Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota[1] in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.
There are two notions of -functions, both introduced by the Sato school. The first is isospectral-functions of the Sato–Segal–Wilson type[2][11] for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectraldeformation equations of Lax type. The second is isomonodromic-functions.[12]
Depending on the specific application, a -function may either be: 1) an analytic function of a finite or infinite number of independent, commuting flow variables, or deformation parameters; 2) a discrete function of a finite or infinite number of denumerable variables; 3) a formal power series expansion in a finite or infinite number of expansion variables, which need have no convergence domain, but serves as generating function for certain enumerative invariants appearing as the coefficients of the series; 4) a finite or infinite (Fredholm) determinant whose entries are either specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite or infinite dimensional) with entries similarly of polynomial or quasi-polynomial type. Examples of all these types are given below.
In the Hamilton–Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a role similar to the -function, serving both as a generating function for the canonical transformation to linearizing canonical coordinates and, when evaluated on simultaneous level sets of a complete set of Poisson commuting invariants, as a complete solution of the Hamilton–Jacobi equation.
^Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 11-12. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN9781108610902. S2CID222379146.
^Pandharipande, R. (2000). "The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere". Letters in Mathematical Physics. 53 (1). Springer Science and Business Media LLC: 59–74. doi:10.1023/a:1026571018707. ISSN0377-9017. S2CID17477158.
^Harnad, J.; Balogh, F. (2021). Tau functions and Their Applications, Chapts. 13-14. Cambridge Monographs on Mathematical Physics. Cambridge, U.K.: Cambridge University Press. doi:10.1017/9781108610902. ISBN9781108610902. S2CID222379146.