Mapping class group actions and their applications to 3D gravity

Abstract

In this thesis we study mapping class group actions of the three-dimensional Reshetikhin-Turaev topological quantum field theory motivated by questions in three-dimensional quantum gravity where mapping class group averages appear as candidates for gravity partition functions. One of the main results is a bulk-boundary correspondence between mapping class group averages and a rational conformal field theory whose chiral mapping class group representations are irreducible and obey a finiteness property. As primary examples we find that Ising-type modular fusion categories and their Reshetikhin-Turaev topological quantum field theories are characterised by these properties. Finally, for a given modular fusion category C we show that if the mapping class group representation on every surface without marked points is irreducible then there is a unique indecomposable C-module category with module trace, namely C itself. Such module categories describe surface defects in three-dimensional Reshetikhin-Turaev topological quantum field theories. This links irreducibility of mapping class group representations and absence of non-trivial surface defects.