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Vertex operator algebras (VOAs) arise in protected subsectors of supersymmetric quantum field theories, notably in 4d ${\mathcal N}=2$ superconformal field theories (SCFT) via the Schur sector and in twisted 3d ${\mathcal N}=4$ theories via…
The SCFT/VOA correspondence provides a powerful framework for studying 4d $\mathcal N=2$ superconformal field theories (SCFTs) through the mathematical machinery of 2d vertex operator algebras (VOAs). It captures the Schur operators of the…
We identify vertex operator algebras (VOAs) of a class of Argyres-Douglas (AD) matters with two types of non-abelian flavor symmetries. They are the $W$ algebra defined using nilpotent orbit with partition $[q^m,1^s]$. Gauging above AD…
We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d $\mathcal{N}=2$ theories and an associative algebra in the Higgs sector of 3d $\mathcal{N}=4$. The natural setting is a 4d $\mathcal{N}=2$ SCFT…
We investigate an alternative approach to the correspondence of four-dimensional $\mathcal{N}=2$ superconformal theories and two-dimensional vertex operator algebras, in the framework of the $\Omega$-deformation of supersymmetric gauge…
There is a well-known map from 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) to 2d vertex operator algebras (VOAs). The 4d Schur index corresponds to the VOA vacuum character, and must be a solution with integral coefficients of…
We study the Higgs branch and associated vertex operator algebra (VOA) of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) from the geometric engineering of IIB superstring on canonical threefold singularities. For terminal…
Superconformal field theories (SCFT) are known to possess solvable yet nontrivial sectors in their full operator algebras. Two prime examples are the chiral algebra sector on a two dimensional plane in four dimensional $\mathcal{N}=2$…
We propose a mirror symmetry for 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) compactified on a circle with finite size. The mirror symmetry involves vertex operator algebra (VOA) describing the Schur sector (containing Higgs…
We initiate the study of boundary Vertex Operator Algebras (VOAs) of topologically twisted 3d $\mathcal{N}=4$ rank-0 SCFTs. This is a recently introduced class of $\mathcal{N}=4$ SCFTs that by definition have zero-dimensional Higgs and…
We explore 3d $ \mathcal{N}=4 $ theories arising from twisted compactification of 4d $ \mathcal{N}=2 $ $ (G, G') $ Argyres-Douglas superconformal field theories (SCFTs), together with the 2d vertex operator algebras (VOAs) supported on the…
The infinite series of 4d $\mathcal{N} = 2$ SCFTs with central charge relation $a_\text{4d} = c_\text{4d}$ are closely related to the $\mathcal{N}=4$ super Yang-Mills. In this paper we study the modular properties of their associated VOAs…
We start with the SCFT/VOA correspondence formulated in the Omega-background approach, and connect it to the boundary VOA in 3d $\mathcal{N}=4$ theories and chiral algebras of 2d $\mathcal{N}=(0,2)$ theories. This is done using the…
The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In…
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…
A well-developed classification program for 4d $\mathcal{N}=2$ super conformal field theories (SCFTs) leverages Seiberg-Witten geometry on the Coulomb branch of vacua; theories are arranged by increasing $\mathfrak{rank}$, the complex…
We define and study a class of $\mathcal{N}=2$ vertex operator algebras $\mathcal{W}_{\mathcal{\mathsf{G}}}$ labelled by complex reflection groups. They are extensions of the $\mathcal{N}=2$ super Virasoro algebra obtained by introducing…
It is typical for a semi-infinite cohomology complex associated with a graded Lie algebra to occur as a vertex operator (or chiral) superalgebra where all the standard operators of cohomology theory, in particular the differential, are…
We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, \Psi, \sigma]$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of…
The Schur index is a powerful tool to probe the spectrum and dualities of 4d $\mathcal{N}=2$ superconformal field theories (SCFTs), deeply related to 2d vertex operator algebras (VOAs). In this paper, we compute the Schur index in closed…