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

Alternative Co-operative Structures for the Agri-food Sector: An Exploratory Study

In a context of internationalisation and concentration, bio-foods co-operatives often face problems of capital access and governance. According to Spear (2001), limited access to capital, management structure, governance conflicts, and the local, regional or national focus of co-operatives limit their expansion. Recent events in Canada, such as Agricore going public and Dairyworld being absorbed by a corporation, have reinforced this perception (Thibault and Dupuis, 2003). In this evolving competitive environment, the traditional co-operative model cannot thrive and succeed if static. New hybrid co-operative models are emerging, and an analysis of these models should contribute to an understanding of the trade-off between co-operative values and other attributes necessary for survival in an increasingly competitive environment.Agribusiness, Marketing,

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