arXiv subjects

Yi-Zhi Huang

Publications and source records attributed to Yi-Zhi Huang.

At least 19 recordsLinked to original sources

arXiv2026

Twisted Intertwining Operators and Tensor Products of (Generalized) Twisted Modules

We study the general twisted intertwining operators (intertwining operators among twisted modules) for a vertex operator algebra $V$. We give the skew-symmetry and contragredient isomorphisms between spaces of twisted intertwining operators and also prove some other properties of twisted intertwining operators. Using twisted intertwining operators,we introduce a notion of $P(z)$-tensor product of two objects for $z\in \mathbb C^{\times}$ in a category of suitable $g$-twisted $V$-modules for $g$ in a group of automorphisms of $V$ and give a construction of such a $P(z)$-tensor product under suitable assumptions. We also construct $G$-crossed commutativity isomorphisms and $G$-crossed braiding isomorphisms. We formulate a $P(z)$-compatibility condition and a $P(z)$-grading-restriction condition and use these conditions to give another construction of the $P(z)$-tensor product.

math.QA↗

arXiv2026

Boson-fermion meromorphic open-string vertex algebras and their canonically twisted modules

We construct a $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra from a finite-dimensional vector space with a nondegenerate symmetric bilinear form, together with its canonically twisted module. This algebra is generated by suitable noncommutative generalizations of bosonic and fermionic fields and is a noncommutative generalization of the free boson-fermion vertex operator superalgebra. Similarly to the purely bosonic and purely fermionic cases in the early works by the second author in [H1], by Fiordalisi and the third author in [FQ], and by the third author in [Q3], the usual super-commutatitive relations between creation and annihilation operators still hold while no relations exist among creation operators. In particular, normal-ordering remains well-defined. As in [H1], [FQ], and [Q3], we prove a generalized Wick's theorem in this case, which gives a formula for a product of two normal ordered products of bosonic and fermionic generating fields. Using this generalized Wick's theorem, we construct the $\frac{\mathbb{Z}}{2}$-graded meromorphic open-string vertex algebra and its canonically twisted module in this case. The construction in this paper is the algebraic part of our construction of suitable Dirac-like operators from spin manifolds.

math.QA↗

arXiv2026

Meromorphic open-string vertex algebras and Riemannian manifolds

Let $M$ be a Riemannian manifold. For $p\in M$, the tensor algebra of the negative part of the (complex) affinization of the tangent space of $M$ at $p$ has a natural structure of a meromorphic open-string vertex algebra. These meromorphic open-string vertex algebras form a vector bundle over $M$ with a connection. We construct a sheaf $\mathcal{V}$ of meromorphic open-string vertex algebras on the sheaf of parallel sections of this vector bundle. Using covariant derivatives, we construct representations on the spaces of complex smooth functions of the algebras of parallel tensor fields. These representations are used to construct a sheaf $\mathcal{W}$ of left $\mathcal{V}$-modules from the sheaf of smooth functions. In particular, we obtain a meromorphic open-string vertex algebra $V_{M}$ of the global sections on $M$ of the sheaf $\mathcal{V}$ and a left $V_{M}$-module $W_{M}$ of the global sections on $M$ of the sheaf $\mathcal{W}$. By the definitions of meromorphic open-string vertex algebra and left module, we obtain, among many other properties, operator product expansion for vertex operators. We also show that the Laplacian on $M$ is in fact a component of a vertex operator for the left $V_{M}$-module $W_{M}$ restricted to the space of smooth functions.

math.DG↗

arXiv2026

Cofiniteness and $P(z)$-tensor product bifunctors in orbifold theories associated to abelian but not-necessarily-finite groups

Let $V$ be a Möbius vertex algebra and $G$ an abelian group of automorphisms of $V$. We construct $P(z)$-tensor product bifunctors for the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules (without $g$-actions) for $g\in G$ and the category of $C_{n}$-cofinite grading-restricted generalized $g$-twisted $V$-modules with $G$-actions for $g\in G$. In this paper, an automorphism $g$ of $V$ can be of infinite order and does not have to act semisimply on $V$, and the group $G$ can be an infinite abelian group containing nonsemisimple automorphisms of $V$.

math.QA↗

arXiv2025

Modular invariance of (logarithmic) intertwining operators

Let $V$ be a $C_2$-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-$q$-traces ($q=e^{2πiτ}$) shifted by $-\frac{c}{24}$ of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized $V$-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-$q$-traces were proved by Fiordalisi in [F1] and [F2] using the method developed in [H2]. The method that we use to prove this conjecture is based on the theory of the associative algebras $A^{N}(V)$ for $N\in \mathbb{N}$, their graded modules and their bimodules introduced and studied by the author in [H8] and [H9]. This modular invariance result gives a construction of $C_2$-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.

math.QA↗

arXiv2025

$C_{1}$-cofiniteness and vertex tensor categories

We first generalize the logarithmic tensor category theory of Huang-Lepowsky-Zhang to the more general case that the module category for a vertex operator algebra $V$ (more generally a Möbius vertex algebra) might not be closed under the contragredient functor. Then by verifying the assumptions to use this generalization, we obtain that (logarithmic) intertwining operators among $C_{1}$-cofinite grading-restricted generalized $V$-modules satisfy the associativity property (operator product expansion) and the category of $C_{1}$-cofinite grading-restricted generalized $V$-modules has a natural vertex tensor category structure. In particular, this category has a natural braided tensor category structure with a twist.

