677 research outputs found
Fusion Rules for Affine Kac-Moody Algebras
This is an expository introduction to fusion rules for affine Kac-Moody
algebras, with major focus on the algorithmic aspects of their computation and
the relationship with tensor product decompositions. Many explicit examples are
included with figures illustrating the rank 2 cases. New results relating
fusion coefficients to tensor product coefficients are proved, and a conjecture
is given which shows that the Frenkel-Zhu affine fusion rule theorem can be
seen as a beautiful generalization of the Parasarathy-Ranga Rao-Varadarajan
tensor product theorem. Previous work of the author and collaborators on a
different approach to fusion rules from elementary group theory is also
explained.Comment: 43 pp, LateX, 18 postscript figures. Paper for my talk at the
Ramanujan International Symposium on Kac-Moody Lie Algebras and Applications,
ISKMAA-2002, Jan. 28-31, 2002, Chennai, India. Important references and
comments added. Final version accepted for publication. Also available from
ftp://ftp.math.binghamton.edu/pub/alex/Madras_Paper_Latex.ps.g
The 3-state Potts model and Rogers-Ramanujan series
We explain the appearance of Rogers-Ramanujan series inside the tensor
product of two basic -modules, previously discovered by the first
author in [F]. The key new ingredients are Virasoro minimal models and
twisted modules for the Zamolodchikov \WW_3-algebra.Comment: 20 pages, published in CEJ
A new perspective on the Frenkel-Zhu fusion rule theorem
In this paper we prove a formula for fusion coefficients of affine Kac-Moody
algebras first conjectured by Walton [Wal2], and rediscovered in [Fe]. It is a
reformulation of the Frenkel-Zhu affine fusion rule theorem [FZ], written so
that it can be seen as a beautiful generalization of the classical
Parasarathy-Ranga Rao-Varadarajan tensor product theorem [PRV].Comment: 19 pages, no figures, uses conm-p-l.cls style fil
Minimal model fusion rules from 2-groups
The fusion rules for the -minimal model representations of the
Virasoro algebra are shown to come from the group G = \boZ_2^{p+q-5} in the
following manner. There is a partition into disjoint
subsets and a bijection between and the sectors
of the -minimal model such that the fusion rules correspond to where .Comment: 8 pages, amstex, v2.1, uses fonts msam, msbm, no figures, tables
constructed using macros: cellular and related files are included. This paper
will be submitted to Communications in Math. Physics. A compressed dvi file
is available at ftp://math.binghamton.edu/pub/alex/fusionrules.dvi.Z , and
compressed postscript at ftp://math.binghamton.edu/pub/alex/fusionrules.ps.
Deformations and dilations of chaotic billiards, dissipation rate, and quasi-orthogonality of the boundary wavefunctions
We consider chaotic billiards in d dimensions, and study the matrix elements
M_{nm} corresponding to general deformations of the boundary. We analyze the
dependence of |M_{nm}|^2 on \omega = (E_n-E_m)/\hbar using semiclassical
considerations. This relates to an estimate of the energy dissipation rate when
the deformation is periodic at frequency \omega. We show that for dilations and
translations of the boundary, |M_{nm}|^2 vanishes like \omega^4 as \omega -> 0,
for rotations like \omega^2, whereas for generic deformations it goes to a
constant. Such special cases lead to quasi-orthogonality of the eigenstates on
the boundary.Comment: 4 pages, 3 figure
Parametric Evolution for a Deformed Cavity
We consider a classically chaotic system that is described by a Hamiltonian
H(Q,P;x), where (Q,P) describes a particle moving inside a cavity, and x
controls a deformation of the boundary. The quantum-eigenstates of the system
are |n(x)>. We describe how the parametric kernel P(n|m) = , also
known as the local density of states, evolves as a function of x-x0. We
illuminate the non-unitary nature of this parametric evolution, the emergence
of non-perturbative features, the final non-universal saturation, and the
limitations of random-wave considerations. The parametric evolution is
demonstrated numerically for two distinct representative deformation processes.Comment: 13 pages, 8 figures, improved introduction, to be published in Phys.
Rev.
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