Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms
Yi-Zhi Huang, James Lepowsky, Lin Zhang
备注：
Part VI of a series of 8 papers generalizing the results in and collectively replacing arXiv:0710:2687, with new titles. 108 pages. Enhanced version. Minor changes
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.
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a number of results on intertwining operator algebras were given in [H9] but some of the proofs were postponed to an unpublished monograph.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…
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.
We prove an $S_{3}$-symmetry of the Jacobi identity for intertwining operator algebras. Since this Jacobi identity involves the braiding and fusing isomorphisms satisfying the genus-zero Moore-Seiberg equations, our proof uses not only the basic properties of intertwining operators, but also the properties of braiding and fusing isomorphisms and the genus-zero Moore-Seiberg equations. Our proof depends heavily on the theory of multivalued analytic functions of several variables, especially the theory of analytic extensions.
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…
For a finite-dimensional simple Lie algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the affine Lie algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\ma…
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 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 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 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.
In this paper, we study intertwining operators between subregular Whittaker modules of $\gl_N$ generalizing, on the one hand, the classical exchange construction of dynamical quantum groups, on the other hand, earlier results for principal W-algebras. We explicitly construct them using the generators of W-algebras introduced by Brundan-Kleshchev. We interpret the fusion on intertwining operators in terms of categorical actions and compute the semi-classical limit of the corresponding monoidal isomorphisms which turn out to depend on dynamical-like parameters.
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…
For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functi…
We study what happens when coherence fails. Categories with a tensor product and a natural associativity isomorphism that does not necessarily satisfy the pentagon coherence requirements (called associative categories) are considered. Categorical versions of associahedra where naturality squares commute and pentagons do not, are constructed (called Catalan groupoids, $\An$). These groupoids are used in the construction of the free associative category. They are also used in the construction of the theory of associative categories (given as a 2-sketch). Generators and relations are given for th…
In several examples it has been observed that a module category of a vertex operator algebra (VOA) is equivalent to a category of representations of some quantum group. The present article is concerned with developing such a duality in the case of the Virasoro VOA at generic central charge; arguably the most rudimentary of all VOAs, yet structurally complicated. We do not address the category of all modules of the generic Virasoro VOA, but we consider the infinitely many modules from the first row of the Kac table. Building on an explicit quantum group method of Coulomb gas integrals, we give …
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 …
Given a category with a bifunctor and natural isomorphisms for associativity, commutativity and left and right identity we do not assume that extra constraining diagrams hold. We introduce groupoids of coupling trees to describe a version of coherence that is weaker than the usual notion of monoidal coherence.
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 prove an associative law of the fusion products $\boxtimes$ of $C_1$-cofinite ${\mathbb N}$-gradable modules for a vertex operator algebra $V$. To be more precise, for $C_1$-cofinite ${\mathbb N}$-gradable $V$-modules $A,B,C$ and their fusion products $(A\!\boxtimes\! B, {\cal Y}^{AB})$, $((A\!\boxtimes\! B)\!\boxtimes\! C, {\cal Y}^{(AB)C})$, $(B\!\boxtimes\! C, {\cal Y}^{BC})$, $(A\!\boxtimes\! (B\!\boxtimes\! C),{\cal Y}^{A(BC)})$ with logarithmic intertwining operators ${\cal Y}^{AB},\ldots,{\cal Y}^{A(BC)}$ satisfying the universal properties for ${\mathbb N}$-gradable modules, we prov…
We incorporate a category of certain modules for an affine Lie algebra, of a certain fixed non-positive-integral level, considered by Kazhdan and Lusztig, into the representation theory of vertex operator algebras, by using the logarithmic tensor product theory for generalized modules for a vertex operator algebra developed by Huang, Lepowsky and the author. We do this by proving that the conditions for applying this general logarithmic tensor product theory hold. As a consequence, we prove that this category has a natural vertex tensor category structure, and in particular we obtain a new, ve…
We generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ``conformal vertex algebra'' or even more generally, for a "Möbius vertex algebra.'' We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is involved. This first part is devoted to the st…
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 generalize the tensor product theory for modules for a vertex operator algebra previously developed in a series of papers by the first two authors to suitable module categories for a ''conformal vertex algebra'' or even more generally, for a "Möbius vertex algebra.'' We do not require the module categories to be semisimple, and we accommodate modules with generalized weight spaces. As in the earlier series of papers, our tensor product functors depend on a complex variable, but in the present generality, the logarithm of the complex variable is required; the general representation theory of…
Interest in weak cubical n-categories arises in various contexts, in particular in topological field theories. In this paper, we describe a concept of double bicategory, namely a strict model of the theory of bicategories in Bicat. We show that in a special case one can reduce this to what we call a Verity double bicategory, after Domenic Verity. This is a weakened version of a double category, in the sense that composition in both horizontal and vertical directions satisfy associativity and unit laws only up to (coherent) isomorphisms. We prove that there are examples in the form of double bi…
Suppose $V^G$ is the fixed-point vertex operator subalgebra of a compact group $G$ acting on a simple abelian intertwining algebra $V$. We show that if all irreducible $V^G$-modules contained in $V$ live in some braided tensor category of $V^G$-modules, then they generate a tensor subcategory equivalent to the category $\mathrm{Rep}\,G$ of finite-dimensional representations of $G$, with associativity and braiding isomorphisms modified by the abelian $3$-cocycle defining the abelian intertwining algebra structure on $V$. Additionally, we show that if the fusion rules for the irreducible $V^G$-m…
We introduce notions of open-string vertex algebra, conformal open-string vertex algebra and variants of these notions. These are ``open-string-theoretic,'' ``noncommutative'' generalizations of the notions of vertex algebra and of conformal vertex algebra. Given an open-string vertex algebra, we show that there exists a vertex algebra, which we call the ``meromorphic center,'' inside the original algebra such that the original algebra yields a module and also an intertwining operator for the meromorphic center. This result gives us a general method for constructing open-string vertex algebras…
Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street's weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl^{(n)}(K), of (n+1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl^{(0)}(K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl^{(n)}(K) contains Mnd^{n+1}(K) as a full 2-subcategory. For the local idempotent closure K^ of K, extending the multiplicati…
