This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra $M(1)_a$, of central charge $1-12a^2$. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for $a=0$ we focus on the vertex operator subalgebra L(1,0) of $M(1)_0$ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of {\em hidden} logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1,0)-module.
This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining operator superalgebras and vertex tensor categories in the N=1 case.
We construct the intertwining operator superalgebras and vertex tensor categories for the N=2 superconformal unitary minimal models and other related models.
This is the second part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part II), we develop logarithmic formal calculus and study logarithmic intertwining operators.
For every $p \geq 2$, we obtained an explicit construction of a family of $\mathcal{W}(2,2p-1)$-modules, which decompose as direct sum of simple Virasoro algebra modules. Furthermore, we classified all irreducible self-dual $\mathcal{W}(2,2p-1)$-modules, we described their internal structure, and computed their graded dimensions. In addition, we constructed certain hidden logarithmic intertwining operators among two ordinary and one logarithmic $\mathcal{W}(2,2p-1)$-modules. This work, in particular, gives a mathematically precise formulation and interpretation of what physicists have been ref…
In vertex algebra theory, fusion rules are described as the dimension of the vector space of intertwining operators between three irreducible modules. We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde algebra. We explicitly construct intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$.
Let $V$ be a vertex operator subalgebra of $U$. Assume that $U$, $V$, and its commutant $V^c$ in $U$ are CFT-type, self-dual, and regular VOAs. Assume also that the double commutant $V^{cc}$ equals $V$. We prove that any intertwining operator of $V$ is a compression of intertwining operators of $U$.
This is the fifth part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part V), we study products and iterates of intertwining maps and of logarithmic intertwining operators and we begin the development of our analytic approach.
This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study intertwining maps and tensor product bifunctors.
We produce counterexamples to show that in the definition of the notion of intertwining operator for modules for a vertex operator algebra, the commutator formula cannot in general be used as a replacement axiom for the Jacobi identity. We further give a sufficient condition for the commutator formula to imply the Jacobi identity in this definition. Using these results we illuminate the crucial role of the condition called the ``compatibility condition'' in the construction of the tensor product module in vertex operator algebra theory, as carried out in work of Huang and Lepowsky. In particul…
We establish an isomorphism between the space of logarithmic intertwining operators among suitable generalized modules for a vertex operator algebra and the space of homomorphisms between suitable modules for a generalization of Zhu's algebra given by Dong-Li-Mason.
We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This is considered as a generalization of Zhu's result for ordinary intertwining operators among ordinary modules.
We define an integral intertwining operator among modules for a vertex operator algebra to be an intertwining operator which respects integral forms in the modules, and we show that an intertwining operator is integral if it is integral when restricted to generators of the integral forms in the modules. We apply this result to classify integral intertwining operators which respect certain natural integral forms in modules for affine Lie algebra and lattice vertex operator algebras.
We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible logarithmic intertwiners at the level $c=0$.
The energy bounds condition for intertwining operators of unitary rational vertex operator algebras (VOAs) was studied, first by A.Wassermann for type $A$ affine VOAs, and later by T.Loke for $c<1$ Virasoro VOAs, and by V.Toledano-Laredo for type $D$ affine VOAs. In this paper, we extend their results to affine VOAs of type $B$, $C$, and $G_2$. As a consequence, the modular tensor categories of these unitary vertex operator algebras are unitary.
We discuss some applications of fusion rules and intertwining operators in the representation theory of cyclic orbifolds of the triplet vertex operator algebra. We prove that the classification of irreducible modules for the orbifold vertex algebra W(p)^{A_m} follows from a conjectural fusion rules formula for the singlet vertex algebra modules. In the p=2 case, we computed fusion rules for the irreducible singlet vertex algebra modules by using intertwining operators. This result implies the classification of irreducible modules for W(2)^{A_m}, conjectured previously in [4]. The main technica…
This is the sixth part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part VI), we construct the appropriate natural associativity isomorphisms between triple tensor product functors. In fact, we establish a "logarithmic operator product expansion" theorem for logarithmic intertwining operators. In this part, a great deal of analytic reasoning is needed; the statements of the main theorems themselves involve convergence assertions.
This is the seventh part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part VII), we give sufficient conditions for the existence of the associativity isomorphisms.
Let V be a simple vertex operator algebra and G a finite automorphism group. We give a construction of intertwining operators for irreducible V^G-modules which occur as submodules of irreducible V-modules by using intertwining operators for V. We also determine some fusion rules for a vertex operator algebra as an application.
We generalize the notion of an intertwining operator to N-graded weak modules over a vertex operator algebra and study their properties. We show a formula for the dimensions of these intertwining operators in terms of modules over the Zhu algebras under some conditions on N-graded weak modules.
