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These notes survey the theory of (twisted) conformal blocks from an algebro-geometric perspective and have two main goals. The first one is to summarize the construction of conformal blocks from vertex operator algebras, and to describe…

Algebraic Geometry · Mathematics 2026-04-02 Chiara Damiolini

A commutative associative algebra $A$ over ${\mathbb C}$ with a derivation is one of the simplest examples of a vertex algebra. However, the differences between the modules for $A$ as a vertex algebra and the modules for $A$ as an…

Quantum Algebra · Mathematics 2013-12-18 Kenichiro Tanabe

This paper studies the twisted representations of vertex operator algebras. Let V be a vertex operator algebra and g an automorphism of V of finite order T. For any m,n in (1/T)Z_+, an A_{g,n}(V)-A_{g,m}(V)-bimodule A_{g,n,m}(V) is…

Quantum Algebra · Mathematics 2007-05-23 Chongying Dong , Cuipo Jiang

The problem of constructing twisted modules for a vertex operator algebra and an automorphism has been solved in particular in two contexts. One of these two constructions is that initiated by the third author in the case of a lattice…

Quantum Algebra · Mathematics 2007-05-23 Katrina Barron , Yi-Zhi Huang , James Lepowsky

We give a complete description of the full automorphism group of a lattice vertex operator algebra, determine the twisted Zhu's algebra for the automorphism lifted from the -1 isometry of the lattice and classify the corresponding…

Quantum Algebra · Mathematics 2007-05-23 Chongying Dong , Kiyokazu Nagatomo

We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{\mathcal{L}}(\ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $\mathcal{L}_{t}$, and show that…

Quantum Algebra · Mathematics 2020-08-04 Hongyan Guo

We construct the twisted Fock module of quantum toroidal $\mathfrak{gl}_1$ algebra with a slope $n'/n$ using vertex operators of quantum affine $\mathfrak{gl}_n$. The proof is based on the $q$-wedge construction of an integrable level-one…

Quantum Algebra · Mathematics 2021-09-28 Mikhail Bershtein , Roman Gonin

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…

High Energy Physics - Theory · Physics 2008-02-03 Yi-Zhi Huang

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.…

Quantum Algebra · Mathematics 2020-04-03 Yi-Zhi Huang

We construct Wakimoto modules for twisted affine Lie algebras, and interpret the construction in terms of vertex algebras and their twisted modules. Using the Wakimoto realization, we prove the Kac-Kazhdan conjecture on the characters of…

Quantum Algebra · Mathematics 2007-05-23 Matthew Szczesny

A representation of the central extension of the unitary Lie algebra coordinated with a skew Laurent polynomial ring is constructed using vertex operators over an integral Z_2-lattice. The irreducible decomposition of the representation is…

Quantum Algebra · Mathematics 2021-03-17 Fulin Chen , Yun Gao , Naihuan Jing , Shaobin Tan

In this paper, we use the twisted regular representation theory of vertex operator algebras to construct bimodules over twisted Zhu algebras, extending Haisheng Li's work in untwisted scenarios. Moreover, a conjecture of Dong and Jiang on…

Quantum Algebra · Mathematics 2025-05-23 Yiyi Zhu

In this paper, we introduce a notion of twisted restricted conformal blocks on totally ramified orbicurves and establish an isomorphism between the space of twisted restricted conformal blocks and the space of twisted conformal blocks. The…

Algebraic Geometry · Mathematics 2024-04-02 Xu Gao , Jianqi Liu , Yiyi Zhu

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,…

Quantum Algebra · Mathematics 2019-08-28 Yi-Zhi Huang

We introduce new concepts in order to develop a general formalism for twisted differential operators in several variables. We investigate the notion of twisted coordinates on Huber rings that allows us to build various rings of twisted…

Algebraic Geometry · Mathematics 2024-10-11 Pierre Houédry

The paper is to classify irreducible integrable modules for the twisted full toroidal Lie algebra with some technical conditions. The twisted full toroidal Lie algebra are extensions of multiloop algebra twisted by sevaral finite order…

Representation Theory · Mathematics 2015-09-10 S. Eswara Rao , Punita Batra

Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to…

Quantum Algebra · Mathematics 2025-01-14 Yuri Bazlov , Edward Jones-Healey

This contribution studies a specific deformation of algebras with anti-involution. Starting with the observation that twisting the multiplication of such an algebra by its anti-involution generates a Hom-associative algebra of type II, it…

Rings and Algebras · Mathematics 2023-10-03 Alexis Langlois-Rémillard

A twisted ring is a ring endowed with a family of endomorphisms satisfying certain relations. One may then consider the notions of twisted module and twisted differential module. We study them and show that, under some general hypothesis,…

Algebraic Geometry · Mathematics 2015-03-18 Bernard Le Stum , Adolfo Quirós

In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FI-modules and Delta-modules. In this paper, we add another example to the growing list: we show that…

Commutative Algebra · Mathematics 2016-03-24 Rohit Nagpal , Steven V Sam , Andrew Snowden