math.QA↗

arXiv2023

Associative algebras and the representation theory of grading-restricted vertex algebras

We introduce an associative algebra $A^{\infty}(V)$ using infinite matrices with entries in a grading-restricted vertex algebra $V$ such that the associated graded space $Gr(W)=\coprod_{n\in \mathbb{N}}Gr_{n}(W)$ of a filtration of a lower-bounded generalized $V$-module $W$ is an $A^{\infty}(V)$-module satisfying additional properties (called a graded $A^{\infty}(V)$-module). We prove that a lower-bounded generalized $V$-module $W$ is irreducible or completely reducible if and only if the graded $A^{\infty}(V)$-module $Gr(W)$ is irreducible or completely reducible, respectively. We also prove that the set of equivalence classes of the lower-bounded generalized $V$-modules are in bijection with the set of the equivalence classes of graded $A^{\infty}(V)$-modules. For $N\in \mathbb{N}$, there is a subalgebra $A^{N}(V)$ of $A^{\infty}(V)$ such that the subspace $Gr^{N}(W)=\coprod_{n=0}^{N}Gr_{n}(W)$ of $Gr(W)$ is an $A^{N}(V)$-module satisfying additional properties (called a graded $A^{N}(V)$-module). We prove that $A^{N}(V)$ are finite dimensional when $V$ is of positive energy (CFT type) and $C_{2}$-cofinite. We prove that the set of the equivalence classes of lower-bounded generalized $V$-modules is in bijection with the set of the equivalence classes of graded $A^{N}(V)$-modules. In the case that $V$ is a Möbius vertex algebra and the differences between the real parts of the lowest weights of the irreducible lower-bounded generalized $V$-modules are less than or equal to $N\in \mathbb{N}$, we prove that a lower-bounded generalized $V$-module $W$ of finite length is irreducible or completely reducible if and only if the graded $A^{N}(V)$-module $Gr^{N}(W)$ is irreducible or completely reducible, respectively.

math.QA↗

arXiv2022

Weight-one elements of vertex operator algebras and automorphisms of categories of generalized twisted modules

Given a weight-one element $u$ of a vertex operator algebra $V$, we construct an automorphism of the category of generalized $g$-twisted modules for automorphisms $g$ of $V$ fixing $u$. We apply this construction to the case that $V$ is an affine vertex operator algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules.

math.QA↗

arXiv2022

Associative algebras and intertwining operators

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 $\hom_{A^{\infty}(V)}(A^{\infty}(W_{1})\otimes_{A^{\infty}(V)}W_{2}, W_{3})$. Assuming that $W_{2}$ and $W_{3}'$ are equivalent to certain universal lower-bounded generalized $V$-modules generated by their $A^{N}(V)$-submodules consisting of elements of levels less than or equal to $N\in \mathbb{N}$, we also prove that the space of (logarithmic) intertwining operators of type $\binom{W_{3}}{W_{1}W_{2}}$ is isomorphic to the space of $\hom_{A^{N}(V)}(A^{N}(W_{1})\otimes_{A^{N}(V)}Ω_{N}^{0}(W_{2}), Ω_{N}^{0}(W_{3}))$.

math.QA↗

arXiv2022

Convergence in conformal field theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. We review some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. We also review the related analytic extension results, conjectures and problems. We discuss 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. We also discuss the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. We then explain conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

math.QA↗

arXiv2021

Logarithmic intertwining operators and associative algebras

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.

math.QA↗

arXiv2020

Lower-bounded and grading-restricted twisted modules for affine vertex (operator) algebras

We apply the construction of the universal lower-bounded generalized twisted modules by the author to construct universal lower-bounded and grading-restricted generalized twisted modules for affine vertex (operator) algebras. We prove that these universal twisted modules for affine vertex (operator) algebras are equivalent to suitable induced modules of the corresponding twisted affine Lie algebra or quotients of such induced modules by explicitly given submodules.

math.QA↗

arXiv2020

The first cohomology, derivations and the reductivity of a (meromorphic open-string) vertex algebra

