H_T Vertex Algebras and the Infinite Toda Lattice

The $W_{1+\infty}(gl_s)$--symmetries of the $S$--component KP hierarchy

Adler, Shiota and van Moerbeke obtained for the KP and Toda lattice hierarchies a formula which translates the action of the vertex operator on tau--functions to an action of a vertex operator of pseudo-differential operators on wave functions. This relates the additional symmetries of the KP and Toda lattice hierarchyto the $W_{1+\infty}$--, respectively $W_{1+\infty}\times W_{1+\infty}$--algebra symmeties. In this paper we generalize the results to the $s$--component KP hierarchy. The vertex operators generate the algebra $W_{1+\infty}(gl_s)$, the matrix version of $W_{1+\infty}$. Since the …

On Infinite Order Simple Current Extensions of Vertex Operator Algebras

We construct a direct sum completion $\mathcal{C}_{\oplus}$ of a given braided monoidal category $\mathcal{C}$ which allows for the rigorous treatment of infinite order simple current extensions of vertex operator algebras as seen in \cite{CKL}. As an example, we construct the vertex operator algebra $V_L$ associated to an even lattice $L$ as an infinite order simple current extension of the Heisenberg VOA and recover the structure of its module category through categorical considerations.

Vertex operators for quantum groups and application to integrable systems

Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{R} provides its integrals of motion. The construction can be applied to any infinite dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N…

Vertex operator approach to semi-infinite spin chain : recent progress

Vertex operator approach is a powerful method to study exactly solvable models. We review recent progress of vertex operator approach to semi-infinite spin chain. (1) The first progress is a generalization of boundary condition. We study $U_q(\widehat{sl}(2))$ spin chain with a triangular boundary, which gives a generalization of diagonal boundary [Baseilhac and Belliard 2013, Baseilhac and Kojima 2014]. We give a bosonization of the boundary vacuum state. As an application, we derive a summation formulae of boundary magnetization. (2) The second progress is a generalization of hidden symmetry…

关于分级酉顶点代数的半无穷上同调

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。

Axiomatic $G_{1}$-vertex algebras

Inspired by the Borcherds' work on ``$G$-vertex algebras,'' we formulate and study an axiomatic counterpart of Borcherds' notion of $G$-vertex algebra for the simplest nontrivial elementary vertex group, which we denote by $G_{1}$. Specifically, we formulate a notion of axiomatic $G_{1}$-vertex algebra, prove certain basic properties and give certain examples, where the notion of axiomatic $G_{1}$-vertex algebra is a nonlocal generalization of the notion of vertex algebra. We also show how to construct axiomatic $G_{1}$-vertex algebras from a set of compatible $G_{1}$-vertex operators. The res…

环面上的量子化六顶点模型

Quantized six-vertex model on a torus

本文研究环面上四参数量子化六顶点模型的可积性，通过四面体方程与反演关系证明层转移矩阵对易，并得到一族对易量子哈密顿量。

Partitions, Vertex Operator Constructions and Multi-component KP equations

For every partition of a positive integer $n$ in $k$ parts and every point of an infinite Grassmannian we obtain a solution of the $k$ component differential-difference KP hierarchy and a corresponding Baker function. A partition of $n$ also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra $gl_\infty$ and hence a $τ$ function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group $Gl_\infty$ and to express the Baker function in terms of $τ$-functions. The reduction to loop algebras is di…

Associating quantum vertex algebras to Lie algebra $\gl_{\infty}$

In this paper, we present a canonical association of quantum vertex algebras and their $ϕ$-coordinated modules to Lie algebra $\gl_{\infty}$ and its 1-dimensional central extension. To this end we construct and make use of another closely related infinite-dimensional Lie algebra.

Representations of Centrally-Extended Lie Algebras over Differential Operators and Vertex Algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vaccum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naru…

A Hilbert theorem for vertex algebras

Given a simple vertex algebra A and a reductive group G of automorphisms of A, the invariant subalgebra A^G is strongly finitely generated in most examples where its structure is known. This phenomenon is subtle, and is generally not true of the classical limit of A^G, which often requires infinitely many generators and infinitely many relations to describe. Using tools from classical invariant theory, together with recent results on the structure of the W_{1+\infty} algebra, we establish the strong finite generation of a large family of invariant subalgebras of βγ-systems, bc-systems, and bcβ…

Some cohomology operators in 2-D field theory

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 modes of vertex operators (fields). Although vertex operator superalgebras -with the inherent Virasoro action- are regarded as part of Conformal Field Theory (CFT), a VOSA may exhibit a square-zero operator (often, but not always, the semi-infinite cohomology differential) for which the Virasoro algebra acts trivially in the cohomology. Capable of shedding its C…

有限生成顶点代数的自同构群的可表示性

Representability of the automorphism group of finitely generated vertex algebras

研究自由代数的自同构群，证明有限生成顶点代数在其noetherian环上的自同构群为仿射群方案。