We determine fusion rules (dimensions of the space of intertwining operators) among simple modules for the vertex operator algebra obtained as an even part of the symplectic fermionic vertex operator superalgebra. By using these fusion rules we show that the fusion algebra of this vertex operator algebra is isomorphic to the group algebra of the Klein four group over Z.
This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and $C_2$-cofinite vertex operator algebra $V$. We consider grading-restricted generalized $V$-modules which admit a right action of some associative algebra $P$, and intertwining operators among such modules which commute with the action of $P$ ($P$-intertwining operators). We obtain duality properties, i.e., suitable associat…
We describe a completely algebraic axiom system for intertwining operators of vertex algebra modules, using algebraic flat connections, thus formulating the concept of a {\em tree algebra}. Using the Riemann-Hilbert correspondence, we further prove that a vertex tensor category in the sense of Huang and Lepowsky gives rise to a tree algebra over $\C$. We also show that the chiral WZW model of a simply connected simple compact Lie group gives rise to a tree algebra over $\Q$.
We construct vertex algebraic intertwining operators among certain generalized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. W…
We derive certain systems of differential equations for matrix elements of products and iterates of logarithmic intertwining operators among strongly graded generalized modules for a strongly graded conformal vertex algebra under suitable assumptions. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory for such strongly graded generalized modules developed by Huang, Lepowsky and Zhang.
In [H5] (q-alg/9512024) and [H7] (q-alg/9704008), the author introduced the notion of intertwining operator algebra, a nonmeromorphic generalization of the notion of vertex operator algebra involving monodromies. The problem of constructing intertwining operator algebras from representations of suitable vertex operator algebras were solved implicitly earlier in [H3] (q-alg/9505019). In the present paper, we generalize the geometric and operadic formulation of the notion of vertex operator algebra given in [H1], [H2], [HL1] (hep-th/9301009), [HL2] and [H6] to the notion of intertwining operator…
We introduce intertwining operators among twisted modules or twisted intertwining operators associated to not-necessarily-commuting automorphisms of a vertex operator algebra. Let $V$ be a vertex operator algebra and let $g_{1}$, $g_{2}$ and $g_{3}$ be automorphisms of $V$. We prove that for $g_{1}$-, $g_{2}$- and $g_{3}$-twisted $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$, respectively, such that the vertex operator map for $W_{3}$ is injective, if there exists a twisted intertwining operator of type ${W_{3}\choose W_{1}W_{2}}$ such that the images of its component operators span $W_{3}$, then $…
We find sufficient conditions for the construction of vertex algebraic intertwining operators, among generalized Verma modules for an affine Lie algebra $\hat{\mathfrak{g}}$, from $\mathfrak{g}$-module homomorphisms. When $\mathfrak{g}=\mathfrak{sl}_2$, these results extend previous joint work with J. Yang, but the method used here is different. Here, we construct intertwining operators by solving Knizhnik-Zamolodchikov equations for three-point correlation functions associated to $\hat{\mathfrak{g}}$, and we identify obstructions to the construction arising from the possible non-existence of …
We show that if every module W for a vertex operator algebra V satisfies the condition that the dimension of W/C_1(W) is less than infinity, where C_1(W) is the subspace of W spanned by elements of the form u_{-1}w for u in V of positive weight and w in W, then matrix elements of products and iterates of intertwining operators satisfy certain systems of differential equations. Moreover, for prescribed singular points, there exist such systems of differential equations such that the prescribed singular points are regular. The finiteness of the fusion rules is an immediate consequence of a resul…
We study the principal subspaces, introduced by B. Feigin and A. Stoyanovsky, of the level 1 standard modules for $\hat{\goth{sl}(l+1)}$ with $l \geq 2$. In this paper we construct exact sequences which give us a complete set of recursions that characterize the graded dimensions of the principal subspaces of these representations. This problem can be viewed as a continuation of a new program to obtain Rogers-Ramanujan-type recursions, which was initiated by S. Capparelli, J. Lepowsky and A. Milas. In order to prove the exactness of the sequences we use intertwining vertex operators and we supp…
We apply the general theory of tensor products of modules for a vertex operator algebra developed in our papers hep-th/9309076, hep-th/9309159, hep-th/9401119, q-alg/9505018, q-alg/9505019 and q-alg/9505020 to the case of the Wess-Zumino-Novikov-Witten models and related models in conformal field theory. We show that for the category of modules for a vertex operator algebra containing a subalgebra isomorphic to a tensor product of rational vertex operator algebras associated to affine Lie algebras, the intertwining operators among the modules have the associativity property, the category has a…
We construct the non-stationary Ruijsenaars functions (affine analogue of the Macdonald functions) in the special case $κ=t^{-1/N}$, using the intertwining operators of the Ding-Iohara-Miki algebra (DIM algebra) associated with $N$-fold Fock tensor spaces. By the $S$-duality of the intertwiners, another expression is obtained for the non-stationary Ruijsenaars functions with $κ=t^{-1/N}$, which can be regarded as a natural elliptic lift of the asymptotic Macdonald functions to the multivariate elliptic hypergeometric series. We also investigate some properties of the vertex operator of the DIM…
In [1], Nakatsu and Takasaki have shown that the melting crystal model behind the topological strings vertex provides a tau-function of the KP hierarchy after an appropriate time deformation. We revisit their derivation with a focus on the underlying quantum $W_{1+\infty}$ symmetry. Specifically, we point out the role played by automorphisms and the connection with the intertwiner - or vertex operator - of the algebra. This algebraic perspective allows us to extend part of their derivation to the refined melting crystal model, lifting the algebra to the quantum toroidal algebra of $\mathfrak{g…
A cohomological criterion for the complete reducibility of modules of finite length satisfying a composability condition for a meromorphic open-string vertex algebra $V$ has been given by Qi and the author. In order to apply this criterion, one needs to determine what types of $V$-modules satisfy the composability condition. In this paper, we prove that the composability condition is satisfied by generalized modules in a suitable category for a grading-restricted vertex algebra satisfying natural conditions.