We give a criterion for the complete reducibility of modules satisfying a composability condition for a meromorphic open-string vertex algebra $V$ using the first cohomology of the algebra. For a $V$-bimodule $M$, let $\hat{H}^{1}_{\infty}(V, M)$ be the first cohomology of $V$ with the coefficients in $M$. Let $\hat{Z}^{1}_{\infty}(V, M)$ be the subspace of $\hat{H}^{1}_{\infty}(V, M)$ canonically isomorphic to the space of derivations obtained from the zero mode of the right vertex operators of weight $1$ elements such that the difference between the skew-symmetric opposite action of the left action and the right action on these elements are Laurent polynomials in the variable. If $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ for every $\Z$-graded $V$-bimodule $M$, then every left $V$-module satisfying a composability condition is completely reducible. In particular, since a lower-bounded $\Z$-graded vertex algebra $V$ is a special meromorphic open-string vertex algebra and left $V$-modules are in fact what has been called generalized $V$-modules with lower-bounded weights (or lower-bounded generalized $V$-modules), this result provides a cohomological criterion for the complete reducibility of lower-bounded generalized modules for such a vertex algebra. We conjecture that the converse of the main theorem above is also true. We also prove that when a grading-restricted vertex algebra $V$ contains a subalgebra satisfying some familiar conditions, the composability condition for grading-restricted generalized $V$-modules always holds and we need $\hat{H}^{1}_{\infty}(V, M)= \hat{Z}^{1}_{\infty}(V, M)$ only for every $\Z$-graded $V$-bimodule $M$ generated by a grading-restricted subspace in our complete reducibility theorem.

math.QA↗

arXiv2020

Generators, spanning sets and existence of twisted modules for a grading-restricted vertex (super)algebra

For a grading-restricted vertex superalgebra $V$ and an automorphism $g$ of $V$, we give a linearly independent set of generators of the universal lower-bounded generalized $g$-twisted $V$-module $\widehat{M}^{[g]}_{B}$ constructed by the author in \cite{H-const-twisted-mod}. We prove that there exist irreducible lower-bounded generalized $g$-twisted $V$-modules by showing that there exists a maximal proper submodule of $\widehat{M}^{[g]}_{B}$ for a one-dimensional space $M$. We then give several spanning sets of $\widehat{M}^{[g]}_{B}$ and discuss the relations among elements of the spanning sets. Assuming that $V$ is a Möbius vertex superalgebra (to make sure that lowest weights make sense) and that $P(V)$ (the set of all numbers of the form $\Re(α)\in [0, 1)$ for $α\in \C$ such that $e^{2πi α}$ is an eigenvalue of $g$) has no accumulation point in $\R$ (to make sure that irreducible lower-bounded generalized $g$-twisted $V$-modules have lowest weights). Under suitable additional conditions, which hold when the twisted zero-mode algebra or the twisted Zhu's algebra is finite dimensional, we prove that there exists an irreducible grading-restricted generalized $g$-twisted $V$-module, which is in fact an irreducible ordinary $g$-twisted $V$-module when $g$ is of finite order. We also prove that every lower-bounded generalized module with an action of $g$ for the fixed-point subalgebra $V^{g}$ of $V$ under $g$ can be extended to a lower-bounded generalized $g$-twisted $V$-module.

math.QA↗

arXiv2020

Representation theory of vertex operator algebras and orbifold conformal field theory

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 operators and their basic properties are also reviewed. The conjectural properties in the basic open problems and conjectures mentioned above are then formulated precisely and explicitly. Some thoughts of the author on further developments of orbifold conformal field theory are also discussed.

math.QA↗

arXiv2019

A construction of lower-bounded generalized twisted modules for a grading-restricted vertex (super)algebra

We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra $V$ associated to an automorphism $g$ of $V$. In particular, when $g$ is the identity, we obtain lower-bounded generalized $V$-modules satisfying a universal property. Let $W$ be a lower-bounded graded vector space equipped with a set of "generating twisted fields" and a set of "generator twist fields" satisfying a weak commutativity for generating twisted fields, a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify. We first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields, one twist generator field and an arbitrary number of generating fields for $V$. Then using the convergence and commutativity, we define a twisted vertex operator map for $W$ and prove that $W$ equipped with this twisted vertex operator map is a lower-bounded generalized $g$-twisted $V$-module. Using this result, we give an explicit construction of lower-bounded generalized $g$-twisted $V$-modules satisfying a universal property starting from vector spaces graded by weights, $\mathbb{Z}_{2}$-fermion numbers and $g$-weights (eigenvalues of $g$) and real numbers corresponding to the lower bounds of the weights of the modules to be constructed. In particular, every lower-bounded generalized $g$-twisted $V$-module (every lower-bounded generalized $V$-module when $g$ is the identity) is a quotient of such a universal lower-bounded generalized $g$-twisted $V$-module (a universal lower-bounded generalized $V$-module).

math.QA↗

arXiv2019

Twist vertex operators for twisted modules

We introduce and study twist vertex operators for a (lower-bounded generalized) twisted modules for a grading-restricted vertex (super)algebra. We prove duality, weak associativity, a Jacobi identity, a generalized commutator formula, generalized weak commutativity, and convergence and commutativity for products of more than two operators involving twist vertex operators. These properties of twist vertex operators play an important role in the author's recent general, direct and explicit construction of (lower-bounded generalized) twisted modules.

math.QA↗

arXiv2018

Affine Lie algebras and tensor categories

We review briefly the existing vertex-operator-algebraic constructions of various tensor category structures on module categories for affine Lie algebras. We discuss the results first conjectured in the work of Moore and Seiberg that led us to the construction of the modular tensor category structure in the positive integral level case. Then we review the existing constructions and results in the following three cases: (i) the level plus the dual Coxeter number is not a nonnegative rational number, (ii) the level is a positive integer and (iii) the level is an admissible number. We also present several open problems.

math.QA↗