It is known that if gauge conditions have Gribov zero modes, then topological symmetry is broken. In this paper we apply it to the Witten type topological gravitational theory in dimension n ≥ 3. Our choice of the gauge condition for conformal invariance is R + α = 0, where R is the Ricci scalar curvature. We find when α ≠ 0, topological symmetry is not broken, but when α = 0 and solutions of the Einstein equations exist then topological symmetry is broken. These 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 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 becomes semiclassical Einstein gravitational theory under a special definition of observables.
Metrics of Eguchi–Hanson types with the negative constant scalar curvature
