Tensor categories for vertex operator superalgebra extensions

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) Amodules 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 A-modules is a vertex tensor functor. Two applications are given:First, we derive Verlinde formulae for regular vertex operator superalgebras and regular 1 2 Z-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. * T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460).

Quantum coordinate ring in WZW model and affine vertex algebra extensions

“…The rest of the argument is completely similar to the above (for more detail, see [CKM1]). Hence, we have:…”

Modularity of Bershadsky-Polyakov minimal models

“…is such an orbit, by (5.38), and conformal weight considerations show that it is a simple twisted B-module for all ≠ 1 6 . Fusion rules for these B-modules may be obtained from the BP(4, 3) fusion rules by induction [10], see also [55]. Those involving the simple current extension B (and its spectral flows) are obvious, so we compute only…”

A Kazhdan-Lusztig correspondence for $L_{-\frac{3}{2}}(\mathfrak{sl}_3)$

“…As L − 3 2 (sl 3 ) is a simple current extension in this direct limit completion [CMY20a,AR18], it now follows from [CKL20a, CKM17] that D is the category of local modules for L − 3 2 (sl 3 ), viewed as a commutative algebra object in the direct limit completion. There is therefore an induction functor that maps any module V in the completion that centralises the algebra object L − 3 2 (sl 3 ) to an object Ind V in D. Moreover, this functor is a vertex tensor functor [CKM17], meaning that it respects the fusion products of the completion and D:…”

“…Another important check of this Kazhdan-Lusztig correspondence is that it is indeed a tensor equivalence: it maps tensor products in C R to fusion rules in E . As the latter are not known, the former will be used to deduce fusion rules in D (again using [CKM17]). These in turn will be compared with the Grothendieck fusion rules (4.6) computed for D in [KRW21].…”