Representation theoretic tools for equivariant and orbifold conformal field theories

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Rational two-dimensional conformal quantum field theories can be described by separable symmetric Frobenius algebras in modular tensor categories – e.g. subcategories of representation categories of affine Kac-Moody algebras – and their representation theory; they are thus particularly amenable to a mathematical treatment. The TFT construction furnishes a constructive proof of existence for a consistent set of correlators for these theories. In the present project, a G-equivariant version of the theory shall be studied. The equivariant version admits, under suitable conditions, an orbifold construction yielding – in a similar way as “passing to the quotient” in other mathematical theories – new examples of non-equivariant theories. The project has three major aims: 1. The construction of examples of G-equivariant modular tensor categories. Typically, such a construction is done in two steps: first, a family of module categories is constructed and endowed, in a second step, by additional structure completing the equivariant data. A particularly tractable example is provided by the action of the permutation group on the tensor product of identical modular tensor categories. A Lie-theoretic example is provided by representation categories based on twisted representations of affine Kac-Moody algebras and the action of the automorphism group of the corresponding compact, connected and simply-connected Lie group on it. 2. A conceptual understanding of the orbifold construction using pseudomonads and their representation theory in a three-categorical setting. A particularly interesting example should be provided by the Brauer three-group. 3. An equivariant generalization of the TFT construction of RCFT correlators.