Knot Theory, Exotic Spheres And Global Gravitational Anomalies

Abstract

Global gravitational anomalies, which may ruin the physical consistency of a quantum field theory, are shown to be absent in D = 2 + 1 Topological Quantum Field Theories. This comes about by establishing the absence of disconnected general coordinate transformations in the Hamiltonian version of the theories. Specifically, we prove the invariance of the Chern-Simons-Witten effective action under the mapping class group, that is, the group of equivalence classes of diffeomorphisms that cannot be smoothly deformed to the identity. A newly found relationship between knot theory, exotic spheres and global anomalies, as exhibited in a joint work with L. H. Kauffman and the author, is described. * Paper based on a talk given in the Special Session on Knots and Topological Quantum Field Theory as part of the American Mathematical Society October 1992 Annual Meeting in Dayton, Ohio t Research Supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy of The United States Department of Energy under Contracts DE-AC03-76-SF00098 and in part by a Grant from the Eppley Foundation for Research, Inc. § Permanent Address