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Yi-Zhi Huang

Publications and source records attributed to Yi-Zhi Huang.

At least 37 records · Page 2Linked to original sources
arXiv2018

Intertwining operators, the composability condition and the complete reducibility of generalized modules

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.

math.QA↗
arXiv2017

Intertwining operators among twisted modules associated to not-necessarily-commuting automorphisms

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 $g_{3}=g_{1}g_{2}$. We also construct what we call the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators among twisted modules of suitable types. The proofs of these results involve careful analysis of the analytic extensions corresponding to the actions of the not-necessarily-commuting automorphisms of the vertex operator algebra.

math.QA↗
arXiv2017

Braided tensor categories of admissible modules for affine Lie algebras

Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is $\widehat{\mathfrak{sl}}_2$, we prove the rigidity and modularity conjectures.

math.QA↗
arXiv2017

On the applicability of logarithmic tensor category theory

We give results and observations which allow the application of the logarithmic tensor category theory of Lepowsky, Zhang and the author ([HLZ1]--[HLZ9]) to more general vertex (operator) algebras and their module categories than those studied in a paper by the author ([H3]).

math.QA↗
arXiv2016

Two constructions of grading-restricted vertex (super)algebras

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.

math.QA↗
arXiv2016

Associative algebras for (logarithmic) twisted modules for a vertex operator algebra

We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called $g$-twisted zero-mode algebra, is a subquotient of what we call $g$-twisted universal enveloping algebra of $V$. These algebras are generalizations of the corresponding algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the (untwisted) case that $g$ is the identity. The other is a generalization of the $g$-twisted version of Zhu's algebra for suitable $g$-twisted modules constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly interested in $g$-twisted $V$-modules introduced by the first author in the case that $g$ is of infinite order and does not act on $V$ semisimply. In this case, twisted vertex operators in general involve the logarithm of the variable. We construct functors between categories of suitable modules for these associative algebras and categories of suitable (logarithmic) $g$-twisted $V$-modules. Using these functors, we prove that the $g$-twisted zero-mode algebra and the $g$-twisted generalization of Zhu's algebra are in fact isomorphic.

math.QA↗
arXiv2014

Braided tensor categories and extensions of vertex operator algebras

Let $V$ be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. We prove that the notions of extension (i.e., enlargement) of $V$ and of commutative associative algebra, with uniqueness of unit and with trivial twist, in the braided tensor category of $V$-modules are equivalent.

math.QA↗
arXiv2013

A cohomology theory of grading-restricted vertex algebras

We introduce a cohomology theory of grading-restricted vertex algebras. To construct the {\it correct} cohomologies, we consider linear maps from tensor powers of a grading-restricted vertex algebra to "rational functions valued in the algebraic completion of a module for the algebra," instead of linear maps from tensor powers of the algebra to a module for the algebra. One subtle complication arising from such functions is that we have to carefully address the issue of convergence when we compose these linear maps with vertex operators. In particular, for each $n\in \mathbb{N}$, we have an inverse system $\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}$ of $n$-th cohomologies and an additional $n$-th cohomology $H_{\infty}^{n}(V, W)$ of a grading-restricted vertex algebra $V$ with coefficients in a $V$-module $W$ such that $H_{\infty}^{n}(V, W)$ is isomorphic to the inverse limit of the inverse system $\{H^{n}_{m}(V, W)\}_{m\in \mathbb{Z}_{+}}$. In the case of $n=2$, there is an additional second cohomology denoted by $H^{2}_{\frac{1}{2}}(V, W)$ which will be shown in a sequel to the present paper to correspond to what we call square-zero extensions of $V$ and to first order deformations of $V$ when $W=V$.

math.QA↗
arXiv2013

First and second cohomologies of grading-restricted vertex algebras

Let $V$ be a grading-restricted vertex algebra and $W$ a $V$-module. We show that for any $m\in \mathbb{Z}_{+}$, the first cohomology $H^{1}_{m}(V, W)$ of $V$ with coefficients in $W$ introduced by the author is linearly isomorphic to the space of derivations from $V$ to $W$. In particular, $H^{1}_{m}(V, W)$ for $m\in \mathbb{N}$ are equal (and can be denoted using the same notation $H^{1}(V, W)$). We also show that the second cohomology $H^{2}_{\frac{1}{2}}(V, W)$ of $V$ with coefficients in $W$ introduced by the author corresponds bijectively to the set of equivalence classes of square-zero extensions of $V$ by $W$. In the case that $W=V$, we show that the second cohomology $H^{2}_{\frac{1}{2}}(V, V)$ corresponds bijectively to the set of equivalence classes of first order deformations of $V$.

math.QA↗
arXiv2013

On functors between module categories for associative algebras and for $\mathbb{N}$-graded vertex algebras

We prove that the weak associativity for modules for vertex algebras are equivalent to a residue formula for iterates of vertex operators, obtained using the weak associativity and the lower truncation property of vertex operators, together with a known formula expressing products of components of vertex operators as linear combinations of iterates of components of vertex operators. By requiring that these two formulas instead of the commutator formula hold, we construct a functor $S$ from the category of modules for Zhu's algebra of a vertex operator algebra $V$ to the category of $\mathbb{N}$-gradable weak $V$-modules. We prove that $S$ has a universal property and the functor $T$ of taking top levels of $\mathbb{N}$-gradable weak $V$-modules is a left inverse of $S$. In particular, $S$ is equal to a functor implicitly given by Zhu and explicitly constructed by Dong, Li and Mason and we obtain a new construction without using relations corresponding to the commutator formula. The hard part of this new construction is a technical theorem stating roughly that in a module for Zhu's algebra, the relation corresponding to the residue formula mentioned above can in fact be obtained from the relations corresponding to the action of Zhu's algebra.

math.QA↗
arXiv2013

Tensor categories and the mathematics of rational and logarithmic conformal field theory

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. For rational and logarithmic conformal field theories, we review the precise results that yield braided tensor categories, and in the rational case, modular tensor categories as well. In the case of rational conformal field theory, we also briefly discuss the construction of the modular tensor categories for the Wess-Zumino-Novikov-Witten models and, especially, a recent discovery concerning the proof of the fundamental rigidity property of the modular tensor categories for this important special case. In the case of logarithmic conformal field theory, we mention suitable categories of modules for the triplet \mathcal{W}-algebras as an example of the applications of our general construction of the braided tensor category structure.

hep-th↗
arXiv2013

Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules

This is the first 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. This theory generalizes the tensor category 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 vertex operator algebras requires logarithmic structure. This work includes the complete proofs in the present generality and can be read independently of the earlier series of papers. Since this is a new theory, we present it in detail, including the necessary new foundational material. In addition, with a view toward anticipated applications, we develop and present the various stages of the theory in the natural, general settings in which the proofs hold, settings that are sometimes more general than what we need for the main conclusions. In this paper (Part I), we give a detailed overview of our theory, state our main results and introduce the basic objects that we shall study in this work. We include a brief discussion of some of the recent applications of this theory, and also a discussion of some recent literature.

math.QA↗
arXiv2012

Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition

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.

math.QA↗
arXiv2012

Logarithmic tensor category theory, VI: Expansion condition, associativity of logarithmic intertwining operators, and the associativity isomorphisms

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.

math.QA↗