Soliton equations, vertex operators, and simple singularities

Vertex operators, semiclassical limit for soliton S-matrices and the number of bound states in Affine Toda Field Theories

Soliton time-delays and the semiclassical limit for soliton S-matrices are calculated for non-simply laced Affine Toda Field Theories. The phase shift is written as a sum over bilinears on the soliton conserved charges. The results apply to any two solitons of any Affine Toda Field Theory. As a by-product, a general expression for the number of bound states and the values of the coupling in which the S-matrix can be diagonal are obtained. In order to arrive at these results, a vertex operator is constructed, in the principal gradation, for non-simply laced affine Lie algebras, extending the pr…

KP 族的顶点算子与奇异代数曲线

Vertex Operators of the KP hierarchy and Singular Algebraic Curves

本文利用 Sato Grassmannian 证明 KP 族顶点算子作用产生的解对应奇异代数曲线上的无挠秩一 sheaf，并说明其可视为拟周期背景上的孤子。

Uniqueness Theorem of W-Constraints for Simple Singularities

In a recent paper [3], Bakalov and Milanov proved that the total descendant potential of a simple singularity satisfies the W-constraints, which come from the W-algebra of the lattice vertex algebra associated to the root lattice of this singularity and a twisted module of this vertex algebra. In the present paper, we prove that the solution of these W-constraints is unique up to a constant factor, as conjectured by Bakalov and Milanov in their paper.

Tau Functions of (n,1) curves and Soliton Solutions on Non-Zero Constant Backgrounds

We study the tau function of the KP-hierarchy associated with an (n,1) curve $y^n=x-α$. If $α=0$ the corresponding tau function is 1. On the other hand if $α\neq 0$ the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.

Hilbert schemes of points on the minimal resolution and soliton equations

The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C^2//G for cyclic G are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C^2//G are tau functions of 2-Toda hierarchies.

超出刚性与有限性条件下，由顶点算子代数构造带奇点的模函子

Modular Functors with Singularities from Vertex Operator Algebras Beyond Rigidity and Finiteness

本文构造超出刚性与有限性条件下由顶点算子代数得到的带奇点开-闭模函子，证明其满足切除性质等，还将其应用于三重态模型的关联函数研究。