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 in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form.
The term tau function, or -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili (or KP) equation and related integrable hierarchies. It is a central ingredient in the theory of solitons. In this setting, given any -function satisfying a Hirota-type system of bilinear equations (see below), the corresponding solutions of the equations of the integrable hierarchy are explicitly expressible in terms of it and its logarithmic derivatives up to a finite order. Tau functions also appear as matrix model partition functions in the spectral theory of random matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers.
There are two notions of -functions, both introduced by the Sato school. The first is isospectral -functions of the Sato–Segal–Wilson type for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectral deformation equations of Lax type. The second is isomonodromic -functions.
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.
Tau functions: isospectral and isomonodromic
A -function of isospectral type is defined as a solution of the Hirota bilinear equations (see below), from which the linear operator undergoing isospectral evolution can be uniquely reconstructed. Geometrically, in the Sato and Segal-Wilson sense, it is the value of the determinant of a Fredholm integral operator, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) Grassmann manifold onto the origin, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group. It typically arises as a partition function, in the sense of statistical mechanics, many-body quantum mechanics or quantum field theory, as the underlying measure undergoes a linear exponential deformation.
Isomonodromic -functions for linear systems of Fuchsian type are defined below in . For the more general case of linear ordinary differential equations with rational coefficients, including irregular singularities, they are developed in reference.
Hirota bilinear residue relation for KP tau functions
A KP (Kadomtsev–Petviashvili) -function
is a function of an infinite collection of variables (called KP flow variables) that satisfies the bilinear formal residue equation
identically in the variables, where is the
coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in , and
As explained below in the section , every such -function determines a set of solutions to the equations of the KP hierarchy.