.