Searcharxiv⌕ Search

arXiv subjects

Chongying Dong

Publications and source records attributed to Chongying Dong.

At least 19 recordsLinked to original sources
arXiv2026

Generalized Symmetries From Fusion Actions

Let $A$ be a condensable algebra in a modular tensor category $\mathcal{C}$. We define an action of the fusion category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ on the morphism space $\mbox{Hom}_{\mathcal{C}}(x,A)$ for any $x$ in $\mathcal{C}$, whose characters are generalized Frobenius-Schur indicators. This fusion action can be considered on $A$, and we prove a categorical generalization of the Schur-Weyl duality for this action. For any fusion subcategory $\mathcal{B}$ of $\mathcal{C}_A$ containing all the local $A$-modules, we prove the invariant subobject $B=A^\mathcal{B}$ is a condensable subalgebra of $A$. The assignment of $\mathcal{B}$ to $A^\mathcal{B}$ defines a Galois correspondence between this kind of fusion subcategories of $\mathcal{C}_A$ and the condensable subalgebras of $A$. In the context of VOAs, we prove for any nice VOAs $U \subset A$, $U=A^{\mathcal{C}_A}$ where $\mathcal{C}=\mathcal{M}_U$ is the category of $U$-modules. In particular, if $U = A^G$ for some finite automorphism group $G$ of $A,$ the fusion action of $\mathcal{C}_A$ on $A$ is equivalent to the $G$-action on $A.$

math.QA↗
arXiv2024

Fusion Products of Twisted Modules in Permutation Orbifolds: II

Let $V$ be a simple, rational, $C_{2}$-cofinite vertex operator algebra of CFT type, and let $k$ be a positive integer. In this paper, we determine the fusion products of twisted modules for $V^{\otimes k}$ and $G = \left\langle g \right\rangle$ generated by any permutation $g \in S_{k}$.

math.QA↗
arXiv2024

Coset constructions and Kac-Wakimoto Hypothesis

Categorical coset constructions are investigated and Kac-Wakimoto Hypothesis associated with pseudo unitary modular tensor categories is proved. In particular, the field identifications are obtained. These results are applied to the coset constructions in the theory of vertex operator algebra.

math.QA↗
arXiv2023

Twisted Verlinde formula for vertex operator algebras

For a rational and $C_2$-cofinite vertex operator algebra $V$ with an automorphism group $G$ of prime order, the fusion rules for twisted $V$-modules are studied, a twisted Verlinde formula which relates fusion rules for $g$-twisted modules to the $S$-matrix in the orbifold theory is established. As an application of the twisted Verlinde formula, a twisted analogue of the Kac-Walton formula is proved, which gives fusion rules between twisted modules of affine vertex operator algebras in terms of Clebsch-Gordan coefficients associated to the corresponding finite dimensional simple Lie algebras.

math.QA↗
arXiv2023

Hopf actions on vertex algebras

In this article, we investigate Hopf actions on vertex algebras. Our first main result is that every finite-dimensional Hopf algebra that inner faithfully acts on a given π_2-injective vertex algebra must be a group algebra. Secondly, under suitable assumptions, we establish a Schur-Weyl type duality for semisimple Hopf actions on Hopf modules of vertex algebras.

math.QA↗
arXiv2023

Permutation orbifolds of vertex operator superalgebra and associative algebras

Let $V$ be a vertex operator superalgebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the tensor product vertex operator superalgebra $V^{\otimes k}$. In this paper, we construct an explicit isomorphism from $A_{g}\left(V^{\otimes k}\right)$ to $A\left(V\right)$ if $k$ is odd and to $A_σ\left(V\right)$ if $k$ is even where $σ$ is the canonical automorphism of $V$ of order 2 determined by the superspace structure of $V.$ These recover previous results by Barron and Barron-Werf that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modules and irreducible $V$-modules (resp. irreducible $σ$-twisted $V$-modules) when $k$ is odd (resp. even). This explicit isomorphism is expected to be useful in our further study on the Zhu algebra of fixed point subalgebra.

