Simple affine vertex algebras at admissible levels are semi-simple in the category O, but beyond the category O they contain interesting categories of representations with many new research challenges. We shall first discuss the existence and explicit realization of logarithmic modules which appear as extensions of weight modules. Next natural task is to include Whittaker modules in the representation category. Although Whittaker modules are constructed using standard Lie-theoretic constructions, we will show that in order to understand the structure of affine Whittaker modules, one needs to apply vertex-algebraic techniques. We also present our explicit lattice realizations of Whittaker modules for simple VOA $L_k(sl(2))$ at arbitrary admissible level $k$, and our recent efforts to generalize this realization in higher rang cases. We will try to predict possible fusion rules and intertwining operators which include Whittaker-type modules.
Recently, Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees have discovered a 4d/2d duality which associates a VOA to any 4d N=2 superconformal field theory.
There is a large class of 4d N=2 superconformal field theories called the theory of class S, and the corresponding VOAs are called chiral algebras of class S.
We give a functorial construction of the chiral algebras of class S, and show that the associated varieties of these vertex algebras are exactly the Moore-Tachikawa symplectic varieties that have been constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian for the Langlands dual group.
In 1994, Ryba proposed the "modular moonshine conjecture" following some numerical experimentation with modular representations of sporadic groups and the modular functions appearing in monstrous moonshine. Borcherds and Ryba reinterpreted the conjecture as a question about the action of centralizing elements on Tate cohomology of a self-dual integral form of the monster vertex operator algebra. The self-dual integral form was not known to exist at the time, but they got an unconditional result for odd primes by using a self-dual form over Z[1/2]. We describe a construction of a self-dual integral form, and speculate on connections to other moonshines.
This is a preliminary report on the application of Zhu's algebra theory for logarithmic vertex operator algebras (joint project with D. Adamovic). The theory of Zhu's algebras can be used for the construction of some logarithmic (projective) modules for triplet vertex algebras W(p). In order to construct and realize all projective and indecomposable modules, it is natural to consider higher Zhu's algebras. In this talk, we present some results in the case p=2. In particular, we study the symplectic fermion vertex superalgebra.
TBA
Motivated by celebrated results of Arakawa-Creutzig-Linshaw (ACL) for coset constructions of W-algebras of type ADE, which was conjectured by Feigin, I will talk about the slight generalizations: type B. Those works are understood as examples of "corner VOAs", general (conjectural) constructions of vertex algebra extensions, investigated by Creutzig-Gaiotto in the study of topologically twisted four-dimensional $\mathcal{N}=4$ gauge theory. I also explain that other coset constructions of W-algebras may be deduced from the decomposition of the CDO for type C and the conjectural commutativity between BRST reductions and tensor products of integrable representations by Arakawa-Creutzig. This is a joint work with Thomas Creutzig.
Unitary VOAs and conformal nets are two main mathematical axiomatizations of unitary CFTs. The VOA approach is purely algebraic, whereas the conformal net approach is mainly functional analytic. Despite the methodological differences, there are many similar and parallel results in the representation theories of these two mathematical objects. So one would like to have a unifying picture, in which these two approaches are conceptually related. A recent work by Carpi-Kawahigashi-Longo-Weiner (CKLW) shows how to relate unitary VOAs and conformal nets using smeared vertex operators. In this talk, I will explain how to extend the method of CKLW to show the equivalence of their representation tensor categories. The main idea is to construct the categorical extensions of VOAs and of conformal nets, and show that these two categorical extensions can be related using smeared intertwining operators.
The tensor category theory for vertex operator algebras has been showing its power in solving long-standing problems. I will give a quick introduction to the construction of various tensor category structures on suitable module categories for a vertex operator algebra. Since a number of mathematically wrong publications about this theory have been circulated and even published in top
journals but have never been (and probably will never be) publicly corrected, I will discuss the mistakes in these publications so that people new to this theory will not be confused and puzzled
anymore. I will also use the mistakes in these papers to explain the reason why many of the subtle conditions, sophisticated techniques and difficult and long proofs in this theory are necessary.
We study description of anyons in terms of matrix product operator algebras due to Bultinck et al. and relate it directly to operator algebraic theory of subfactors and unitary fusion categories.
I will survey a breadth of recent searches undertaken to discover q-series identities possibly related to affine Lie algebras. I will focus on three philosophically different directions: first of which yielded conjectures related to affine Lie algebra A_9^2 (which have been proved recently by Bringmann, Jennings-Shaffer and Mahlburg), second which is based on a certain Z-algebraic mechanism, and third based on modularity considerations (which has not uncovered any new identities). This talk is based on joint works with Matthew C. Russell and Debajyoti Nandi.
Conjecturally, critical statistical mechanics in two dimensions can be described by conformal field theories (CFT). The CFT description has in particular lead to exact and correct (albeit mostly non-rigorous) predictions of critical exponents and scaling limit correlation functions in many lattice models. The main ingredient of CFT is the Virasoro algebra, accounting for the effect of infinitesimal conformal transformations on local fields. In this talk we show that an exact Virasoro algebra action exists on the probabilistic local fields of two discrete models: the discrete Gaussian free field and the critical Ising model on the square lattice. The talk is based on joint work with Clément Hongler and Fredrik Viklund.
