Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 086, 64 pages      arXiv:2501.15003      https://doi.org/10.3842/SIGMA.2026.086
Contribution to the Special Issue on Recent Advances in Vertex Operator Algebras in honor of James Lepowsky

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

Jishen Du and Yi-Zhi Huang
Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA

Received July 09, 2025, in final form July 30, 2026; Published online September 02, 2026

Abstract
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.

Key words: (generalized) twisted module; twisted intertwining operator; $P(z)$-tensor product; $G$-crossed commutativity isomorphism; $G$-crossed braiding isomorphism; $P(z)$-compatibility condition.

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