We use vertex operator algebras and intertwining operators to study certain substructures of standard $A_1^{(1)}$--modules, allowing us to conceptually obtain the classical Rogers--Ramanujan recursion. As a consequence we recover Feigin-Stoyanovsky's character formulas for the principal subspaces of the level 1 standard $A_1^{(1)}$--modules.
We extend the modular invariance property of the trace functions of vertex operator algebra on the set of irreducible modules (Zhu's theory) to the case of trace functions of intertwining operators.
We prove a generalized rationality property and a new identity that we call the ``Jacobi identity'' for intertwining operator algebras. Most of the main properties of genus-zero conformal field theories, including the main properties of vertex operator algebras, modules, intertwining operators, Verlinde algebras, and fusing and braiding matrices, are incorporated into this identity. Together with associativity and commutativity for intertwining operators proved by the author in \cite{H2} and \cite{H3}, the results of the present paper solve completely the problem of finding a natural purely al…
A new formula connecting the elliptic $6j$-symbols and the fusion of the vertex-face intertwining vectors is given. This is based on the identification of the $k$ fusion intertwining vectors with the change of base matrix elements from Sklyanin's standard base to Rosengren's natural base in the space of even theta functions of order $2k$. The new formula allows us to derive various properties of the elliptic $6j$-symbols, such as the addition formula, the biorthogonality property, the fusion formula and the Yang-Baxter relation. We also discuss a connection with the Sklyanin algebra based on t…
Using the theory of intertwining operators for vertex operator algebras we show that the graded dimensions of the principal subspaces associated to the standard modules for $\hat{\goth{sl}(2)}$ satisfy certain classical recursion formulas of Rogers and Selberg. These recursions were exploited by Andrews in connection with Gordon's generalization of the Rogers--Ramanujan identities and with Andrews' related identities. The present work generalizes the authors' previous work on intertwining operators and the Rogers--Ramanujan recursion.
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 also clarify some misunderstandings on vertex operator algebras, modular functors and intertwining operator algebras. Then we discuss some basic open problems and conjectures in mathematical orbifold conformal field theory. Generalized twisted modules and their variants, their constructions and some existence results are reviewed. Twisted intertwining op…
As for an elliptic $R$-operator which satisfies the Yang--Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex--IRF correspondence for the elliptic $R$-operator is obtained. The vertex--IRF correspondence implies that the Boltzmann weights of the IRF model satisfy the star--triangle relation. By means of these intertwining vectors, the factorized L-operators for the elliptic $R$-operator are also constructed. The vertex--IRF correspondence and the factorized L-operators for Belavin's $R$-matrix are reproduced from those of the elliptic $R$-operator.
We solve the problem of constructing a genus-zero full conformal field theory (a conformal field theory on genus-zero Riemann surfaces containing both chiral and antichiral parts) from representations of a simple vertex operator algebra satisfying certain natural finiteness and reductive conditions. We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For two vertex operator algebras, their tensor product is naturally a full field algebra and we introduce a notion of full field algebra over such a ten…
We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to chiral vertex operators, and more generally, it establishes the logarithmic operator product expansion for logarithmic intertwining operators. We review the main ideas in the construction of the tensor product bifunctors and the associativity isomorphisms. F…
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 step gives a mathematical structure of independent interest. These mathematical structures are intertwining operator algebras, genus-zero modular functors, genus-zero holomorphic weakly conformal field theories, and genus-zero conformal field theories.
