Mapping class group representations and Morita classes of algebras (Q6149227)

Let \(\mathcal{C}\)\ be a modular fusion category, that is to say, a finitely semisimple ribbon category with simple tensor unit whose braiding is non-degenerate. Such categories give rise to three-dimensional topological quantum field theories [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007); \textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. 2nd revised ed. Berlin: Walter de Gruyter (2010; Zbl 1213.57002)] and consequently also to (projective) representations of surface mapping class groups. This paper aims to establish the following main result (Theorem 5.1). Theorem. Let \(\mathcal{C}\)\ be a modular fusion category over an algebraically closed field of characteristic zero. If the projective mapping class group representations \(V_{g}^{\mathcal{C}}\)\ are irreducible for all \(g\geq0\), then every simple non-degenerate algebra in \(\mathcal{C}\)\ is Morita-equivalent to the tensor unit. A result closely related to the above theorem was established in [\textit{J. E. Andersen} and \textit{J. Fjelstad}, Lett. Math. Phys. 91, No. 3, 215--239 (2010; Zbl 1197.57030)], where it was shown that if there is a \(g\geq1\)\ such that \(V_{g}^{\mathcal{C}}\)\ is irreducible, then for every simple non-degenerate algebra \(A\), its full center \(Z\left( A\right) \in\mathcal{C}\boxtimes \mathcal{C}^{\mathrm{rev}}\)\ has underlying object \[ \bigoplus_{i\in I}i^{\ast}\times i \] The converse of the above theorem does not hold. For \(\mathcal{C}\left( \mathrm{sl}\left( 2\right) ,k\right) \)\ with \(k\)\ odd, there is a unique such Morita class [\textit{V. Ostrik}, Transform. Groups 8, No. 2, 177--206 (2003; Zbl 1044.18004)], but for \(k+2\)\ odd and not prime or the square of a prime, \(V_{g=1}^{\mathcal{C}}\)\ is reducible. The synopsis of the paper goes as follows. \begin{itemize} \item[\S 2] reviews how to obtain projective representations of surface mapping class groups from a modular fusion category [\textit{N. Reshetikhin} and \textit{V. G. Turaev}, Invent. Math. 103, No. 3, 547--597 (1991; Zbl 0725.57007); \textit{V. G. Turaev}, Quantum invariants of knots and 3-manifolds. 2nd revised ed. Berlin: Walter de Gruyter (2010; Zbl 1213.57002)]. \item[\S 3] describes how to obtain mapping class group invariants from a modular invariant symmetric Frobenius algebra [\textit{J. Fjelstad} et al., Theory Appl. Categ. 16, 342--433 (2006; Zbl 1151.81038); \textit{L. Kong} and \textit{I. Runkel}, Commun. Math. Phys. 292, No. 3, 871--912 (2009; Zbl 1214.81251); \textit{L. Kong} et al., Adv. Math. 262, 604--681 (2014; Zbl 1301.81254)]. \item[\S 4] recalls the relation between Morita classes of algebras and their full centers [\textit{P. Etingof} et al., Ann. Math. (2) 162, No. 2, 581--642 (2005; Zbl 1125.16025); \textit{L. Kong} and \textit{I. Runkel}, Adv. Math. 219, No. 5, 1548--1576 (2008; Zbl 1156.18003)]. \item[\S 5] gives the proof of the main theorem by reducing the difference between the algebra structures of the full center of a given algebra and that of the tensor unit to a symmetric 2-cocycle on the universal grading group, which must be a coboundary. \end{itemize}