Quantum Gravity and World Topology

Abstract

The path integral of quantum gravity probes space-times with nontrivial topology and induces chiral symmetry breaking via an anomaly proportional to ${\ensuremath{\epsilon}}^{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}\ensuremath{\sigma}}R^{\ensuremath{\alpha}}_{\ensuremath{\beta}\ensuremath{\mu}\ensuremath{\nu}}R^{\ensuremath{\beta}}_{\ensuremath{\alpha}\ensuremath{\rho}\ensuremath{\sigma}}$ in the divergence of the axial fermion number current. The corresponding classical instanton solution of Einstein's equations with cosmological term is found. The Euler-Poincar\'e characteristic and ordinary instanton number are also discussed.