UC Berkeley

Topological and conformal field theories and integrable systems can be described by the algebraic structures of quantum groups and quantum affine algebras. Boundary conditions and defects for these theories are described via algebraic constructions from these quantum groups or quantum affine algebras.

In the first third of this thesis we describe constructions associated with three dimensional topological field theories. First we compute some Brauer-Picard groups which characterize nontrivial invertible structures that can be assigned using the cobordism hypothesis with singularities. Then we give a construction of bimodule categories using the procedure of covariantization(transmutation) of coquasitriangular Hopf algebras. The first third closes with a reduction by taking $K_0$ which results in algebraic K-theory of fusion rings.

The next third describes spin chains which are governed by quantum affine algebras. The first chapter gives a proof of the asymptotic completeness of the algebraic Bethe Ansatz for the XXZ chain. Then we describe a similar reduction of taking $K_0$ which results in algebraic K-theory of cluster algebras.

The last third of this thesis describes the appearance of these defects in topological field theories constructed by the AKSZ procedure. In the three dimensional case, this gives perturbative perspectives on the theories covered in the first third of the thesis. The last chapter gives the connection with topological insulators as they fit into the general AKSZ paradigm.