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In recent work with Damiolini and Tarasca, extending previous results, we have shown that simple modules over a vertex operator algebra V of CFT-type determine sheaves of coinvariants, and dual sheaves of conformal blocks on certain moduli spaces of stable pointed curves. If V is strongly rational, these are vector bundles, with Chern classes in the tautological ring. The factorization formula, which relies on rationality of V, played a crucial role in proving these results. In this talk I will discuss recent work with Damiolini and Krashen, where we introduce factorization presentations, applicable to C_1-cofinite V. As I'll explain, this new perspective simplifies the original proof of factorization and gives evidence that modules over strongly finite VOAs may determine vector bundles.
It was conjectured that modular categories were determined by its modular data (S- and T-matrices). In 2017, Mignard and Schauenburg presented a family of counterexamples to this conjecture, which led to the study of link invariants beyond modular data to distinguish these categories. In this talk we will discuss ribbon zesting, which is a construction of modular categories, and how it is related to the family of Mignard-Schauenburg counterexamples. To better understand this relation, we look into how zesting affects link invariants such as the W-matrix and the B-tensor. This talk is based on joint work with Colleen Delaney and Sung Kim (https://arxiv.org/abs/2107.11374).
A major challenge in VOA theory is to show that a given category of modules admits a vertex tensor category structure. It turns out that C_1-cofiniteness plus a few additional conditions is sufficient to ensure the existence of vertex tensor category. I will illustrate this in examples and explain its use beyond C_1-cofinite modules.
I will report on a joint work with K.Coulembier and P.Etingof. We give a characterization of symmetric tensor categories over fields of positive characteristic which admit an exact tensor functor to the Verlinde category; in particular we give a characterization of Tannakian categories. A crucial ingredient of this characterization is exactness of the Frobenius twist functor which mimics the Frobenius twist for representations of algebraic groups. We will also discuss some applications to modular representation theory.
Hook type W-superalgebras are W-superalgebras whose affine cosets appear at junctions of supersymmetric interfaces in N = 4 Super Yang Mills gauge theory. Their affine cosets enjoy a Feigin-Frenkel type duality as proven by Creutzig and Linshaw by the uniqueness property of these algebras. I will explain how this duality is enhanced to a reconstruction theorem for the W-superalgebra themselves via convolution operation with ``shifted" chiral differential operators. If time permits, I will talk about its representation theoretic applications and module characters. The talk is based on my joint work with Thomas Creutzig, Andrew Linshaw and Ryo Sato.
We talk about coset constructions of principal W-superalgebras of osp(1|2n), which are analogs of coset constructions of principal W-algebras of type ADE by Arakawa-Creutzig-Linshaw. The cosets are useful not only to study the category of modules at non-degenerate admissible levels, but also to prove the existence of embeddings of the affine vertex superalgebras of osp(1|2n) into the equivariant W-algebras of sp(2n) times 2n+1 free fermions. This leads to the rigidity of the category O of affine sp(2n) at admissible levels as a corollary. This is joint work with Thomas Creutzig and Andrew Linshaw.
In this talk, we summarize our recent work on the tensor categories for the singlet algebras, including the tensor structure on the category of the atypical modules, as well as on the full category of C_1-cofinite modules. We will also apply these results to study representation theory of vertex algebras that are extensions of the singlet algebras, especially the B_p-algebra. This talk is based on a series of joint work with T. Creutzig and R. McRae.
I will discuss the combinatorics of qq-characters as a tool for constructing deformed W-algebras and their extensions.
This is a report on a joint work in progress with B. Feigin and M. Jimbo.
Given a weight-one element u of a vertex operator algebra V , we construct an automorphism of the category of generalized g-twisted modules for automorphisms of g fixing u. We apply these results to the case that V is an affine vertex algebra to obtain explicit results on these automorphisms of categories. In particular, we give explicit constructions of certain generalized twisted modules from generalized twisted modules associated to diagram automorphisms of finite-dimensional simple Lie algebras and generalized (untwisted) modules. This talk is based on a joint work with Yi-Zhi Huang.
In this talk, we present several results on the rationality of W-algebras associated with subregular nilpotent elements of the Lie algebra so(2n+1) as well as applications to W-superalgebras of type osp(2|2n). The case n=2 was studied in my thesis; the generalization for higher ranks and the « super » cases is an on going work with Shigenori Nakatsuka (Alberta).
I will present upcoming joint work with Jinwei Yang on the non-semisimple Kazhdan-Lusztig categories KL^k(sl_2) of affine sl_2 at admissible levels k = −2 + p/q, where p > 1 and q > 0 are relatively prime integers. KL^k(sl_2) is the category of finite-length modules for affine sl_2 at level k whose composition factors are irreducible highest-weight modules whose highest weights are dominant integral for the finite-dimensional subalgebra sl_2. We show that KL^k(sl_2) admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang, but that it is not rigid. Instead, an object of KL^k(sl_2) is rigid if and only if it is projective and, moreover, KL^k(sl_2) has enough projectives; most of the indecomposable projective objects are logarithmic modules, which means that the Virasoro L(0) operator acts non-semisimply. We show also that the monoidal subcategory of rigid and projective objects is tensor equivalent to tilting modules for quantum sl_2 at the root of unity e^{pi i/(k+2)}. This leads to a universal property for KL^k(sl_2), which allows us to construct an essentially surjective (but not fully faithful) exact tensor functor from KL^k(sl2) to the category of finite-dimensional weight modules for quantum sl_2 at e^{pi i/(k+2)}.
In this talk, I will present a uniform description of the linkage classes for finite dimensional and affine Kac-Moody superalgebras. These linkage classes are then used to parametrize the blocks in the category O. I will describe the interaction between these linkage classes and the Duflo-Serganova functors. The latter are homological functors from the category of representations of a Lie superalgebra to the category of representations of a Lie superalgebra of the same type, but smaller rank.
A large portion of the literature in both mathematics and physics is concerned with vertex algebras V that are of CFT-type and rational. It is natural to ask whether there are other significant classes of vertex algebras that well-behaved from the representation theoretic point of view such as
Non CFT-type, and rational for some category of modules or
Non CFT-type and irrational but the module category has other nice properties such as C_2-condition.
For this talk, I will focus on a study of representation theory of N-graded vertex algebras. To be precise, I will describe an algebraic structure of rational N-graded vertex algebras, provide some tools to determine when N-graded vertex algebras are irrational and describe roles of Leibniz algebras on the study of N-graded vertex algebras.