math.QA↗
arXiv2023

Orbifold theory for vertex algebras and Galois correspondence

Let $V$ be a simple vertex algebra of countable dimension, $G$ be a finite automorphism group of $V$ and $σ$ be a central element of $G$. Assume that ${\cal S}$ is a finite set of inequivalent irreducible $σ$-twisted $V$-modules such that ${\cal S}$ is invariant under the action of $G$. Then there is a finite dimensional semisimple associative algebra ${\cal A}_α(G,{\cal S})$ for a suitable $2$-cocycle $α$ naturally determined by the $G$-action on ${\cal S}$ such that $({\cal A}_α(G,{\cal S}),V^G)$ form a dual pair on the sum $\cal M$ of $σ$-twisted $V$-modules in ${\cal S}$ in the sense that (1) the actions of ${\cal A}_α(G,{\cal S})$ and $V^G$ on $\cal M$ commute, (2) each irreducible ${\cal A}_α(G,{\cal S})$-module appears in $\cal M,$ (3) the multiplicity space of each irreducible ${\cal A}_α(G,{\cal S})$-module is an irreducible $V^G$-module, (4) the multiplicitiy spaces of different irreducible ${\cal A}_α(G,{\cal S})$-modules are inequivalent $V^G$-modules. As applications, every irreducible $σ$-twisted $V$-module is a direct sum of finitely many irreducible $V^G$-modules and irreducible $V^G$-modules appearing in different $G$-orbits are inequivalent. This result generalizes many previous ones. We also establish a bijection between subgroups of $G$ and subalgebras of $V$ containing $V^G.$

math.QA↗
arXiv2021

$S$-matrix in permutation orbifolds

For a fixed positive integer $k$, any element $g$ of the permutation group $S_{k}$ acts on the tensor product vertex operator algebra $V^{\otimes k}$ in the obvious way. In this paper, we determine the $S$-matrix of $\left(V^{\otimes k}\right)^{G}$ if $G=\left\langle g\right\rangle $ is the cyclic group generated by $g=\left(1,\ 2,\cdots,k\right).$

math.QA↗
arXiv2021

Orbifolds and minimal modular extensions

Let $V$ be a simple, rational, $C_2$-cofinite vertex operator algebra and $G$ a finite group acting faithfully on $V$ as automorphisms, which is simply called a rational vertex operator algebra with a $G$-action. It is shown that the category ${\cal E}_{V^G}$ generated by the $V^G$-submodules of $V$ is a symmetric fusion category braided equivalent to the $G$-module category ${\cal E}={\rm Rep}(G)$. If $V$ is holomorphic, then the $V^G$-module category ${\cal C}_{V^G}$ is a minimal modular extension of ${\cal E},$ and is equivalent to the Drinfeld center ${\cal Z}({\rm Vec}_G^α)$ as modular tensor categories for some $α\in H^3(G,S^1)$ with a canonical embedding of ${\cal E}$. Moreover, the collection ${\cal M}_v({\cal E})$ of equivalence classes of the minimal modular extensions ${\cal C}_{V^G}$ of ${\cal E}$ for holomorphic vertex operator algebras $V$ with a $G$-action form a group, which is isomorphic to a subgroup of $H^3(G,S^1).$ Furthermore, any pointed modular category ${\cal Z}({\rm Vec}_G^α)$ is equivalent to ${\cal C}_{V_L^G}$ for some positive definite even unimodular lattice $L.$ In general, for any rational vertex operator algebra $U$ with a $G$-action, ${\cal C}_{U^G}$ is a minimal modular extension of the braided fusion subcategory ${\cal F}$ generated by the $U^G$-submodules of $U$-modules. Furthermore, the group ${\cal M}_v({\cal E})$ acts freely on the set of equivalence classes ${\cal M}_v({\cal F})$ of the minimal modular extensions ${\cal C}_{W^G}$ of ${\cal F}$ for any rational vertex operators algebra $W$ with a $G$-action.

math.QA↗
arXiv2021

Fusion products of twisted modules in permutation orbifolds

Let $V$ be a vertex operator algebra, $k$ a positive integer and $σ$ a permutation automorphism of the vertex operator algebra $V^{\otimes k}$. In this paper, we determine the fusion product of any $V^{\otimes k}$-module with any $σ$-twisted $V^{\otimes k}$-module.

