23 research outputs found
Semi-Lorentz invariance, unitarity, and critical exponents of symplectic fermion models
We study a model of N-component complex fermions with a kinetic term that is
second order in derivatives. This symplectic fermion model has an Sp(2N)
symmetry, which for any N contains an SO(3) subgroup that can be identified
with rotational spin of spin-1/2 particles. Since the spin-1/2 representation
is not promoted to a representation of the Lorentz group, the model is not
fully Lorentz invariant, although it has a relativistic dispersion relation.
The hamiltonian is pseudo-hermitian, H^\dagger = C H C, which implies it has a
unitary time evolution. Renormalization-group analysis shows the model has a
low-energy fixed point that is a fermionic version of the Wilson-Fisher fixed
points. The critical exponents are computed to two-loop order. Possible
applications to condensed matter physics in 3 space-time dimensions are
discussed.Comment: v2: Published version, minor typose correcte
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
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
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
Minimal Superstrings and Loop Gas Models
We reformulate the matrix models of minimal superstrings as loop gas models
on random surfaces. In the continuum limit, this leads to the identification of
minimal superstrings with certain bosonic string theories, to all orders in the
genus expansion. RR vertex operators arise as operators in a Z_2 twisted sector
of the matter CFT. We show how the loop gas model implements the sum over spin
structures expected from the continuum RNS formulation. Open string boundary
conditions are also more transparent in this language.Comment: 36 pages, 3 figure
Extended multiplet structure in Logarithmic Conformal Field Theories
We use the process of quantum hamiltonian reduction of SU(2)_k, at rational
level k, to study explicitly the correlators of the h_{1,s} fields in the
c_{p,q} models. We find from direct calculation of the correlators that we have
the possibility of extra, chiral and non-chiral, multiplet structure in the
h_{1,s} operators beyond the `minimal' sector. At the level of the vacuum null
vector h_{1,2p-1}=(p-1)(q-1) we find that there can be two extra non-chiral
fermionic fields. The extra indicial structure present here permeates
throughout the entire theory. In particular we find we have a chiral triplet of
fields at h_{1,4p-1}=(2p-1)(2q-1). We conjecture that this triplet algebra may
produce a rational extended c_{p,q} model. We also find a doublet of fields at
h_{1,3p-1}=(\f{3p}{2}-1)(\f{3q}{2}-1). These are chiral fermionic operators if
p and q are not both odd and otherwise parafermionic.Comment: 24 pages LATEX. Minor corrections and extra reference
Extended chiral algebras in the SU(2)_0 WZNW model
We investigate the W-algebras generated by the integer dimension chiral
primary operators of the SU(2)_0 WZNW model. These have a form almost identical
to that found in the c=-2 model but have, in addition, an extended Kac-Moody
structure. Moreover on Hamiltonian reduction these SU(2)_0 W-algebras exactly
reduce to those found in c=-2. We explicitly find the free field
representations for the chiral j=2 and j=3 operators which have respectively a
fermionic doublet and bosonic triplet nature. The correlation functions of
these operators accounts for the rational solutions of the
Knizhnik-Zamolodchikov equation that we find. We explicitly compute the full
algebra of the j=2 operators and find that the associativity of the algebra is
only guaranteed if certain null vectors decouple from the theory. We conjecture
that these algebras may produce a quasi-rational conformal field theory.Comment: 18 pages LATEX. Minor corrections. Full j=2 algebra adde
On the Completeness of the Set of Classical W-Algebras Obtained from DS Reductions
We clarify the notion of the DS --- generalized Drinfeld-Sokolov ---
reduction approach to classical -algebras. We first strengthen an
earlier theorem which showed that an embedding can be associated to every DS reduction. We then use the fact that a
\W-algebra must have a quasi-primary basis to derive severe restrictions on
the possible reductions corresponding to a given embedding. In the
known DS reductions found to date, for which the \W-algebras are denoted by
-algebras and are called canonical, the
quasi-primary basis corresponds to the highest weights of the . Here we
find some examples of noncanonical DS reductions leading to \W-algebras which
are direct products of -algebras and `free field'
algebras with conformal weights . We also show
that if the conformal weights of the generators of a -algebra
obtained from DS reduction are nonnegative (which isComment: 48 pages, plain TeX, BONN-HE-93-14, DIAS-STP-93-0
A paradigm of open/closed duality: Liouville D-branes and the Kontsevich model
We argue that topological matrix models (matrix models of the Kontsevich
type) are examples of exact open/closed duality. The duality works at finite N
and for generic `t Hooft couplings. We consider in detail the paradigm of the
Kontsevich model for two-dimensional topological gravity. We demonstrate that
the Kontsevich model arises by topological localization of cubic open string
field theory on N stable branes. Our analysis is based on standard worldsheet
methods in the context of non-critical bosonic string theory. The stable branes
have Neumann (FZZT) boundary conditions in the Liouville direction. Several
generalizations are possible.Comment: v2: References added; a new section with generalization to non-zero
bulk cosmological constant; expanded discussion on topological localization;
added some comment
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