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 $V_{ev}$.Comment: 75pp; typos fixed, references update

Reflection and Transmission for Conformal Defects

We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the reflectivity and transmissivity of the defect. Their properties are investigated and they are computed in a number of examples. We obtain a complete answer for all defects in the Ising model and between certain pairs of minimal models. In the case of two conformal field theories with an enhanced symmetry we restrict ourselves to non-trivial defects that can be obtained by a coset construction.Comment: 32 pages + 13 pages appendix, 12 figures; v2: added eqns (2.7), (2.8) and refs [6,7,39,40], version published in JHE

Symplectic fermions and a quasi-Hopf algebra structure on Uˉisl(2)\bar{U}_i sl(2)

We consider the (finite-dimensional) small quantum group $\bar{U}_q sl(2)$ at $q=i$. We show that $\bar{U}_i sl(2)$ does not allow for an R-matrix, even though $U \otimes V \cong V \otimes U$ holds for all finite-dimensional representations $U,V$ of $\bar{U}_i sl(2)$. We then give an explicit coassociator $\Phi$ and an R-matrix $R$ such that $\bar{U}_i sl(2)$ becomes a quasi-triangular quasi-Hopf algebra. Our construction is motivated by the two-dimensional chiral conformal field theory of symplectic fermions with central charge $c=-2$. There, a braided monoidal category, $\mathcal{SF}$, has been computed from the factorisation and monodromy properties of conformal blocks, and we prove that $\mathrm{Rep}\,(\bar{U}_i sl(2),\Phi,R)$ is braided monoidally equivalent to $\mathcal{SF}$.Comment: 40pp, 11 figures; v2: few very minor corrections for the final version in Journal of Algebr

Perturbed Defects and T-Systems in Conformal Field Theory

Defect lines in conformal field theory can be perturbed by chiral defect fields. If the unperturbed defects satisfy su(2)-type fusion rules, the operators associated to the perturbed defects are shown to obey functional relations known from the study of integrable models as T-systems. The procedure is illustrated for Virasoro minimal models and for Liouville theory.Comment: 24 pages, 13 figures; v2: typos corrected, in particular in (2.10) and app. A.2, version to appear in J.Phys.

Topological defects for the free boson CFT

Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be described by the action of a topological defect, and hence that T-duality can be understood as a special type of order-disorder duality.Comment: 43 pages, 4 figure