Vanda Farsad | Searcharxiv

arXiv2018

SL(2,Z)-action for ribbon quasi-Hopf algebras

We study the universal Hopf algebra L of Majid and Lyubashenko in the case that the underlying ribbon category is the category of representations of a finite dimensional ribbon quasi-Hopf algebra A. We show that L=A* with coadjoint action and compute the Hopf algebra structure morphisms of L in terms of the defining data of A. We give explicitly the condition on A which makes Rep(A) factorisable and compute Lyubashenko's projective SL(2,Z)-action on the centre of A in this case. The point of this exercise is to provide the groundwork for the applications to ribbon categories arising in logarithmic conformal field theories - in particular symplectic fermions and W_p-models - and to test a conjectural non-semisimple Verlinde formula.

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arXiv2017

The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras $Q(N)$ for $N$ a positive integer: as an algebra, $Q(N)$ is the semidirect product of $\mathbb{C}\mathbb{Z}_2$ with the direct sum of a Grassmann and a Clifford algebra in $2N$ generators. We show that $Rep Q(N)$ is ribbon equivalent to the symplectic fermion category $SF(N)$ that was computed by the third author from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of $V_{ev}$, the even part of the symplectic fermion vertex operator super algebra. Using the formalism developed in our previous paper we compute the projective $SL(2,\mathbb{Z})$-action on the centre of $Q(N)$ as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the $SL(2,\mathbb{Z})$-action computed from $Q(N)$ agrees projectively with that on pseudo trace functions of $V_{ev}$.

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