Fusion of twisted representations

The comultiplication formula for fusion products of untwisted representations of the chiral algebra is generalised to include arbitrary twisted representations. We show that the formulae define a tensor product with suitable properties, and determine the analogue of Zhu's algebra for arbitrary twisted representations. As an example we study the fusion of representations of the Ramond sector of the N=1 and N=2 superconformal algebra. In the latter case, certain subtleties arise which we describe in detail.Comment: 24 pages, LATE

Fusion in conformal field theory as the tensor product of the symmetry algebra

Following a recent proposal of Richard Borcherds to regard fusion as the ring-like tensor product of modules of a {\em quantum ring}, a generalization of rings and vertex operators, we define fusion as a certain quotient of the (vector space) tensor product of representations of the symmetry algebra ${\cal A}$. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of ${\cal A}$ on it, under which the central extension is preserved. \\ Having given a precise meaning to fusion, determining the fusion rules is now a well-posed algebraic problem, namely to decompose the tensor product into irreducible representations. We demonstrate how to solve it for the case of the WZW- and the minimal models and recover thereby the well-known fusion rules. \\ The action of the symmetry algebra on the tensor product is given in terms of a comultiplication. We calculate the $R$-matrix of this comultiplication and find that it is triangular. This seems to shed some new light on the possible r\^{o}le of the quantum group in conformal field theory.Comment: 21 pages, Latex, DAMTP-93-3

Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models

We introduce a Kazhdan--Lusztig-dual quantum group for (1,p) Virasoro logarithmic minimal models as the Lusztig limit of the quantum sl(2) at pth root of unity and show that this limit is a Hopf algebra. We calculate tensor products of irreducible and projective representations of the quantum group and show that these tensor products coincide with the fusion of irreducible and logarithmic modules in the (1,p) Virasoro logarithmic minimal models.Comment: 19 page

A Local Logarithmic Conformal Field Theory

The local logarithmic conformal field theory corresponding to the triplet algebra at c=-2 is constructed. The constraints of locality and crossing symmetry are explored in detail, and a consistent set of amplitudes is found. The spectrum of the corresponding theory is determined, and it is found to be modular invariant. This provides the first construction of a non-chiral rational logarithmic conformal field theory, establishing that such models can indeed define bona fide conformal field theories.Comment: 29 pages, LaTeX, minor changes, reference adde

Strings and branes in plane waves

An overview of string theory in the maximally supersymmetric plane-wave background is given, and some supersymmetric D-branes are discussed.Comment: 12 pages, LateX, needs fortschritte.sty; to appear in the proceedings of the 36th International Symposium Ahrenshoop on the Theory of Elementary Particles: Recent Developments in String/M-Theory and Field Theory, Berlin, Germany, 26-30 Aug 200

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi quantum group $A_{g,t}$, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The $R$ matrix describes the braiding of the chiral vertex operators and the coassociator $\Phi$ gives rise to a modification of the duality property. For generic $q$ the quasi quantum group is isomorphic to the coassociative quantum group $U_{q}(g)$ and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasi quantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case of $g=su(2)$ is worked out in detail.Comment: 28 pages, LATEX; a remark about other possible symmetry algebras and some references are adde

Stable non-BPS D-particles

It is shown that the orbifold of type IIB string theory by (-1)^{F_L} I_4 admits a stable non-BPS Dirichlet particle that is stuck on the orbifold fixed plane. It is charged under the SO(2) gauge group coming from the twisted sector, and transforms as a long multiplet of the D=6 supersymmetry algebra. This suggests that it is the strong coupling dual of the perturbative stable non-BPS state that appears in the orientifold of type IIB by \Omega I_4.Comment: 10 pages, LaTe