Tensor Categories for Vertex Operator Superalgebra Extensions

Creutzig T, Kanade S, McRae R (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Memoirs of the American Mathematical Society American Mathematical Society

Book Volume: 295

Pages Range: 1-181

Journal Issue: 1472

DOI: 10.1090/memo/1472

Abstract

Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V. We employ tensor categories to study untwisted (also called local) A-modules in C, using results of Huang-Kirillov-Lepowsky that show that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V-modules to Amodules is a vertex tensor functor. Two applications are given: First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 12Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively. Second, we analyze parafermionic cosets C = Com(VL,V) where L is a positive definite even lattice and V is regular. If the vertex tensor category of either V-modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

Involved external institutions

University of Alberta CA Canada (CA) University of Denver US United States (USA) (US) Vanderbilt University US United States (USA) (US)

How to cite

APA:

Creutzig, T., Kanade, S., & McRae, R. (2024). Tensor Categories for Vertex Operator Superalgebra Extensions. Memoirs of the American Mathematical Society, 295(1472), 1-181. https://doi.org/10.1090/memo/1472

MLA:

Creutzig, Thomas, Shashank Kanade, and Robert McRae. "Tensor Categories for Vertex Operator Superalgebra Extensions." Memoirs of the American Mathematical Society 295.1472 (2024): 1-181.

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