Tensor Categories for Vertex Operator Superalgebra Extensions: Volume: 295 Number: 1472 (Memoirs of the American Mathematical Society) - Softcover

Creutzig, Thomas; Kanade, Shashank; McRae, Robert

Softcover

ISBN 10: 1470467240 ISBN 13: 9781470467241

Publisher: American Mathematical Society, 2024

Synopsis

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 1 2Z-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and Z-graded components, respectively.

About this title

About the Author

Thomas Creutzig, University of Alberta, Edmonton, Alberta, Canada.

Shashank Kanade, University of Denver, CO.

Robert McRae, Vanderbilt University, Nashville, TN.

Edition details

• Publisher: American Mathematical Society
• Publication date: 2024
• Language: English
• ISBN 10: 1470467240
• ISBN 13: 9781470467241
• Binding: Paperback
• Number of pages: 187