Skein theoretic topological quantum field theories associated to quantum groups

Abstract

This thesis is concerned with topological quantum field theories associated to the representation theory of quantum groups, both at and away from root of unity quantum parameters. These TQFTs are closely related to skein-theoretic constructions.

The first part of the thesis describes computations when the quantum parameter is generic. We determine the dimension of the Kauffman bracket skein module for mapping tori of the 2-torus, which is the skein module associated to the representation theory of quantum SL_2. In the process, we give a decomposition of the twisted Hochschild homology of the skein algebra for GL_N or SL_N, which is a direct summand of the whole skein module, and from which the dimensions follow easily in the cases of SL_2 and GL_1. Skein modules for 3-manifolds form part of an oriented (3, 2)-TQFT first defined by Walker, which assigns to a surface its skein category, a natural generalization of the skein algebra. The computations of this part show that Walker’s skein TQFT does not extend to an oriented 4-dimensional theory which extends the natural action of SL_3(Z) on the skein module of the 3-torus.

In the second part of the thesis we treat the root of unity case. We construct a relative version of the Crane--Yetter topological quantum field theory in four dimensions, from non-semisimple data. Our theory is defined relative to the classical G-gauge theory in five dimensions -- this latter theory assigns to each manifold M the appropriate linearization of the moduli stack of G-local systems, called the character stack. Our main result is to establish a relative invertibility property for our construction. This invertibility echoes -- recovers and greatly generalizes -- the key invertibility property of the original Crane--Yetter theory which allowed it to capture the framing anomaly of the celebrated Witten--Reshetikhin--Turaev theory. In particular our invertibilty statement at the level of surfaces implies a categorical, stacky version of the unicity theorem for skein algebras; at the level of 3-manifolds it equips the character stack with a canonical line bundle. Regarded as a topological symmetry defect of classical gauge theory, our work establishes invertibility of this defect by a gauging procedure.