Integrable Hierarchies and Contact Terms in u-plane Integrals of Topologically Twisted Supersymmetric Gauge Theories

The $u$-plane integrals of topologically twisted $N = 2$ supersymmetric gauge theories generally contain contact terms of nonlocal topological observables. This paper proposes an interpretation of these contact terms from the point of view of integrable hierarchies and their Whitham deformations. This is inspired by Mari\~no and Moore's remark that the blowup formula of the $u$-plane integral contains a piece that can be interpreted as a single-time tau function of an integrable hierarchy. This single-time tau function can be extended to a multi-time version without spoiling the modular invariance of the blowup formula. The multi-time tau function is comprised of a Gaussian factor $e^{Q(t_1,t_2,...)}$ and a theta function. The time variables $t_n$ play the role of physical coupling constants of 2-observables $I_n(B)$ carried by the exceptional divisor $B$. The coefficients $q_{mn}$ of the Gaussian part are identified to be the contact terms of these 2-observables. This identification is further examined in the language of Whitham equations. All relevant quantities are written in the form of derivatives of the prepotential.Comment: latex, 17 pages, no figures; final version for publicatio

A qq-anaolg of the sixth Painlev\'e equation

A $q$-difference analog of the sixth Painlev\'e equation is presented. It arises as the condition for preserving the connection matrix of linear $q$-difference equations, in close analogy with the monodromy preserving deformation of linear differential equations. The continuous limit and special solutions in terms of $q$-hypergeometric functions are also discussed.Comment: 8 pages, LaTeX file (Two misprints corrected

Toda Lattice Hierarchy and Zamolodchikov's Conjecture

In this letter, we show that certain Fredholm determinant $D(\lambda;t)$, introduced by Zamolodchikov in his study of 2D polymers, is a continuum limit of soliton solution for the Toda lattice hierarchy with 2-periodic reduction condition.Comment: 6 pages, LaTeX file, no figure

Fusion of the qq-Vertex Operators and its Application to Solvable Vertex Models

We diagonalize the transfer matrix of the inhomogeneous vertex models of the 6-vertex type in the anti-ferroelectric regime intoducing new types of q-vertex operators. The special cases of those models were used to diagonalize the s-d exchange model\cite{W,A,FW1}. New vertex operators are constructed from the level one vertex operators by the fusion procedure and have the description by bosons. In order to clarify the particle structure we estabish new isomorphisms of crystals. The results are very simple and figure out representation theoretically the ground state degenerations.Comment: 35 page

Annihilation poles of a Smirnov-type integral formula for solutions to quantum Knizhnik--Zamolodchikov equation

We consider the recently obtained integral representation of quantum Knizhnik-Zamolodchikov equation of level 0. We obtain the condition for the integral kernel such that these solutions satisfy three axioms for form factor \'{a} la Smirnov. We discuss the relation between this integral representation and the form factor of XXZ spin chain.Comment: 14 pages, latex, no figures

Difference equations for the higher rank XXZ model with a boundary

The higher rank analogue of the XXZ model with a boundary is considered on the basis of the vertex operator approach. We derive difference equations of the quantum Knizhnik-Zamolodchikov type for 2N-point correlations of the model. We present infinite product formulae of two point functions with free boundary condition by solving those difference equations with N=1.Comment: LaTEX 16 page

Notes on highest weight modules of the elliptic algebra Aq,p(sl^2){\cal A}_{q,p}\left(\widehat{sl}_2\right)

We discuss a construction of highest weight modules for the recently defined elliptic algebra ${\cal A}_{q,p}(\widehat{sl}_2)$, and make several conjectures concerning them. The modules are generated by the action of the components of the operator $L$ on the highest weight vectors. We introduce the vertex operators $\Phi$ and $\Psi^*$ through their commutation relations with the $L$-operator. We present ordering rules for the $L$- and $\Phi$-operators and find an upper bound for the number of linearly independent vectors generated by them, which agrees with the known characters of $\widehat{sl}_2$-modules.Comment: Nonstandard macro package eliminate