246 research outputs found

### Twisted modules for vertex operator algebras

This contribution is mainly based on joint papers with Lepowsky and Milas,
and some parts of these papers are reproduced here. These papers further
extended works by Lepowsky and by Milas. Following our joint papers, I explain
the general principles of twisted modules for vertex operator algebras in their
powerful formulation using formal series, and derive general relations
satisfied by twisted and untwisted vertex operators. Using these, I prove new
"equivalence" and "construction" theorems, identifying a set of sufficient
conditions in order to have a twisted module for a vertex operator algebra, and
a simple way of constructing the twisted vertex operator map. This essentially
combines our general relations for twisted modules with ideas of Li (1996), who
had obtained similar construction theorems using different relations. Then, I
show how to apply these theorems in order to construct twisted modules for the
Heisenberg vertex operator algebra. I obtain in a new way the explicit twisted
vertex operator map, and in particular give a new derivation and expression for
the formal operator $\Delta_x$ constructed some time ago by Frenkel, Lepowsky
and Meurman. Finally, I reproduce parts of our joint papers. The untwisted
relations in the Heisenberg vertex operator algebra are employed to explain
properties of a certain central extension of a Lie algebra of differential
operators on the circle, in relation to the Riemann Zeta function at negative
integers. A family of representations for this algebra are constructed from
twisted modules for the vertex operator algebra, and are related to the
Bernoulli polynomials at rational values.Comment: 41 pages, contribution to proceedings of the workshop "Moonshine -
the First Quarter Century and Beyond, a Workshop on the Moonshine Conjectures
and Vertex Algebras" (Edinburgh, 2004) v2: 43 pages, presentation, discussion
and proofs improve

### Hypotrochoids in conformal restriction systems and Virasoro descendants

A conformal restriction system is a commutative, associative, unital algebra
equipped with a representation of the groupoid of univalent conformal maps on
connected open sets of the Riemann sphere, and a family of linear functionals
on subalgebras, satisfying a set of properties including conformal invariance
and a type of restriction. This embodies some expected properties of
expectation values in conformal loop ensembles CLE. In the context of conformal
restriction systems, we study certain algebra elements associated with
hypotrochoid simple curves (including the ellipse). These have the CLE
interpretation of being "renormalized random variables" that are nonzero only
if there is at least one loop of hypotrochoid shape. Each curve has a center w,
a scale \epsilon\ and a rotation angle \theta, and we analyze the renormalized
random variable as a function of u=\epsilon e^{i\theta} and w. We find that it
has an expansion in positive powers of u and u*, and that the coefficients of
pure u (u*) powers are holomorphic in w (w*). We identify these coefficients
(the "hypotrochoid fields") with certain Virasoro descendants of the identity
field in conformal field theory, thereby showing that they form part of a
vertex operator algebraic structure. This largely generalizes works by the
author (in CLE), and the author with his collaborators V. Riva and J. Cardy (in
SLE 8/3 and other restriction measures), where the case of the ellipse, at the
order u^2, led to the stress-energy tensor of CFT. The derivation uses in an
essential way the Virasoro vertex operator algebra structure of conformal
derivatives established recently by the author. The results suggest in
particular the exact evaluation of CLE expectations of products of hypotrochoid
fields as well as non-trivial relations amongst them through the vertex
operator algebra, and further shed light onto the relationship between CLE and
CFT.Comment: 1 figure, 39 page

### Thermalization and pseudolocality in extended quantum systems

Recently, it was understood that modified concepts of locality played an
important role in the study of extended quantum systems out of equilibrium, in
particular in so-called generalized Gibbs ensembles. In this paper, we
rigorously study pseudolocal charges and their involvement in time evolutions
and in the thermalization process of arbitrary states with strong enough
clustering properties. We show that the densities of pseudolocal charges form a
Hilbert space, with inner product determined by thermodynamic susceptibilities.
Using this, we define the family of pseudolocal states, which are determined by
pseudolocal charges. This family includes thermal Gibbs states at high enough
temperatures, as well as (a precise definition of) generalized Gibbs ensembles.
We prove that the family of pseudolocal states is preserved by finite time
evolution, and that, under certain conditions, the stationary state emerging at
infinite time is a generalized Gibbs ensemble with respect to the evolution
dynamics. If the evolution dynamics does not admit any conserved pseudolocal
charges other than the evolution Hamiltonian, we show that any stationary
pseudolocal state with respect to this dynamics is a thermal Gibbs state, and
that Gibbs thermalization occurs. The framework is that of
translation-invariant states on hypercubic quantum lattices of any
dimensionality (including quantum chains) and finite-range Hamiltonians, and
does not involve integrability.Comment: v1: 43 pages. v2: corrections and clarifications, references added,
46 pages. v3: 48 pages, further corrections made, accepted for publication in
Commun. Math. Phy

### Finite-Temperature Form Factors: a Review

We review the concept of finite-temperature form factor that was introduced
recently by the author in the context of the Majorana theory.
Finite-temperature form factors can be used to obtain spectral decompositions
of finite-temperature correlation functions in a way that mimics the
form-factor expansion of the zero temperature case. We develop the concept in
the general factorised scattering set-up of integrable quantum field theory,
list certain expected properties and present the full construction in the case
of the massive Majorana theory, including how it can be applied to the
calculation of correlation functions in the quantum Ising model. In particular,
we include the ''twisted construction'', which was not developed before and
which is essential for the application to the quantum Ising model.Comment: This is a contribution to the Proc. of the O'Raifeartaigh Symposium
on Non-Perturbative and Symmetry Methods in Field Theory (June 2006,
Budapest, Hungary), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA

### Conformal field theory out of equilibrium: a review

We provide a pedagogical review of the main ideas and results in
non-equilibrium conformal field theory and connected subjects. These concern
the understanding of quantum transport and its statistics at and near critical
points. Starting with phenomenological considerations, we explain the general
framework, illustrated by the example of the Heisenberg quantum chain. We then
introduce the main concepts underlying conformal field theory (CFT), the
emergence of critical ballistic transport, and the CFT scattering construction
of non-equilibrium steady states. Using this we review the theory for energy
transport in homogeneous one-dimensional critical systems, including the
complete description of its large deviations and the resulting (extended)
fluctuation relations. We generalize some of these ideas to one-dimensional
critical charge transport and to the presence of defects, as well as beyond
one-dimensional criticality. We describe non-equilibrium transport in
free-particle models, where connections are made with generalized Gibbs
ensembles, and in higher-dimensional and non-integrable quantum field theories,
where the use of the powerful hydrodynamic ideas for non-equilibrium steady
states is explained. We finish with a list of open questions. The review does
not assume any advanced prior knowledge of conformal field theory,
large-deviation theory or hydrodynamics.Comment: 50 pages + 10 pages of references, 5 figures. v2: minor
modifications. Review article for special issue of JSTAT on nonequilibrium
dynamics in integrable quantum system

### Drude Weight for the Lieb-Liniger Bose Gas

Based on the method of hydrodynamic projections we derive a concise formula
for the Drude weight of the repulsive Lieb-Liniger $\delta$-Bose gas. Our
formula contains only quantities which are obtainable from the thermodynamic
Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear
functional: it is bilinear in the currents, and each current may refer to a
general linear combination of the conserved charges of the model. As a
by-product we obtain the dynamical two-point correlation functions involving
charge and current densities at small wavelengths and long times, and in
addition the scaled covariance matrix of charge transfer. We expect that our
formulas extend to other integrable quantum models.Comment: 23 pages. v2: improved discussion, typos corrected, references added.
v3: 26 pages, further improved discussion, references adde

- …