Given a vertex operator algebra, one obtains screening operators by taking the residue of suitable vertex operators. Frequently, in this way we obtain an action of an interesting Lie algebra. More generally, one can study screening operators associated to (non-local) intertwining operators, they appear for example in free-field realizations of logarithmic vertex algebra, such as the triplet algebra or the symplectic fermions.
In a previous work I have shown, that certain such screening operators on lattice vertex algebras give an action of the Borel part of the quantum group (or more general diagonal Nichols algebras). Here, the non-trivial braiding in contrast to a Lie algebra is due to the non-locality of the intertwining operator. After explaining this example, I report on work in progress with Y.-Z. Huang, which should show that this is a general picture: Under certain finiteness- and smallness-assumptions, the screening operators of any set of intertwining operators on any vertex algebra should give an action of a Nichols algebra, namely those associated to the braiding of the intertwining operators in the HLZ-construction.
Many families of generalized Rogers-Ramanujan identities have been interpreted by means of vertex operator algebra theory, and in fact, classical q-series theory has stimulated a lot of fundamental
research in the representation theory of vertex operator algebras. We expect that there will be a theory of twisted intertwining operators that will categorify a family of proofs termed ``motivated proofs'' of such identities. I will discuss ongoing work toward this goal. The spirit of this work is to develop a rich theory of twisted modules for intertwining algebras. Intertwining algebra theory, developed largely by Yi-Zhi Huang, has long been known to be fundamental. The goal of the current ideas is to enhance this theory from a new point of view.
I will sketch and motivate a selection of Yi-Zhi Huang's fundamental work.
Previously, we introduced a notion of vertex bialgebra and a notion of module vertex algebra for a vertex bialgebra, and gave a smash product construction of nonlocal vertex algebras. Here, we introduce a notion of right comodule vertex algebra for a vertex bialgebra. Then we give a construction of quantum vertex algebras from vertex algebras with a right comodule vertex algebra structure and a compatible (left) module vertex algebra structure for a vertex bialgebra. As an application, we obtain a family of deformations of the lattice vertex algebras. This is based on a joint work with Naihuan Jing, Fei Kong, and Shaobin Tan.
We study the effective \BV quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. The generating functions are proven to have modular property with mild holomorphic anomaly. As an application, we construct an exact solution of open-closed quantum B-model in complex one dimension. This can be viewed as an example of Koszul duality for vertex algebras.
The logarithmic tensor category theory of Huang, Lepowsky, and Zhang has opened the door to using tensor-categorical tools for studying non-semisimple representations of vertex operator algebras. In this talk, I will discuss new results on twisted modules associated to a vertex operator algebra V and a finite automorphism group G, under the assumption that the (possibly non-semisimple) representation category of the fixed-point subalgebra VG admits Huang-Lepowsky-Zhang braided tensor category structure. Specifically, I will show: (1) There is a braided tensor equivalence between finite-dimensional representations of G and a tensor subcategory of VG-modules generated by V. (2) Every VG-module admitting a suitably associative, unital action of V is a direct sum of g-twisted V-modules for automorphisms g in G. (3) The category of twisted V-modules is a braided G-crossed tensor category, and its “G-equivariantization” is a braided tensor category equivalent to the category of VG-modules. If time permits, I will discuss applications to symplectic fermion vertex operator (super)algebras and the orbifold rationality problem for arbitrary finite groups.
Borcherds-Kac-Moody algebras are natural generalisations of finite-dimensional simple Lie algebras. There are exactly 10 Borcherds-Kac-Moody algebras whose denominator identities are completely
reflective automorphic products of singular weight on lattices of squarefree level (classified by Scheithauer). These belong to a larger class of Borcherds-Kac-Moody (super)algebras obtained by Borcherds by twisting the denominator identity of the Fake Monster Lie algebra. For the 10 Lie algebras we prove a conjecture by Borcherds that they can be realised uniformly as the physical states of bosonic strings moving on suitable spacetimes. This amounts to applying the BRST formalism to certain vertex algebras of central charge 26 obtained as graded tensor products of abelian intertwining algebras.
We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde formula. We explicitly construct the intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{gl(1 \vert 1)}$. This is joint work with Drazen Adamovic.
For a number of lattice models in 2D statistical physics, it has been proven that the scaling limit of an interface at criticality (with suitable boundary conditions) is a Schramm-Loewner evolution (SLE). Similarly, collections of several interfaces converge to families of interacting random curves, multiple SLEs. Connection probabilities of these interfaces encode crossing probabilities in the lattice models, which should also be related to correlation functions of appropriate fields in the corresponding conformal field theory (CFT).