math.QA↗
arXiv2021

Super orbifold theory

Let $V$ be a simple vertex operator superalgebra and $G$ a finite automorphism group of $V$ containing the canonical automorphism $σ$ such that $V^G$ is regular. It is proved that every irreducible $V^G$-module occurs in an irreducible $g$-twisted $V$-module for some $g\in G$ and the irreducible $V^G$-modules are classified. Moreover, the quantum dimensions of irreducible $V^G$-modules are determined, a global dimension formula for $V$ in terms of twisted modules is obtained and a super quantum Galois theory is established. In addition, the $S$-matrix of $V^G$ is computed

math.QA↗
arXiv2021

Cliffold algebras, modular Virasoro vertex operator algebras and Z[1/2]-forms

This paper consists of two parts: (1) Using a Z[1/2]-form of Virasoro vertex operator algebra L(1/2,0) with central charge 1/2, we obtain a modular vertex operator algebra over any field F of finite characteristic different from 2. We determine the generators and classify the irreducible modules for this vertex operator algebra. (2) We investigate modular framed vertex operator algebras. In particular, the rationality of modular framed vertex operator algebras is established. For a modular code vertex operator algebra, the irreducible modules are constructed and classified. Moreover, a Z[1/2]-form for any framed vertex operator algebra over complex field C is constructed. As a result, one can obtain a modular framed vertex operator algebra from any framed vertex operator algebra over C.

math.QA↗
arXiv2020

S-matrix in orbifold theory

The restricted $S$-matrix of $V^G$ is determined for any regular vertex operator algebra $V$ and finite automorphism group $G$ of $V.$ As an application, the $S$-matrices for cyclic permutation orbifolds of prime orders are computed.

math.QA↗
arXiv2020

Permutation orbifolds and associative algebras

Let $V$ be a vertex operator algebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the vertex operator algebra $V^{\otimes k}$. It is proved that Dong-Li-Mason's associated associative algebra $A_{g}\left(V^{\otimes k}\right)$ is isomorphic to Zhu's algebra $A\left(V\right)$ explicitly. This result recovers a previous result that there is a one-to-one correspondence between irreducible $g$-twisted $V^{\otimes k}$-modules and irreducible $V$-modules.

math.QA↗
arXiv2020

Uniqueness of VOA structure of 3C-algebra and 5A-algebra

The structure of 3C-algebra and 5A-algebra constructed by Lam-Yamada-Yamauchi is studied and the uniqueness of the vertex operator algebra structure of these two algebras is established. We also give the fusion rules for these two algebras.

math.QA↗
arXiv2020

Vertex operator superalgebras and 16-fold way

Let $V$ be a vertex operator superalgebra with the natural order 2 automorphism $σ$. Under suitable conditions on $V$, the $σ$-fixed subspace $V_{\bar 0}$ is a vertex operator algebra and the category $C_{V_{\bar 0}}$ of $V_{\bar 0}$-modules is modular tensor category. In this paper, we prove that $C_{V_{\bar 0}}$ is a fermionic modular tensor category and the Müger centralizer $C_{V_{\bar 0}}^0$ of the fermion in $C_{V_{\bar 0}}$ is generated by the irreducible $V_{\bar 0}$-submodules of the $V$-modules. In particular, $C_{V_{\bar 0}}^0$ is a super-modular tensor category and $C_{V_{\bar 0}}$ is a minimal modular extension of $C_{V_{\bar 0}}^0$. We provide a construction of a vertex operator $V^l$ for each positive integer $l$ such that $C_{V^l_{\bar 0}}$ is minimal modular extension of $C_{V_{\bar 0}}^0$. We prove that these modular tensor categories $C_{V^l_{\bar 0}}$ are uniquely determined, up to equivalence, by the congruence class of $l$ modulo 16.

math.QA↗
arXiv2019

$6A$-Algebra and its representations

In this paper, we study the structure and representation of a $6A$-algebra which is a vertex operator algebra generated by two Ising vectors $e,f$ with inner product $\left\langle e,f\right\rangle =\frac{5}{2^{10}}.$ In particular, we prove the uniqueness of the vertex operator algebra structure of this 6A-algebra, classify the irreducible modules, and determine the fusion rules.

math.QA↗