775 research outputs found

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

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    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 A{\cal A}. We prove that this tensor product is associative and symmetric up to equivalence. We also determine explicitly the action of A{\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 RR-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

    Fusion of twisted representations

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    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

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

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    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 Ag,tA_{g,t}, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The RR matrix describes the braiding of the chiral vertex operators and the coassociator Φ\Phi gives rise to a modification of the duality property. For generic qq the quasi quantum group is isomorphic to the coassociative quantum group Uq(g)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)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

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    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

    Non-BPS States in Heterotic - Type IIA Duality

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    The relation between some perturbative non-BPS states of the heterotic theory on T^4 and non-perturbative non-BPS states of the orbifold limit of type IIA on K3 is exhibited. The relevant states include a non-BPS D-string, and a non-BPS bound state of BPS D-particles (`D-molecule'). The domains of stability of these states in the two theories are determined and compared.Comment: 17 pages LaTex, 1 figure; Minor correction in subsection 4.

    Dualities of Type 0 Strings

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    It is conjectured that the two closed bosonic string theories, Type 0A and Type 0B, correspond to certain supersymmetry breaking orbifold compactifications of M-theory. Various implications of this conjecture are discussed, in particular the behaviour of the tachyon at strong coupling and the existence of non-perturbative fermionic states in Type 0A. The latter are shown to correspond to bound states of Type 0A D-particles, thus providing further evidence for the conjecture. We also give a comprehensive description of the various Type 0 closed and open string theories.Comment: 23 pages LaTex, 1 figure. Error corrected in table 1. Version to appear in JHE

    Triality in Minimal Model Holography

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    The non-linear W_{\infty}[\mu] symmetry algebra underlies the duality between the W_N minimal model CFTs and the hs[\mu] higher spin theory on AdS_3. It is shown how the structure of this symmetry algebra at the quantum level, i.e. for finite central charge, can be determined completely. The resulting algebra exhibits an exact equivalence (a`triality') between three (generically) distinct values of the parameter \mu. This explains, among other things, the agreement of symmetries between the W_N minimal models and the bulk higher spin theory. We also study the consequences of this triality for some of the simplest W_{\infty}[\mu] representations, thereby clarifying the analytic continuation between the`light states' of the minimal models and conical defect solutions in the bulk. These considerations also lead us to propose that one of the two scalar fields in the bulk actually has a non-perturbative origin.Comment: 29 pages; v2. Typos correcte

    Logarithmic intertwining operators and vertex operators

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    This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra M(1)aM(1)_a, of central charge 112a21-12a^2. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for a=0a=0 we focus on the vertex operator subalgebra L(1,0) of M(1)0M(1)_0 and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of {\em hidden} logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1,0)-module.Comment: 32 pages. To appear in CM
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