The twistor theory of Whitham hierarchy

We have generalized the approach of M. Dunajski, L. Mason and P. Tod [Einstein–Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001) 63–93] and established a 1–1 correspondence between a solution of the Universal Whitham hierarchy [I.M. Krichever, The ττ-function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994) 437–475] and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Universal Whitham hierarchy is given by deforming the curve in the surface. By treating the family of algebraic curves in CP1×CP1CP1×CP1 as a twistor space, we were able to express the deformations of the isomonodromic spectral curve in terms of the deformations generated by the Universal Whitham hierarchy.