Any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image.
@cite_13: In this paper, we describe the relation between the Turaev--Viro TQFT and the string-net space introduced in the papers of Levin and Wen. In particular, the case of surfaces with boundary is considered in detail.
@cite_1: We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category ( C ) , an equivariant indicator of an object in ( C ) is defined as a functional on the Grothendieck algebra of the quantum double ( Z( C ) ) via generalized Frobenius-Schur indicators. The set of all equivariant indicators admits a natural action of the modular group. Using the properties of equivariant indicators, we prove a congruence subgroup theorem for modular categories. As a consequence, all modular representations of a modular category have finite images, and they satisfy a conjecture of Eholzer. In addition, we obtain two formulae for the generalized indicators, one of them a generalization of Bantay’s second indicator formula for a rational conformal field theory. This formula implies a conjecture of Pradisi-Sagnotti-Stanev, as well as a conjecture of Borisov-Halpern-Schweigert.
@cite_7: A category N of labeled (oriented) trivalent graphs (nets) or ribbon graphs is extended by new generators called fusing, braiding, twist and switch with relations which can be called Moore-Seiberg relations. A functor to N is constructed from the category Surf of oriented surfaces with labeled boundary and their homeomorphisms. Given a (eventually non-semisimple) k-linear abelian ribbon braided category @math with some finiteness conditions we construct a functor from a central extension of N with the set of labels @math ObC to k-vector spaces. Composing the functors we get a modular functor from a central extension of Surfto k-vector spaces.
@cite_3: We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian.
Prior to the current work, certain specific cases had already been solved. In the case of the torus, Ng and Schauenburg's Congruence Subgroup Theorem implies the much stronger result that any Reshitikhin-Turaev representation of the mapping class group of the torus has finite image @cite_1 . Another related result is due to Fjelstad and Fuchs @cite_3 . They showed that, given a surface with at most one boundary component, the mapping class group representations corresponding to the untwisted (i.e. @math ) Dijkgraaf-Witten theory have finite image. Their paper uses an algebraic method of Lyubashenko @cite_7 that gives a projective mapping class group representation to any factorizable ribbon Hopf algebra, in their case, the double @math . In our case, we instead consider the mapping class group action on a vector space of @math -colored embedded graphs defined by Kirillov @cite_13 , yielding a simpler, geometric proof of the more general twisted case.