775 research outputs found
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 . We prove that this tensor product is associative and symmetric up to
equivalence. We also determine explicitly the action of 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 -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
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
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 , which commutes with the action of
the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The
matrix describes the braiding of the chiral vertex operators and the
coassociator gives rise to a modification of the duality property.
For generic the quasi quantum group is isomorphic to the coassociative
quantum group 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 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
Non-BPS States in Heterotic - Type IIA Duality
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
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
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
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 , of
central charge . We classify these operators in terms of {\em depth}
and provide explicit constructions in all cases. Furthermore, for we
focus on the vertex operator subalgebra L(1,0) of 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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