Lax matrices for Yang-Baxter maps

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff

Constraints and Period Relations in Bosonic Strings at Genus-g

We examine some of the implications of implementing the usual boundary conditions on the closed bosonic string in the hamiltonian framework. Using the KN formalism, it is shown that at the quantum level, the resulting constraints lead to relations among the periods of the basis 1-forms. These are compared with those of Riemanns' which arise from a different consideration.Comment: 16 pages, (Plain Tex), NUS/HEP/9320

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