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Vertex algebras are certain noncommutative, nonassociative algebraic structures that arose out of physics in the 1980s. They were axiomatized by Borcherds in his proof of the Moonshine Conjecture, and in the last 35 years they have become important in a diverse range of subjects. A fruitful perspective is that many vertex algebras can be viewed as quantizations of coordinate rings of arc spaces. In this talk, I will give an introduction to vertex algebras, arc spaces, and their interconnections. This is based on joint work with Bailin Song.
In joint work with S. Capparelli, A. Meurman, and A. Primc (arXiv:2106.06262) we conjecture combinatorial Rogers-Ramanujan type identities for colored partitions, related to standard representations of symplectic affine Lie algebras. The conjecture is stated in purely combinatorial terms, and it is supported by numerical evidence. In my talk, I would state the conjecture and then explain the representation theory background.
Rigidity of tensor categories plays an important role, in quantum topology and in the representation theory of many algebraic objects, in particular of Hopf algebras and vertex algebras. In this talk, we discuss inherent restrictions of the notion of rigidity. We then explain why rigidity is so useful in the study of bulk fields of conformal field theories.
As a SUSY analogue of vertex algebras, Heluani and Kac introduced SUSY vertex algebras. On the other hand, in physics literature, SUSY counterpart of Toda theory has been studied. In particular, Madsen and Ragoucy described an N=1 SUSY analogue of the quantum Drinfeld-Sokolov reduction. In this talk, I will explain the SUSY Hamiltonian reduction process in terms of supersymmetric vertex algebras. This is based on the joint work with Molev and Ragoucy.
The orbifold theory studies a vertex operator algebra under the action of a finite group. The goal is to understand the representation theory for the fixed point vertex operator subalgebra. The main feature in orbifold theory is the appearance of the twisted modules. The permutation orbifolds study the action of the symmetric group of degree k on the k-tensor product of a vertex operator algebra. In [Dong-Li-Xu-Yu; 2019], we determined the fusion product of any untwisted module with any twisted module for permutation orbifolds. In this talk I will talk about fusion products of twisted modules for permutation orbifolds. This is a joint work with C. Dong and F. Xu.
Recently, a program for mathematically realizing a ubiquitous relationship in physics called holography in terms of Koszul duality has been proposed. In this talk I will explain how three exceptional super Lie algebras appear in a (twisted) version of this correspondence in the context of M-theory. One of these Lie algebras, which Kac calls E(3|6), plays a particular important role related to the AGT correspondence and we will argue how its representation theory sheds light on the holographic story and beyond.
Given any vertex operator algebra V, Zhu defined an associative algebra A(V), and showed that to any A(V)-module, one can associate an admissible V-module. This gives rise to a functor taking n-tuples of A(V)-modules to a sheaf of coinvariants (and its dual sheaf of conformal blocks) on the moduli space of stable n-pointed curves of genus g. If V is rational and C_2-cofinite (and so A(V) is finite and semi-simple), much is known about these sheaves, including that they are coherent (fibers are finite dimensional) and satisfy a factorization property. Factorization ultimately allows one to show these sheaves are vector bundles. In this talk I will describe a program in which we are aiming for analogous results after removing the assumption of rationality, and weakening C_2-cofiniteness. As a first step, we replace the standard factorization formula with an inductive one that holds for sheaves defined by modules over any VOA of CFT-type. As an application, we show that if V is of CFT-type and A(V) is finite, then sheaves of coinvariants and conformal blocks are coherent. This is a preliminary description of new and ongoing joint work with Chiara Damiolini and Daniel Krashen.
Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of $n$-ary chiral operations. This allows us to introduce the cohomology of a vertex algebra $V$ as a Lie algebra cohomology. When $V$ is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of $V$ to the variational Poisson cohomology studied previously by De Sole and Kac. This talk is based on joint work with Alberto De Sole, Reimundo Heluani, Victor Kac, and Veronica Vignoli.
In this talk, I will explain our study of the category of modules of the beta-gamma VOA from the point of view of 3d mirror symmetry. I will introduce a category of modules of the beta-gamma VOA, containing the category studied by Ridout-Wood and Allen-Wood. We propose that this category is the category of line operators for a twisted 3d N=4 theory. I will explain that using a relation of beta-gamma and affine Lie superalgebra of \mathfrak{gl}(1|1), we can show that this category has the structure of a braided tensor category. This relation is an example of 3d abelian mirror symmetry. If time permits, I will talk about a relation to matrix factorizations. This is based on joint work with Andrew Ballin.
We study the effect of linear transformations on quantum fields, with the main example of application to vertex operator presentations of Hall-Littlewood polynomials. The construction is illustrated with examples that include certain versions of multiparameter symmetric functions, dual Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature. Linear transformations of quantum fields effectively describe preservation of commutation relations of operators, stability of symmetric polynomials, polynomial tau functions of the KP and the BKP hierarchy.
A modular tensor category is pointed if every simple object is a simple current. We show that any pointed modular tensor category is equivalent to the module category of a lattice vertex operator algebra. Moreover, if the pointed modular tensor category C is the module category of a twisted Drinfeld double associated to a finite abelian group G and a 3-cocycle with coefficients in U(1), then there exists a self dual positive definite even lattice L such that G can be realized an automorphism group of lattice vertex operator algebra $V_L,$ $V_L^G$ is also a lattice vertex operator algebra and C is equivalent to the module category of $V_L^G.$ This is a joint work with S. Ng and L. Ren.
For any complex manifold M, the chiral de Rham complex is a sheaf of vertex algebras on M that was introduced in 1998 by Malikov, Schechtman, and Vaintrob. It is N-graded by conformal weight, and the weight zero piece coincides with the ordinary de Rham sheaf. When M is a Calabi-Yau manifold with holonomy group SU(d), it was shown by Ekstrand, Heluani, Kallen and Zabzine that the algebra of global sections \Omega^{ch}(M) contains a certain vertex algebra defined by Odake which is an extension of the N=2 superconformal algebra. When M is a hyperkahler manifold, it was shown by Ben-Zvi, Heluani, and Szczesny that \Omega^{ch}(M) contains the small N=4 superconformal algebra. In this talk, we will show that in both cases, these subalgebras are actually the full algebras of global sections. In an earlier work, Bailin Song has shown that the global section algebra can be identified with a certain subalgebra of a free field algebra which is invariant under the action of an infinite-dimensional Lie algebra of Cartan type. They key observation is that this algebra can be described using the arc space analogue of Weyl's first and second fundamental theorems of invariant theory for the special linear and symplectic groups. This is a joint work with Bailin Song.