360 research outputs found
From boundary to bulk in logarithmic CFT
The analogue of the charge-conjugation modular invariant for rational
logarithmic conformal field theories is constructed. This is done by
reconstructing the bulk spectrum from a simple boundary condition (the analogue
of the Cardy `identity brane'). We apply the general method to the c_1,p
triplet models and reproduce the previously known bulk theory for p=2 at c=-2.
For general p we verify that the resulting partition functions are modular
invariant. We also construct the complete set of 2p boundary states, and
confirm that the identity brane from which we started indeed exists. As a
by-product we obtain a logarithmic version of the Verlinde formula for the
c_1,p triplet models.Comment: 35 pages, 2 figures; v2: minor corrections, version to appear in
J.Phys.
The logarithmic triplet theory with boundary
The boundary theory for the c=-2 triplet model is investigated in detail. In
particular, we show that there are four different boundary conditions that
preserve the triplet algebra, and check the consistency of the corresponding
boundary operators by constructing their OPE coefficients explicitly. We also
compute the correlation functions of two bulk fields in the presence of a
boundary, and verify that they are consistent with factorisation.Comment: 43 pages, LaTeX; v2: references added, typos corrected, footnote 4
adde
Infinite Symmetry in the Fractional Quantum Hall Effect
We have generalized recent results of Cappelli, Trugenberger and Zemba on the
integer quantum Hall effect constructing explicitly a for
the fractional quantum Hall effect such that the negative modes annihilate the
Laughlin wave functions. This generalization has a nice interpretation in
Jain's composite fermion theory. Furthermore, for these models we have
calculated the wave functions of the edge excitations viewing them as area
preserving deformations of an incompressible quantum droplet, and have shown
that the is the underlying symmetry of the edge
excitations in the fractional quantum Hall effect. Finally, we have applied
this method to more general wave functions.Comment: 15pp. LaTeX, BONN-HE-93-2
Bits and Pieces in Logarithmic Conformal Field Theory
These are notes of my lectures held at the first School & Workshop on
Logarithmic Conformal Field Theory and its Applications, September 2001 in
Tehran, Iran.
These notes cover only selected parts of the by now quite extensive knowledge
on logarithmic conformal field theories. In particular, I discuss the proper
generalization of null vectors towards the logarithmic case, and how these can
be used to compute correlation functions. My other main topic is modular
invariance, where I discuss the problem of the generalization of characters in
the case of indecomposable representations, a proposal for a Verlinde formula
for fusion rules and identities relating the partition functions of logarithmic
conformal field theories to such of well known ordinary conformal field
theories.
These two main topics are complemented by some remarks on ghost systems, the
Haldane-Rezayi fractional quantum Hall state, and the relation of these two to
the logarithmic c=-2 theory.Comment: 91 pages, notes of lectures delivered at the first School and
Workshop on Logarithmic Conformal Field Theory and its Applications, Tehran,
September 2001. Amendments in Introductio
Generalized twisted modules associated to general automorphisms of a vertex operator algebra
We introduce a notion of strongly C^{\times}-graded, or equivalently,
C/Z-graded generalized g-twisted V-module associated to an automorphism g, not
necessarily of finite order, of a vertex operator algebra. We also introduce a
notion of strongly C-graded generalized g-twisted V-module if V admits an
additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a
vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let
u be an element of V of weight 1 such that L(1)u=0. Then the exponential of
2\pi \sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this case, a
strongly C-graded generalized g_{u}-twisted V-module is constructed from a
strongly C-graded generalized V-module with a compatible action of g_{u} by
modifying the vertex operator map for the generalized V-module using the
exponential of the negative-power part of the vertex operator Y(u, x). In
particular, we give examples of such generalized twisted modules associated to
the exponentials of some screening operators on certain vertex operator
algebras related to the triplet W-algebras. An important feature is that we
have to work with generalized (twisted) V-modules which are doubly graded by
the group C/Z or C and by generalized eigenspaces (not just eigenspaces) for
L(0), and the twisted vertex operators in general involve the logarithm of the
formal variable.Comment: Final version to appear in Comm. Math. Phys. 38 pages. References on
triplet W-algebras added, misprints corrected, and expositions revise
Logarithmic operators in SL(2, R) WZNW model, Singletons and AdS_3/(L)CFT_2 correspondence
We discuss the role of singletons and logarithmic operators in AdS_3 string
theory in the context of AdS_3/CFT_2 correspondence.Comment: Revised version, more references added, to appear in Journal of Phys.
