Boundaries, defects and Frobenius algebras

Abstract

The interpretation of D-branes in terms of open strings has lead to much interest in boundary conditions of two-dimensional conformal field theories (CFTs). These studies have deepened our understanding of CFT and allowed us to develop new computational tools. The construction of CFT correlators based on combining tools from topological field theory and non-commutative algebra in tensor categories, which we summarize in this contribution, allows e.g. to discuss, apart from boundary conditions, also defect lines and disorder fields.

Diagrammatics for Bose condensation in anyon theories

“…This is important to the classification problem for CFTs 40,41 , and possibly even mostreous moonshine 42 or generalizations thereof 43,44 , but we will not dwell on such abstract topics here. The same structures have also been used in the description of boundary conditions and topological defects in CFT [31][32][33]45 . On the other hand there have been studies on the effects of nontrivial boundaries in lattice realizations of topological order based on Kitaev's quantum double model 46,47 and Levin-Wen models 48 .…”

Symmetry breaking, permutation D-branes on group manifolds: boundary states and geometric description

“…Instead, let us recall that in order to have a well defined Lagrangian action of the WZW theory on a world-sheet with boundary, the restriction of the WZW three-form to the D-brane worldvolume has to belong to the trivial cohomology class [14]. 4 More precisely, there should exist a globally well-defined two-form ω (2) on the brane worldvolume, satisfying the equation…”

“…The construction which we present does not in general require these manifolds to be identical, though most of our explicit formulas and examples will be given for these cases. There are two main tools which are used in our construction: the permutation symmetry between the group manifolds as it was used in [1,2,3,4,5,6,7] 1 and the construction of symmetry breaking branes using orbifolds in combination with T-duality, as proposed in [8,9]. The resulting branes of our construction are symmetry breaking, i.e.…”

A geometric construction for permutation equivariant categories from modular functors

“…, N } of N elements, one obtains from permutation equivariant categories permutation orbifolds. These categories [FRS03] enter in the construction of boundary conditions for tensor product theories that break permutation symmetries, so-called permutation branes [Rec03]. Moreover, they conveniently encode refined aspects of the family of representations of mapping class groups that is associated to the modular tensor category C. This explains the role of permutation orbifolds in Bantay's approach to the congruence subgroup conjecture ( [Ban02,Ban03]).…”