Related papers: Nichols algebras, tensor categories and Kazhdan-Lu…
Two related constructions are associated with screening operators in models of two-dimensional conformal field theory. One is a local system constructed in terms of the braided vector space X spanned by the screening species in a given CFT…
Let $\mathcal{U}$ be a braided tensor category, typically unknown, complicated and in particular non-semisimple. We characterize $\mathcal{U}$ under the assumption that there exists a commutative algebra $A$ in $\mathcal{U}$ with certain…
We prove that certain screening operators in conformal field theory obey the algebra relations of a corresponding Nichols algebra with diagonal braiding. Our result proves in particular a long-standing expectation that the Borel parts of…
We incorporate a category of certain modules for an affine Lie algebra, of a certain fixed non-positive-integral level, considered by Kazhdan and Lusztig, into the representation theory of vertex operator algebras, by using the logarithmic…
We prove the Kazhdan-Lusztig correspondence for a class of vertex operator superalgebras which, via the work of Costello-Gaiotto, arise as boundary VOAs of topological B twist of 3d $\mathcal{N}=4$ abelian gauge theories. This means that we…
We introduce the main concepts and announce the main results in a theory of tensor products for module categories for a vertex operator algebra. This theory is being developed in a series of papers including hep-th 9309076 and hep-th…
These are lecture notes for an introductory course on Nichols algebras. As a main reference, I work with the book by Heckenberger and Schneider, but I want to take a distinct categorical perspective and try to develop the topic for an…
By a pointed vertex operator algebra (VOA) we mean one whose modules are all simple currents (i.e. invertible), e.g. lattice VOAs. This paper systematically explores the interplay between their orbifolds and tensor category theory. We begin…
In a vertex algebra setting, we consider non-local screening operators associated to the basis of any non-integral lattice. We have previously shown that, under certain restrictions, these screening operators satisfy the relations of a…
This paper is the first in a series of papers developing a functional-analytic theory of vertex (operator) algebras and their representations. For an arbitrary Z-graded finitely-generated vertex algebra (V, Y, 1) satisfying the standard…
This paper provides a unified framework resolving two long-standing problems: the intrinsic construction of global quantum gauge groups for braided tensor $C^*$-categories (the Doplicher-Roberts problem) and the direct proof of the…
Certain deformable families of vertex algebras acquire at a limit of the deformation parameter a large center, similar to affine Lie algebras at critical level. Then the vertex algebra and its representation category become a bundle over…
The logarithmic Kazhdan-Lusztig correspondence is a conjectural equivalence between braided tensor categories of representations of small quantum groups and representations of certain vertex operator algebras. In this article we prove such…
Every four-dimensional ${\cal N}=2$ superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected…
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a…
This is the third part in a series of papers developing a tensor product theory for modules for a vertex operator algebra. The goal of this theory is to construct a ``vertex tensor category'' structure on the category of modules for a…
We formulate a conjectural relation between the category of line defects in topologically twisted 3d ${\cal N} = 4$ supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local operators for…
We formally define and study the distinguished pre-Nichols algebra $\widetilde{\mathcal{B}}(V)$ of a braided vector space of diagonal type $V$ with finite-dimensional Nichols algebra $\mathcal{B}(V)$. The algebra…
The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair…
This is the first part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. This theory generalizes the tensor category theory for…