Tau函数与单值辛同胚
Tau-functions and monodromy symplectomorphisms
摘要：
本文利用动力学$r$ -矩阵结构导出了Schlesinger方程的一种新的哈密顿形式。相应的辛形式示出的拉回，下的monodromy映射，一个自然的辛形式的扩展monodromy流形。我们证明了Fock-Goncharov坐标对于扩展的单值流形上的辛形式是对数正则的。使用这些坐标，我们定义的辛势的monodromy流形和解释的isomonodromic tau函数作为生成函数的monodromy地图。这特别解决了A.Its，O.Lisovyy和A. Prokhorov最近提出的一个猜想。
英文摘要：
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock-Goncharov coordinates are log-canonical for the symplectic form on the extended monodromy manifold. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the isomonodromic tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A.Its, O.Lisovyy and A.Prokhorov.
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