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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 ?

Semiclassical
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Julien Roussillon
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    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 08:20

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A good starting point for learning how $\tau$ functions are defined explicitly is Okamoto's paper: Okamoto, K. (1981). On the $\tau$-function of the Painlevé equations. Physica D: Nonlinear Phenomena, 2(3), 525-535. Okamoto starts from a Hamiltonian viewpoint.

A $\tau$ function associated with a Painlevé equation has (movable) zeroes where the solution of the original equation has (movable) poles. This analytic relationship is explained, e.g., in Hietarinta, J., & Kruskal, M. D. (1992). Hirota forms for the six Painlevé equations from singularity analysis. In Painlevé Transcendents (pp. 175-185). Springer, Boston, MA.

If you are familiar with elliptic functions, it may be useful to know that the same role is played by Weierstrass' sigma function $\sigma(x)$ or Jacobi's theta functions. These associated functions have zeroes where the elliptic function has poles. See for example, Chapter 23 of DLMF, where Weierstrass' sigma function $\sigma(x)$ is defined with zeroes where $\wp(x)$ has poles. Similarly, theta functions associated with Jacobi's elliptic functions are entire functions.

I am not familiar with the term $\tau$-structure, but I am familiar with what is often called a $\tau$-lattice. This is a fascinating lattice of interrelationships between $\tau$ functions corresponding to different values of parameters in a Painlevé equation. These relationships are also related to isomonodromy problems. See for example Section 5 of Jimbo, M., & Miwa, T. (1981). Monodromy perserving deformation of linear ordinary differential equations with rational coefficients. II. Physica D: Nonlinear Phenomena, 2(3), 407-448.

It is impossible to answer your question "why is it important?" in finite terms. One answer is that $\tau$ functions are functions of several variables and satisfy differential equations, differential-difference equations and partial difference equations with very rich properties. They are special cases of the Hirota's DAGTE equation, which includes almost all the known integrable equations (including the Kadomtsev-Petviashvili equation) and their difference versions. See Hirota, R. (1981). Discrete analogue of a generalized Toda equation. Journal of the Physical Society of Japan, 50(11), 3785-3791.

I hope this helps.

S. Kowalevsky
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