Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations

An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows $(\lambda_t=\lambda ^l, l\ge0)$ from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.Comment: 24 pages, LaTex, revise

Isomonodromic deformations and supersymmetric gauge theories

Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories possess rich but involved integrable structures. The goal of this paper is to show that an isomonodromy problem provides a unified framework for understanding those various features of integrability. The Seiberg-Witten solution itself can be interpreted as a WKB limit of this isomonodromy problem. The origin of underlying Whitham dynamics (adiabatic deformation of an isospectral problem), too, can be similarly explained by a more refined asymptotic method (multiscale analysis). The case of $N=2$ SU($s$) supersymmetric Yang-Mills theory without matter is considered in detail for illustration. The isomonodromy problem in this case is closely related to the third Painlev\'e equation and its multicomponent analogues. An implicit relation to t\tbar fusion of topological sigma models is thereby expected.Comment: Several typos are corrected, and a few sentenses are altered. 19 pp + a list of corrections (page 20), LaTe

Gauging of Geometric Actions and Integrable Hierarchies of KP Type

This work consist of two interrelated parts. First, we derive massive gauge-invariant generalizations of geometric actions on coadjoint orbits of arbitrary (infinite-dimensional) groups $G$ with central extensions, with gauge group $H$ being certain (infinite-dimensional) subgroup of $G$. We show that there exist generalized ``zero-curvature'' representation of the pertinent equations of motion on the coadjoint orbit. Second, in the special case of $G$ being Kac-Moody group the equations of motion of the underlying gauged WZNW geometric action are identified as additional-symmetry flows of generalized Drinfeld-Sokolov integrable hierarchies based on the loop algebra {\hat \cG}. For {\hat \cG} = {\hat {SL}}(M+R) the latter hiearchies are equivalent to a class of constrained (reduced) KP hierarchies called {\sl cKP}_{R,M}, which contain as special cases a series of well-known integrable systems (mKdV, AKNS, Fordy-Kulish, Yajima-Oikawa etc.). We describe in some detail the loop algebras of additional (non-isospectral) symmetries of {\sl cKP}_{R,M} hierarchies. Apart from gauged WZNW models, certain higher-dimensional nonlinear systems such as Davey-Stewartson and $N$-wave resonant systems are also identified as additional symmetry flows of {\sl cKP}_{R,M} hierarchies. Along the way we exhibit explicitly the interrelation between the Sato pseudo-differential operator formulation and the algebraic (generalized) Drinfeld-Sokolov formulation of {\sl cKP}_{R,M} hierarchies. Also we present the explicit derivation of the general Darboux-B\"acklund solutions of {\sl cKP}_{R,M} preserving their additional (non-isospectral) symmetries, which for R=1 contain among themselves solutions to the gauged $SL(M+1)/U(1)\times SL(M)$ WZNW field equations.Comment: LaTeX209, 47 page

Hierarchy of QM SUSYs on a Bounded Domain

We systematically formulate a hierarchy of isospectral Hamiltonians in one-dimensional supersymmetric quantum mechanics on an interval and on a circle, in which two successive Hamiltonians form N=2 supersymmetry. We find that boundary conditions compatible with supersymmetry are severely restricted. In the case of an interval, a hierarchy of, at most, three isospectral Hamiltonians is possible with unique boundary conditions, while in the case of a circle an infinite tower of isospectral Hamiltonians can be constructed with two-parameter family of boundary conditions.Comment: 15 pages, 3 figure