The symplectic fermion ribbon quasi-Hopf algebra and the [formula omitted]-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 CZ2 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 in [54] from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of Vev, the even part of the symplectic fermion vertex operator super algebra.Using the formalism developed in [20] we compute the projective SL(2,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,Z)-action computed from Q(N) agrees projectively with that on pseudo trace functions of Vev.