Conformal designs based on vertex operator algebras

We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of an extremal self-dual vertex operator algebra of fixed degree form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus and Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6- and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.     

Toward a vertex operator construction of quantum affine algebras

We describe a construction of the quantum-deformed affine algebras using vertex operators in the free field theory. We prove the Serre relations for the Borel subalgebras of quantum affine algebras; in particular, we consider the case in detail. We also construct the generators corresponding to the positive roots of . __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 240–248, February, 2008.     

Parafermion vertex operator algebras

This paper reviews some recent results on the parafermion vertex operator algebra associated to the integrable highest weight module L(k, 0) of positive integer level k for any affine Kac-Moody Lie algebra ĝ, where g is a finite dimensional simple Lie algebra. In particular, the generators and the C ₂-cofiniteness of the parafermion vertex operator algebras are discussed. A proof of the well-known fact that the parafermion vertex operator algebra can be realized as the commutant of a lattice vertex operator algebra in L(k, 0) is also given.     

A family of regular vertex operator algebras with two generators

For every m ∈ ℂ ∖ {0, −2} and every nonnegative integer k we define the vertex operator (super)algebra D _m,k having two generators and rank . If m is a positive integer then D _m,k can be realized as a subalgebra of a lattice vertex algebra. In this case, we prove that D _m,k is a regular vertex operator (super) algebra and find the number of inequivalent irreducible modules.      

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.     

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras for which we present a short proof. The need to get such a proof was formulated in Semrl's paper.

     

Laplaza sets, or how to select coherence diagrams for pseudo algebras

We define a general concept of pseudo algebras over theories and 2-theories. A more restrictive such a notion was introduced in [Po Hu, Igor Kriz, Conformal field theory and elliptic cohomology, Adv. Math. 189 (2) (2004) 325-412, http://www.math.lsa.umich.edu/~ikriz/], but as noticed by M. Gould, did not capture the desired examples. The approach taken in this paper corrects the mistake by introducing a more general concept, allowing more flexibility in selecting coherence diagrams for pseudo algebras.     

The Notion of Vertex Operator Coalgebra and a Geometric Interpretation

The notion of vertex operator coalgebra is presented and motivated via the geometry of conformal field theory. Specifically, we describe the category of geometric vertex operator coalgebras, whose objects have comultiplicative structures meromorphically induced by conformal equivalence classes of worldsheets. We then show this category is isomorphic to the category of vertex operator coalgebras, which is defined in the language of formal algebra. The latter has several characteristics which give it the flavor of a coalgebra with respect to the structure of a vertex operator algebra and several characteristics that distinguish it from a standard dual—both of them will be highlighted.     

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.     

Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.     

Cardy algebras and sewing constraints,II

This is the part II of a two-part work started in [18]. In part I, Cardy algebras were studied, a notion which arises from the classification of genus-0, 1 open–closed rational conformal field theories. In this part, we prove that a Cardy algebra also satisfies the higher genus factorisation and modular-invariance properties formulated in [7] in terms of the notion of a solution to the sewing constraints. We present the proof by showing that the latter notion, which is defined as a monoidal natural transformation, can be expressed in terms of generators and relations, which correspond exactly to the defining data and axioms of a Cardy algebra.     

Conformal Quantum Field Theory and Subfactors

We survey a recent progress on algebraic quantum field theory in connection with subfactor theory. We mainly concentrate on one-dimensional conformal quantum field theory.     

Monogenic functions and representations of nilpotent Lie groups in quantum mechanics

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford‐valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal–Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Copyright © 1999 John Wiley & Sons, Ltd.