Related papers: A proof of the fusion rules theorem

We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories.…

We describe the construction of the genus-zero parts of conformal field theories in the sense of G. Segal from representations of vertex operator algebras satisfying certain conditions. The construction is divided into four steps and each…

It is shown that a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V) is semisimple and each irreducible admissible V-module is ordinary. A contravariant form on a Verma type admissible V-module is constructed…

"Fusion rules" are laws of multiplication among eigenspaces of an idempotent. This terminology is relatively new and is closely related to axial algebras, introduced recently by Hall, Rehren and Shpectorov. Axial algebras, in turn, are…

Derivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with…

For a vertex operator algebra V, a V-module M and a nonnegative integer n, an A_n(V)-bimodule A_n(M) is constructed and studied. The connection between A_n(M) and intertwining operators are discussed. In the case that V is rational, A_n(M)…

Verlinde's formula for rational vertex operator algebras computes the fusion rules from the modular transformations of characters. In the non semisimple and non finite case, a logarithmic Verlinde formula has been proposed together with…

Let $V$ be a simple vertex operator algebra, and $G$ a finite automorphism group of $V$ such that $V^G$ is regular. The definition of entries in $S$-matrix on $V^G$ is discussed, and then is extended. The set of $V^G$-modules can be…

We prove a result which gives sufficient conditions for a conformal annulus which is a countable union of nested conformal annuli to have bounded modulus. Our theorem also gives estimates for the modulus of such an annulus and is proved…

We prove that under a mild condition, a multiset of tagged permissible arcs over a skew-tiling is uniquely determined by its intersection vector. As an application, it is proved that -- up to isomorphism -- different $\tau$-rigid modules…

We prove the existence of a homomorphism of vertex algebras, from the vacuum Verma module over the loop algebra of an untwisted affine algebra, whose construction is analogous to that of the Feigin-Frenkel homomorphism from the vacuum Verma…

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…

We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The…

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice…

Using the representation theory of the subgroups SL_2(Z_p) of the modular group we investigate the induced fusion algebras in some simple examples. Only some of these representations lead to 'good' fusion algebras. Furthermore, the…

Let V be a vertex operator algebra. We construct a sequence of associative algebras A_n(V) (n=0,1,2,...) such that A_{n}(V) is a quotient of A_{n+1}(V) and a pair of functors between the category of A_n(V)-modules which are not…

These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe…

We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which…

We show that two operator algebras are strongly Morita equivalent (in the sense of Blecher, Muhly and Paulsen) if and only if their categories of operator modules are equivalent via completely contractive functors. Moreover, any such…

In this paper we show that for a large natural class of vertex operator algebras (VOAs) and their modules, the Zhu algebras and bimodules (and their $g$-twisted analogs) are Noetherian. These carry important information about the…