110 research outputs found
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
Resolutions and Characters of Irreducible Representations of the N=2 Superconformal Algebra
We evaluate characters of irreducible representations of the N=2
supersymmetric extension of the Virasoro algebra. We do so by deriving the
BGG-resolution of the admissible N=2 representations and also a new
3,5,7...-resolution in terms of twisted massive Verma modules. We analyse how
the characters behave under the automorphisms of the algebra, whose most
significant part is the spectral flow transformations. The possibility to
express the characters in terms of theta functions is determined by their
behaviour under the spectral flow. We also derive the identity expressing every
character as a linear combination of spectral-flow transformed
N=2 characters; this identity involves a finite number of N=2 characters in the
case of unitary representations. Conversely, we find an integral representation
for the admissible N=2 characters as contour integrals of admissible
characters.Comment: LaTeX2e: amsart, 34pp. An overall sign error corrected in (4.33) and
several consequent formulas, and the presentation streamlined in Sec.4.2.3.
References added. To appear in Nucl. Phys.
On the Equivalence of Affine sl(2) and N=2 Superconformal Representation Theories
There exist two different languages, the ^sl(2) and N=2 ones, to describe
similar structures; a dictionary is given translating the key
representation-theoretic terms related to the two algebras. The main tool to
describe the structure of ^sl(2) and N=2 modules is provided by diagrams of
extremal vectors. The ^sl(2) and N=2 representation theories of certain
highest-weight types turn out to be equivalent modulo the respective spectral
flows.Comment: 14 pages, LaTeX209, needs bezier.sty. Contribution to the proceedings
of the 30th Int. Symposium Ahrenshoop on the theory of elementary particles,
Buckow, Germany, August 27--31, 199
Lattice fusion rules and logarithmic operator product expansions
The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing
over the last few years thanks to recent developments coming from various
approaches. A particularly fruitful point of view consists in considering
lattice models as regularizations for such quantum field theories. The
indecomposability then encountered in the representation theory of the
corresponding finite-dimensional associative algebras exactly mimics the
Virasoro indecomposable modules expected to arise in the continuum limit. In
this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor
introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using
quantum group results, we provide rigorous calculations of the fusion of
various TL modules. Our results are illustrated by many explicit examples
relevant for physics. We discuss how indecomposability arises in the "lattice"
fusion and compare the mechanisms involved with similar observations in the
corresponding field theory. We also discuss the physical meaning of our lattice
fusion rules in terms of indecomposable operator-product expansions of quantum
fields.Comment: 54pp, many comments adde
BRST Analysis of Physical States for 2D (Super) Gravity Coupled to (Super) Conformal Matter
We summarize some recent results on the BRST analysis of physical states of
2D gravity coupled to c<=1 conformal matter and the supersymmetric
generalization.Comment: 11 page
Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
We generalize the string functions C_{n,r}(tau) associated with the coset
^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau)
associated with the coset W(k)/u(1) of the W-algebra of the logarithmically
extended ^sl(2)_k conformal field model with positive integer k. The higher
string functions occur in decomposing W(k) characters with respect to level-k
theta and Appell functions and their derivatives (the characters are neither
quasiperiodic nor holomorphic, and therefore cannot decompose with respect to
only theta-functions). The decomposition coefficients, to be considered
``logarithmic parafermionic characters,'' are given by A_{n,r}(tau),
B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra
characters of the (p=k+2,1) logarithmic model. We study the properties of
A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string
functions C_{n,r}, and evaluate the modular group representation generated from
A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular
transformations of the higher-level Appell functions and the associated
transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some
nonsense in B.3.3. correcte
Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models
The subject of our study is the Kazhdan-Lusztig (KL) equivalence in the
context of a one-parameter family of logarithmic CFTs based on Virasoro
symmetry with the (1,p) central charge. All finite-dimensional indecomposable
modules of the KL-dual quantum group - the "full" Lusztig quantum sl(2) at the
root of unity - are explicitly described. These are exhausted by projective
modules and four series of modules that have a functorial correspondence with
any quotient or a submodule of Feigin-Fuchs modules over the Virasoro algebra.
Our main result includes calculation of tensor products of any pair of the
indecomposable modules. Based on the Kazhdan-Lusztig equivalence between
quantum groups and vertex-operator algebras, fusion rules of Kac modules over
the Virasoro algebra in the (1,p) LCFT models are conjectured.Comment: 40pp. V2: a new introduction, corrected typos, some explanatory
comments added, references adde
Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra
We develop in this paper the principles of an associative algebraic approach
to bulk logarithmic conformal field theories (LCFTs). We concentrate on the
closed spin-chain and its continuum limit - the symplectic
fermions theory - and rely on two technical companion papers, "Continuum limit
and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013)
245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl.
Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local
Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the
continuum limit to a bigger algebra than the product of the left and right
Virasoro algebras. This algebra, S - which we call interchiral, mixes the left
and right moving sectors, and is generated, in the symplectic fermions case, by
the additional field , with
a symmetric form and conformal weights (1,1). We discuss in details
how the Hilbert space of the LCFT decomposes onto representations of this
algebra, and how this decomposition is related with properties of the finite
spin-chain. We show that there is a complete correspondence between algebraic
properties of finite periodic spin chains and the continuum limit. An important
technical aspect of our analysis involves the fundamental new observation that
the action of JTL_N in the spin chain is in fact isomorphic to an
enveloping algebra of a certain Lie algebra, itself a non semi-simple version
of . The semi-simple part of JTL_N is represented by ,
providing a beautiful example of a classical Howe duality, for which we have a
non semi-simple version in the full JTL image represented in the spin-chain. On
the continuum side, simple modules over the interchiral algebra S are
identified with "fundamental" representations of .Comment: 69 pp., 10 figs, v2: the paper has been substantially modified - new
proofs, new refs, new App C with inductive limits construction, et
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