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A geometric construction for permutation equivariant categories from modular functors. (English) Zbl 1255.18005

Note that the tensor categories of two-dimensional topological field theories and of commutative Frobenius algebras are equivalent as symmetric tensor categories. Various generalizations of this assertion have been addressed. On the one hand side, given any (finite) group \(G\), \(G\)-equivariant two-dimensional topological field theories have lead to the notion of a \(G\)-Frobenius algebra. On the other hand, for three-dimensional topological field theories, one is lead to the algebraic structure of a modular tensor category. A structure related to three-dimensional topological field theories that is more appropriate for the authors’ purposes is given by the notion of a modular functor. For any abelian category \(\mathcal{C}\) satisfying suitable finiteness conditions, the following correspondences have been established: (1) \(\mathcal{C}\)-extended genus \(0\) modular functors correspond to structures of (weakly) ribbon categories on \(\mathcal{C}\); (2) Higher genus modular functors correspond to structures of a modular category on \(\mathcal{C}\).
Let \(G\) be a finite group. A partial generalization of the preceding statements asserts that the structure of a (weakly) \(G\)-equivariant fusion category on a given \(G\)-equivariant abelian category \(\mathcal{C}^G\) is equivalent to a \(\mathcal{C}^G\)-extended \(G\)-equivariant genus \(0\) modular functor.
The present paper is devoted to the construction of a \(G\)-equivariant fusion category \(\mathcal{C}^{\mathcal{X}}\) called the permutation equivariant tensor category, from a finite \(G\)-set \(\mathcal{X}\) and a modular tensor category \(\mathcal{C}\). The construction is geometric and uses the formalism of modular functors. The authors addressed the problem by constructing from these data a \(G\)-modular functor. Given a finite \(G\)-set \(\mathcal{X}\), one can construct a symmetric monoidal functor \(\mathcal{F}_{\mathcal{X}}\) from the category Gcob(d) of \(G\)-cobordisms to the category cob(d) of cobordisms. This functor assigns to a principal \(G\)-cover \((P \to M)\) the total space of the associated bundle \[ \mathcal{F}_{\mathcal{X}}(P\to M):=\mathcal{X} \times_GP=\mathcal{X}\times P/((g^{-1}x,p)\sim (x,gp)), \] Pulling back topological field theories along this functor \(\mathcal{F}_{\mathcal{X}}\), one can find \(G\)-equivariant theories. This functor is used in order to study geometric and algebraic properties of two-dimensional \(G\)-equivariant field theories. The authors used the cover functor \(\mathcal{F}_{\mathcal{X}}\) to obtain a \(G\)-equivariant modular functor for every \(G\)-set \(\mathcal{X}\) and modular tensor category \(\mathcal{C}\).
As an application, the authors concretely worked out a complete set of structure morphisms for a \(\mathbb{Z}/2\)-permutation equivariant fusion category, obtained from the permutation action of the group \(\mathbb{Z}/2\) on the set of two elements, completing thus the program initiated by the authors in an earlier paper. It is worth mentioning that the geometric structure unraveled in this paper provides clear guiding principles to write down a consistent set of constraint morphisms.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
81T99 Quantum field theory; related classical field theories
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References:

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