Topological and conformal interfaces in two-dimensional quantum field theories
Abstract
Description
We study conformal boundary conditions, topological defects as well as various concepts
related to their presence in two-dimensional Rational Conformal Field Theories (RCFTs). In
this situation, one new concept for example is a topological defect line with one of its ends
attached to a boundary. Such junctions are called open topological defects. The first goal of
this thesis is to consider new fusing matrices that arise from the existence of such junctions
in our theory. For instance, one type of fusing matrices is related to the fusion of two open
defects while another arises when an open defect junction passes through a boundary operator.
The former appears in the structure constants of an associative algebra known as the boundary
tube algebra while the latter plays an important role in constraining boundary Renormalisation
Group (RG) flows triggered by that boundary operator. We use the Topological Field Theory
(TFT) approach to RCFTs based on Frobenius algebra objects in Modular Tensor Categories
(MTCs) to describe the general structure associated with such matrices and how to compute
them from a given MTC, Frobenius algebra object and its representation theory. We illustrate
the computational process on the rational free boson theories where we also discuss applications
to boundary RG flows.
Our second goal, motivated by applications in RG flows, is to classify pairs consisting of a
local operator and a topological defect which commutes or anticommutes with it. We discuss
both bulk and boundary versions of the problem. In the case of the charge conjugation modular
invariant (anti-)commuting configurations in each problem can be obtained when a certain
condition on the fusion rules is realised. We study the corresponding fusion rule problems in
detail. While in the bulk case it reduces to realising the a × b = c fusion rule which was studied
in [1], in the boundary it leads to a new type of problem. We obtain a full solution to this
problem for the SU(2) and SU(3) Wess–Zumino–Witten (WZW) models and minimal models.
We present a way to obtain solutions in the case of general WZW models using simple currents.