The notion of the genus of a quadratic form is generalized to vertex operator algebras. We define it as the modular braided tensor category associated to a suitable vertex operator algebra together with the central charge. Statements similar as known for quadratic forms are formulated.
We further explain how extension problems for vertex operator algebras can bedescribed in terms of the associated modular braided tensor category.
We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d N=4 gauge theories. We conjecture various relations between these boundary VOA's and properties of the (topologically twisted) bulk theories. We discuss applications to the Symplectic Duality and Geometric Langlands programs.
A large class of supersymmetric quantum field theories, including all theories with $\mathcal{N} = 2$ supersymmetry in three dimensions and theories with $\mathcal{N} = 2$ supersymmetry in four dimensions, possess topological-holomorphic sectors. We formulate Poisson vertex algebras in such topological-holomorphic sectors and discuss some examples. For a four-dimensional $\mathcal{N} = 2$ superconformal field theory, the associated Poisson vertex algebra is the classical limit of a vertex algebra generated by a subset of local operators of the theory.
We prove an equivalence between the following notions: (i) unitary Möbius vertex algebras, and (ii) Wightman conformal field theories on the circle (with finite-dimensional conformal weight spaces) satisfying an additional condition that we call uniformly bounded order. Reading this equivalence in one direction, we obtain new analytic and operator-theoretic information about vertex operators. In the other direction we characterize OPEs of Wightman fields and show they satisfy the axioms of a vertex algebra. As an application we establish new results linking unitary vertex operator algebras wit…
The Schur-index of the $(A_1, X_n)$-Argyres-Douglas theory is conjecturally a character of a vertex operator algebra. Here such vertex algebras are found for the $A_{\text{odd}}$ and $D_{\text{even}}$-type Argyres-Douglas theories. The vertex operator algebra corresponding to $A_{2p-3}$-Argyres-Douglas theory is the logarithmic $\mathcal B_p$-algebra of [1], while the one corresponding to $D_{2p}$, denoted by $\mathcal W_p$, is realized as a non-regular Quantum Hamiltonian reduction of $L_{k}(\mathfrak{sl}_{p+1})$ at level $k=-(p^2-1)/p$. For all $n$ one observes that the quantum Hamiltonian r…
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 the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.
Families of vertex algebras associated to nilpotent elements of simply-laced Lie algebras are constructed. These algebras are close cousins of logarithmic W-algebras of Feigin and Tipunin and they are also obtained as modifications of semiclassical limits of vertex algebras appearing in the context of $S$-duality for four-dimensional gauge theories. In the case of type $A$ and principal nilpotent element the character agrees precisely with the Schur-Index formula for corresponding Argyres-Douglas theories with irregular singularities. For other nilpotent elements they are identified with Schur…
We study aspects of the vertex operator algebra (VOA) corresponding to Argyres-Douglas (AD) theories engineered using the 6d N=(2, 0) theory of type $J$ on a punctured sphere. We denote the AD theories as $(J^b[k],Y)$, where $J^b[k]$ and $Y$ represent an irregular and a regular singularity respectively. We restrict to the `minimal' case where $J^b[k]$ has no associated mass parameters, and the theory does not admit any exactly marginal deformations. The VOA corresponding to the AD theory is conjectured to be the W-algebra $\mathcal{W}^{k_{2d}}(J,Y)$, where $k_{2d}=-h+ \frac{b}{b+k}$ with $h$ b…
Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories
Research Institute for Mathematical Sciences, Kyoto University（京都大学数学科学研究所）；Okinawa Institute of Science and Technology（冲绳科学技术大学院大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器）；School of Mathematics, University of Edinburgh（爱丁堡大学数学学院）；DAMTP, University of Cambridge（剑桥大学应用数学与理论物理系）；Fachbereich Mathematik, Universität Hamburg（汉堡大学数学系）；Department of Mathematics and Statistics, Maynooth University（梅努斯大学数学与统计系）
We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories. We also clarify some misunderstandings on vertex operator algebras, modular functors and intertwining operator algebras. Then we discuss some basic open problems and conjectures in mathematical orbifold conformal field theory. Generalized twisted modules and their variants, their constructions and some existence results are reviewed. Twisted intertwining op…
Rational chiral conformal field theories are organized according to their genus, which consists of a modular tensor category $\mathcal{C}$ and a central charge $c$. A long-term goal is to classify unitary rational conformal field theories based on a classification of unitary modular tensor categories. We conjecture that for any unitary modular tensor category $\mathcal{C}$, there exists a unitary chiral conformal field theory $V$ so that its modular tensor category $\mathcal{C}_V$ is $\mathcal{C}$. In this paper, we initiate a mathematical program in and around this conjecture. We define a cla…
We explore new connections between the fields and local observables in two dimensional chiral conformal field theory. We show that in a broad class of examples, the von Neumann algebras of local observables (a conformal net) can be obtained from the fields (a unitary vertex operator algebra) via a continuous geometric interpolation procedure involving Graeme Segal's functorial definition of conformal field theory. In particular, we construct conformal nets from these unitary vertex operator algebras by showing that 'geometrically mollified' versions of the fields yield bounded, local operators…
Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOPPZ] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed.
The $\mathcal{B}_p$-algebras are a family of vertex operator algebras parameterized by $p\in \mathbb Z_{\geq 2}$. They are important examples of logarithmic CFTs and appear as chiral algebras of type $(A_1, A_{2p-3})$ Argyres-Douglas theories. The first member of this series, the $\mathcal{B}_2$-algebra, are the well-known symplectic bosons also often called the $βγ$ vertex operator algebra. We study categories related to the $\mathcal{B}_p$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $\mathfrak{sl}_2$. These categories are braided, rigid …
Some BPS quantities of $\mathcal{N}=1$ 5D quiver gauge theories, like instanton partition functions or qq-characters, can be constructed as algebraic objects of the Ding-Iohara-Miki (DIM) algebra. This construction is applied here to $\mathcal{N}=2$ super Yang-Mills theories in four dimensions using a degenerate version of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the first one being defined using vertex operators acting on a free boson's Fock space, while the second one is essentially equivalent to the action of Vasserot-Shiffmann's Spherical Heck…
Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(ν_1,ν_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathf…
In the context of 4D/2D dualities, SH$^c$ algebra, introduced by Schiffmann and Vasserot, provides a systematic method to analyse the instanton partition functions of $\mathcal{N}=2$ supersymmetric gauge theories. In this paper, we rewrite the SH$^c$ algebra in terms of three holomorphic fields $D_0(z)$, $D_{\pm1}(z)$ with which the algebra and its epresentations are simplified. The instanton partition functions for arbitrary $\mathcal{N}=2$ super Yang-Mills theories with $A_n$ and $A^{(1)}_n$ type quiver diagrams are compactly expressed as a product of four building blocks: Gaiotto state, dil…