Extended chiral algebras and the emergence of SU(2) quantum numbers in the Coulomb gas
We study a set of chiral symmetries contained in degenerate operators beyond
the `minimal' sector of the c(p,q) models. For the operators
h_{(2j+2)q-1,1}=h_{1,(2j+2)p-1} at conformal weight [ (j+1)p-1 ][ (j+1)q -1 ],
for every 2j \in N, we find 2j+1 chiral operators which have quantum numbers of
a spin j representation of SU(2). We give a free-field construction of these
operators which makes this structure explicit and allows their OPEs to be
calculated directly without any use of screening charges. The first non-trivial
chiral field in this series, at j=1/2, is a fermionic or para-fermionic
doublet. The three chiral bosonic fields, at j=1, generate a closed W-algebra
and we calculate the vacuum character of these triplet models.Comment: 23 pages Late
A differential U-module algebra for U=U_q sl(2) at an even root of unity
We show that the full matrix algebra Mat_p(C) is a U-module algebra for U =
U_q sl(2), a 2p^3-dimensional quantum sl(2) group at the 2p-th root of unity.
Mat_p(C) decomposes into a direct sum of projective U-modules P^+_n with all
odd n, 1<=n<=p. In terms of generators and relations, this U-module algebra is
described as the algebra of q-differential operators "in one variable" with the
relations D z = q - q^{-1} + q^{-2} z D and z^p = D^p = 0. These relations
define a "parafermionic" statistics that generalizes the fermionic commutation
relations. By the Kazhdan--Lusztig duality, it is to be realized in a
manifestly quantum-group-symmetric description of (p,1) logarithmic conformal
field models. We extend the Kazhdan--Lusztig duality between U and the (p,1)
logarithmic models by constructing a quantum de Rham complex of the new
U-module algebra.Comment: 29 pages, amsart++, xypics. V3: The differential U-module algebra was
claimed quantum commutative erroneously. This is now corrected, the other
results unaffecte
Nonlinear Dynamics of Parity-Even Tricritical Gravity in Three and Four Dimensions
Recently proposed "multicritical" higher-derivative gravities in Anti de
Sitter space carry logarithmic representations of the Anti de Sitter isometry
group. While generically non-unitary already at the quadratic, free-theory
level, in special cases these theories admit a unitary subspace. The simplest
example of such behavior is "tricritical" gravity. In this paper, we extend the
study of parity-even tricritical gravity in d = 3, 4 to the first nonlinear
order. We show that the would-be unitary subspace suffers from a linearization
instability and is absent in the full non-linear theory.Comment: 22 pages; v2: references added, published versio
Topologically Massive Gravity and the AdS/CFT Correspondence
We set up the AdS/CFT correspondence for topologically massive gravity (TMG)
in three dimensions. The first step in this procedure is to determine the
appropriate fall off conditions at infinity. These cannot be fixed a priori as
they depend on the bulk theory under consideration and are derived by solving
asymptotically the non-linear field equations. We discuss in detail the
asymptotic structure of the field equations for TMG, showing that it contains
leading and subleading logarithms, determine the map between bulk fields and
CFT operators, obtain the appropriate counterterms needed for holographic
renormalization and compute holographically one- and two-point functions at and
away from the 'chiral point' (mu = 1). The 2-point functions at the chiral
point are those of a logarithmic CFT (LCFT) with c_L = 0, c_R = 3l/G_N and b =
-3l/G_N, where b is a parameter characterizing different c = 0 LCFTs. The bulk
correlators away from the chiral point (mu \neq 1) smoothly limit to the LCFT
ones as mu \to 1. Away from the chiral point, the CFT contains a state of
negative norm and the expectation value of the energy momentum tensor in that
state is also negative, reflecting a corresponding bulk instability due to
negative energy modes.Comment: 54 pages, v2: added comments and reference
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