In this talk, I discuss results towards conformal invariance of critical lattice models in terms of interfaces and crossing probabilities. In general, there are no explicit formulas for the crossing probabilities, but their scaling limits can still be completely understood in terms of so-called pure partition functions of multiple SLEs. In particular, all of the expected CFT properties: conformal invariance, null-field equations, and fusion rules, are satisfied.
In the theory of associative algebras, the first order deformation is characterized by the second Hochschild cohomology group. For commutative associative algebras, the second Harrison cohomology is used. Yi-Zhi Huang defined a cohomology theory for vertex algebras in 2010 that is analogous to Harrison cohomology, and used the second cohomology to characterize the first order deformation of vertex algebras. The result is generalized to meromorphic open-string vertex algebras, where the commutativity does not hold.
Affine vertex algebras form an important class of examples from which an enormous range of new examples may be constructed. Unfortunately, their representation theories are not particularly well understood. I will review what is known (rigorously and conjecturally), including classifications, modularity and fusion, aiming to describe a recently developed approach based on Mathieu's theory of coherent families.
The categorical Verlinde formula for a - possibly non-semisimple - modular tensor category C relates the action of the modular group on a certain hom-space of C to the structure constants of the Grothendieck ring of C (the `fusion rules'). If C arises as the category of representations of a suitable vertex operator algebra, one can formulate a precise conjecture for the relation between the above categorical modular group action and the transformation properties of pseudo-trace functions. In the semi-simple case, this conjecture amounts to a theorem by Yi-Zhi Huang. In the non-semisimple case we provide support for the conjecture in the example of the symplectic fermion VOA. This is joint work with Azat Gainutdinov.
Supermanifolds are known to admit both differential forms and integral forms, thus any appropriate super analogue of the de Rham theory should take both types of forms into account. However, the cohomology of Lie superalgebras studied so far in the literature involves only differential forms when interpreted as a de Rham theory for Lie supergroups. Thus a new cohomology theory of Lie superalgebras is needed to fully incorporate differential-integral forms, and we investigate such a theory here. This new cohomology is defined by a BRST complex of Lie superalgebra modules, and includes the standard Lie superalgebra cohomology as a special case. General properties expected of a cohomology theory are established for the new cohomology, and examples of the new cohomology groups are computed. This is a joint work with R.B. Zhang.
K3 theories, by definition, are certain conformal field theories with N=(4,4) supersymmetry and central charges (6,6). Much is known about their 80-dimensional moduli space and about the theories parametrized by a finite number of special subvarieties of dimensions at most 16 within this moduli space. To access K3 theories away from these special subvarieties, as part of our "symmetry surfing programme" with Anne Taormina, we have proposed and established existence of an infinite dimensional space of primaries with respect to the N=(4,4) superconformal algebra which is common to all K3 theories. The representation of the N=(4,4) superconformal arising from this space can be identified with a superconformal vertex operator algebra by a procedure that we call "reflection".
In the talk, we will give a status report of this "generic space of states".
In light of Yi-Zhi Huang's celebrated proof of the Verlinde formula and subsequent influential work on its generalisations, I will present recent results on the βγ Ghost vertex algebra (also known as the Weyl vertex algebra). This vertex algebra fails almost every assumption in Yi-Zhi Huang's proof of the Verlinde formula, yet still gives rise to remarkably similar structures and the conclusion is, therefore, that there is nothing to fear.
Kapustin and Witten showed that a twisted version of N=4 gauge theory in four dimensions compactifies to a two-dimensional sigma-model whose target space is the Hitchin moduli space. In this talk, I consider the reduction of the gauge theory on a four dimensional orientable spacetime manifold which is not a global product of two surfaces but contains embedded non-orientable surfaces. The low energy theory is a sigma-model on a two dimensional worldsheet whose boundary components end on branes constructed from the Hitchin moduli space associated to a non-orientable surface. I will also compare the discrete topological fluxes in four and two dimensional theories and verify the mirror symmetry on branes as predicted by the S-duality in gauge theory. This provides another non-trivial test of S-duality using reduction along possibly non-orientable surfaces.
We provide a rigorous theory of noncommutative metrics and curvatures in frame of deformation quantization. In terms of them, we are able to propose the noncommutative Einstein
field equations. We show that the deformation of classical pp-waves are exact solutions of vacuum field equations. We also find that the quantization of spherically symmetric metrics is renormalizable and the the deformation of classical Schwarzschild solution is the quantum black hole solution which does not depend on time and can not be evaporated. The talk is based on the early joint works with Chaichian, Tureanu, D. Wang, R.B. Zhang as well as further consideration recently.
A moonshine type VOA is a CFT type VOA whose weight one subspace is a zero space. For a moonshine type VOA the weight two subspace has a commutative but in general not associative algebra structure, which is called the Greiss algebra of the VOA. In this talk, we discuss some VOAs whose Greiss algebras are Jordan algebras. We make an analysis of the VOAs constructed by Ashihara and Miyamoto, and we further construct their simple quotients. The work of Niibori and Sagaki on the simplicity of the VOAs can have simple explanations using these constructions. We also note that some of these simple VOAs were constructed long before in the work of Ching Hung Lam.
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