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

Abstract

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 $RepQ(N)$ 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 $V_{ev}$.

81pp; v3: extended introduction; new Remark 5.4 about decomposition of SL(2,Z) representations; new Remarks 6.2 and 6.7 on pseudo-trace functions; typos fixed, references updated; version for publication in Advances in Mathematics

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• From boundary to bulk in logarithmic CFT
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• A quasi-Hopf algebra for the triplet vertex operator algebra
• SL(2,Z)-action for ribbon quasi-Hopf algebras
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Cited by in corpus (3)

• Parity and Spin CFT with boundaries and defects
• Davydov-Yetter cohomology and relative homological algebra
• Non-semisimple link and manifold invariants for symplectic fermions