Boris Feigin, Sergei Gukov
We take a peek at a general program that associates vertex (or, chiral) algebras to smooth 4-manifolds in such a way that operations on algebras mirror gluing operations on 4-manifolds and, furthermore, equivalent constructions of 4-manifolds give rise to equivalences (dualities) of the corresponding algebras.
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In this paper we study the first cohomologies for the following three examples of vertex operator algebras: (i) the simple affine VOA associated to a simple Lie algebra with positive integral level; (ii) the Virasoro VOA corresponding to minimal models; (iii) the lattice VOA associated to a positive definite even lattice. We prove that in all these cases, the first cohomology $H^1(V, W)$ are given by the zero-mode derivations when $W$ is any $V$-module with an $\N$-grading (not necessarily by the operator $L(0)$). This agrees with the conjecture made by Yi-Zhi Huang and the author in 2018. For…
Non-Chiral Vertex Operator Algebra Associated To Lorentzian Lattices And Narain CFTs
该研究定义非手征顶点算子代数，构造偶整洛伦兹格点对应的非手征VOA及其不可约模，得到相关模不变非手征CFT的模空间，证明Narain CFT是其例子并给出手征代数与配分函数分解。
On bosonic vertex algebras associated with 3D reductions of Argyres-Douglas theories
本文研究由四维Argyres-Douglas理论紧致化得到的三维阿贝尔箭图规范理论所对应的玻色子顶点算子代数，猜想其完备强生成元集合，并发现其中包含c=-2的W3顶点代数作为子代数。
Higgs Branch and VOA of 4d $\mathcal{N}=2$ SCFTs from IIB
本文研究了IIB弦理论中4d $\mathcal{N}=2$ SCFTs的希格斯分支及VOA，通过小解析推导希格斯分支，提出首个E型Kleinian奇点的SCFT例子，并推测其对应VOA为仿射E型W代数，同时发现新的VOA及BPS拟环预测。
On some simple orbifold affine VOAs at non-admissible level arising from rank one 4D SCFTs
本文研究源自秩一4D SCFT的非容许水平仿射顶点代数，分类$L_{-2}(G_2)$与$L_{-2}(B_3)$的不可约最高权模，并证明前者的伴生簇是$L_{-2}(D_4)$伴生簇在三次对称群下的轨形。
We define a quasiclassical limit of the Lian-Zuckerman homotopy BV algebra (quasiclassical LZ algebra) on the subcomplex, corresponding to "light modes", i.e. the elements of zero conformal weight, of the semi-infinite (BRST) cohomology complex of the Virasoro algebra associated with vertex operator algebra (VOA) with a formal parameter. We also construct a certain deformation of the BRST differential parametrized by a constant two-component tensor, such that it leads to the deformation of the $A_{\infty}$-subalgebra of the quasiclassical LZ algebra. Altogether this gives a functor the categor…
Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal N=4$ non-Abelian quiver gauge theories
本文研究三维 $\mathcal{N}=4$ 线性箭图规范理论 H-扭转得到的顶点算子代数，揭示其与希格斯分支及仿射 $\mathcal{W}$-代数的关系，并提出箭图约化算法以系统构造新自由场实现。
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…
We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs are equipped with two $\mathfrak{g}$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation an…
We analyze the N=2 superconformal field theories that arise when a pair of D3-branes probe an F-theory singularity from the perspective of the associated vertex operator algebra. We identify these vertex operator algebras for all cases; we find that they have a completely uniform description, parameterized by the dual Coxeter number of the corresponding global symmetry group. We further present free field realizations for these algebras in the style of recent work by three of the authors. These realizations transparently reflect the algebraic structure of the Higgs branches of these theories. …
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 placed on $S^3\times S^1$: by sending the radius of $S^1$ to zero, we recover the 3d $\mathcal{N}=4$ theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on th…
In his landmark paper, Zhu associated two associative algebras to a vertex operator algebra: what are now called Zhu's algebra and the C_2-algebra. The former has a nice interpretation in terms of the representation theory of the VOA, while the latter only serves as a finiteness condition. In this paper we undertake first steps to unravel the interpretation of the C_2 vector space. In particular we suggest that it seems and controls the twisted representations of the VOA.
Non-Simply Laced Class-S Vertex Operator Algebras
本文分析了非单链 class-S VOA 及其 Drinfeld-Sokolov 约化，发现部分约化 VOA 对应真实 4d $\mathcal{N}=2$ SCFT，并利用扭曲构造为 $F_4$ 与 $G_2$ 瞬子 VOA 非四维起源提供证据。
On intermediate Lie algebra $E_{7+1/2}$
本文研究例外序列中的中间李代数 $E_{7+1/2}$，提出相关新顶点算子代数，给出有理水平猜想、Weyl 维数公式猜想，构造小水平VOA并推广已有结果，预测更高水平的共形权。
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.
Rank-one 4d $\mathcal N=3$ SCFTs: Schur index, VOA modules, and modularity
研究秩一4d $\mathcal{N}=3$ SCFT的VOA表示论，通过MLDE和自由场实现获得Schur指标与特征标的闭式解，并推广到离散商理论。
On $E_{7+1/2}$ gauge theory
本文提出基于$E_{7+1/2}$李代数的规范理论，并通过6维SCFT和模形式bootstrap验证其与VOA的对应关系。
Classification of Rational $c=1$ Vertex Operator Algebras and Vertex Operator Superalgebras
该文严格分类了有理c=1顶点算子代数与顶点算子超代数，证实其为秩1整格格VO(S)A或其有限群自同构的orbifold，为后续完整CFT分类奠定基础。
We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type $\mathfrak{a}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ outer automorphism that has been observed in recent…
Towards a classification of graded unitary ${\mathcal W}_3$ algebras
本文研究四维单位性对从四维通过SCFT/VOA对应得到的${\mathcal W}_3$顶点代数的约束，证明除了$(3,q+4)$极小模型外，其他中央荷值不兼容四维单位性。
Vertex algebras, topological defects, and Moonshine
本文研究全纯顶点算子代数中的拓扑缺陷线，给出魔群顶点算子代数中对偶缺陷的 McKay-Thompson 级数公式，并建立 Conway 模中缺陷与 Leech 格自同态之间的满射环同态，进而探讨包含拓扑缺陷的月光猜想推广。
On the extensions of the left modules for a meromorphic open-string vertex algebra, I
本文研究亚纯开弦顶点代数左模的扩张，证明其与第一上同调类一一对应，并给出Ext^1维数的上界及一个Virasoro VOA的例子。
On the semi-infinite cohomology of graded-unitary vertex algebras
本文研究了具有手性量子动量图的分级酉顶点代数的相对半无穷上同调，证明其具有类似紧致Kähler流形微分形式的结构且本身分级酉，为源自超对称量子场论的顶点算子代数确立了分级酉性，并建立了类似Lefschetz作用的USp(2)作用及形式性结果。
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