In this paper, given a module $W$ for a vertex operator algebra $V$ and a nonzero complex number $z$ we construct a canonical (weak) $V\otimes V$-module ${\cal{D}}_{P(z)}(W)$ (a subspace of $W^{*}$ depending on $z$). We prove that for $V$-modules $W, W_{1}$ and $W_{2}$, a $P(z)$-intertwining map of type ${W'\choose W_{1}W_{2}}$ ([H3], [HL0-3]) exactly amounts to a $V\otimes V$-homomorphism from $W_{1}\otimes W_{2}$ into ${\cal{D}}_{P(z)}(W)$. Using Huang and Lepowsky's one-to-one linear correspondence between the space of intertwining operators and the space of $P(z)$-intertwining maps of the …
Let $V$ be a vertex operator algebra and $A^{\infty}(V)$ and $A^{N}(V)$ for $N\in \mathbb{N}$ the associative algebras introduced by the author in [H5]. For a lower-bounded generalized $V$-module $W$, we give $W$ a structure of graded $A^{\infty}(V)$-module and we introduce an $A^{\infty}(V)$-bimodule $A^{\infty}(W)$ and an $A^{N}(V)$-bimodule $A^{N}(W)$. We prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ for lower-bounded generalized $V$-modules $W_{1}$, $W_{2}$ and $W_{3}$ is isomorphic to the space $\hom_{A^{\infty}(V)}(A^{\infty}(W_{1})\otim…
Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an "evenness" problem that naturally vanishes for 2-div…
We give two constructions of grading-restricted vertex (super)algebras. We first give a new construction of a class of grading-restricted vertex (super)algebras originally obtained by Meurman and Primc using a different method. This construction is based on a new definition of vertex operators and a new method. Our second construction is a generalization of the author's construction of the moonshine module vertex operator algebra and a related vertex operator superalgebra. This construction needs properties of intertwining operators formulated and proved by the author.
Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are constructed explicitly in terms of free bosonic fields.
We describe a natural structure of an abelian intertwining algebra (in the sense of Dong and Lepowsky) on the direct sum of the untwisted vertex operator algebra constructed {}from the Leech lattice and its (unique) irreducible twisted module. When restricting ourselves to the moonshine module, we obtain a new and conceptual proof that the moonshine module has a natural structure of a vertex operator algebra. This abelian intertwining algebra also contains an irreducible twisted module for the moonshine module with respect to the obvious involution. In addition, it contains a vertex operator s…
We describe a logarithmic tensor product theory for certain module categories for a ``conformal vertex algebra.'' In this theory, which is a natural, although intricate, generalization of earlier work of Huang and Lepowsky, we do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. The corresponding intertwining operators contain logarithms of the variables.
A general method for constructing logarithmic modules in vertex operator algebra theory is presented. By utilizing this approach, we give explicit vertex operator construction of certain indecomposable and logarithmic modules for the triplet vertex algebra W(p) and for other subalgebras of lattice vertex algebras and their N=1 super extensions. We analyze in detail indecomposable modules obtained in this way, giving further evidence for the conjectural equivalence between the category of W(p)-modules and the category of modules for the restricted quantum group $\bar{\mathcal{U}}_q(sl_2)$ at ro…
We give an analogue for vertex operator algebras and superalgebras of the notion of endomorphism ring of a vector space by means of a notion of ``local system of vertex operators'' for a (super) vector space. We first prove that any local system of vertex operators on a (super) vector space $M$ has a natural vertex (super)algebra structure with $M$ as a module. Then we prove that for a vertex (operator) superalgebra $V$, giving a $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of vertex operators on $M$. As applications, we pro…
We study intertwining operator algebras introduced and constructed by Huang. In the case that the intertwining operator algebras involve intertwining operators among irreducible modules for their vertex operator subalgebras, a number of results on intertwining operator algebras were given in [H9] but some of the proofs were postponed to an unpublished monograph. In this paper, we give the proofs of these results in [H9] and we formulate and prove results for general intertwining operator algebras without assuming that the modules involved are irreducible. In particular, we construct fusing and…
Fusion rules and rigidity for weight modules over the simple admissible affine $\mathfrak{sl}(2)$ and $\mathcal{N}=2$ superconformal vertex operator superalgebras
Osaka Metropolitan University（大阪公立大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Universitat de València（瓦伦西亚大学）；Cardiff University（卡迪夫大学）；Research Institute for Mathematical Sciences Kyoto University（京都大学数学科学研究所）
We provide a ribbon tensor equivalence between the representation category of small quantum SL(2), at parameter q=exp($π$ i/p), and the representation category of the triplet vertex operator algebra at integral parameter p>1. We provide similar quantum group equivalences for representation categories associated to the Virasoro, and singlet vertex operator algebras at central charge c=1-6(p-1)^2/p. These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type A_1.
We consider the extension of the Heisenberg vertex operator algebra by all its irreducible modules. We give an elementary construction for the intertwining vertex operators and show that they satisfy a complex parametrized generalized vertex operator algebra. We illustrate some of our results with the example of integral lattice vertex operator superalgebras.
