# $\tau$ structure of the sixth Painlevé equation

I am studying the isomonodromic deformations theory, which leads in the case of a $\mathcal{C}_{0,4}$ Riemann surface to the sixth Painlevé equation.

I read that this equation had a $\tau$-structure, so we can determine a $\tau$ function associated to isomonodromic deformations.

Here is my silly question : what is a $\tau$-structure, why is it important ?

• I don't even know what a $\tau$ function means! there seems to be connections with symplectic geometry or hamiltonian formulation in physics, but it's quite mysterious to me. – Julien Roussillon Jun 29 '15 at 8:20