Tau function (integrable systems)

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, 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 the Pfaffian of a skew symmetric matrix 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, 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 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 -function
is a function of an infinite collection of 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.

Kadomtsev–Petviashvili equation

If is a KP -function satisfying
the Hirota residue equation and we identify the first three flow variables as
it follows that the function
satisfies the dimensional nonlinear partial differential equation
known as the Kadomtsev-Petviashvili (KP) equation. This equation plays a prominent role in plasma physics and in shallow water ocean waves.
Taking further logarithmic derivatives of gives an infinite sequence of functions that satisfy further systems of nonlinear autonomous PDE's, each involving partial derivatives of finite order with respect to a finite number of the KP flow parameters. These are collectively known as the KP hierarchy.

Formal Baker–Akhiezer function and the KP hierarchy

If we define the Baker-Akhiezer function
by Sato's formula
and expand it as a formal series in the powers of the variable
this satisfies an infinite sequence of compatible evolution equations
where is a linear ordinary differential operator of degree
in the variable, with coefficients that are functions of the flow variables
, defined as follows
where is the formal pseudo-differential operator
with,
is the wave operator and
denotes the projection to the part of containing
purely non-negative powers of ; i.e. the differential operator part of .
The pseudodifferential operator satisfies the infinite system of isospectral deformation equations
and the compatibility conditions for both the system and
are
This is a compatible infinite system of nonlinear partial differential equations, known as the KP hierarchy, for the functions, with respect to the set of independent variables, each of which contains only a finite number of 's, and derivatives only with respect to the three independent variables. The first nontrivial case of these
is the Kadomtsev-Petviashvili equation.
Thus, every KP -function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations.

Isomonodromic systems. Isomonodromic tau functions

Fuchsian isomonodromic systems. Schlesinger equations

Consider the overdetermined system of first order matrix partial differential equations
where are a set of traceless matrices,
a set of complex parameters, a complex variable, and is an invertible matrix valued function of and.
These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group
of the Riemann sphere punctured at
the points corresponding to the rational covariant derivative operator
to be independent of the parameters ; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations

Isomonodromic $\tau$ -function

Defining functions
the Schlesinger equations imply that the differential form
on the space of parameters is closed:
and hence, locally exact. Therefore, at least locally, there exists a function
of the parameters, defined within a multiplicative constant, such that
The function is called the isomonodromic -function
associated to the fundamental solution of the system,.

Hamiltonian structure of the Schlesinger equations

Defining the Lie Poisson brackets on the space of -tuples of matrices:
and viewing the functions defined in as Hamiltonian functions on this Poisson space, the Schlesinger equations
may be expressed in Hamiltonian form as
for any differentiable function.

Reduction of $r=2$ , $n=3$ case to $P_{VI}$ ">Painleve equations"> $P_{VI}$

The simplest nontrivial case of the Schlesinger equations is when and. By applying a Möbius transformation to the variable,
two of the finite poles may be chosen to be at and, and the third viewed as the independent variable.
Setting the sum of the matrices appearing in
, which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under conjugation, we obtain a system equivalent to the most generic case of the six Painlevé transcendent equations, for which many detailed classes of explicit solutions are known.

Non-Fuchsian isomonodromic systems

For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic -functions may be defined in a similar way, using differentials on the extended parameter space.
There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.
Taking all possible confluences of the poles appearing in for the and case, including the one at, and making the corresponding reductions, we obtain all other instances
of the Painlevé transcendents, for which
numerous special solutions are also known.

Fermionic VEV (vacuum expectation value) representations

The fermionic Fock space, is a semi-infinite exterior product space
defined on a Hilbert space with basis elements
and dual basis elements
for.
The free fermionic creation and annihilation operators
act as endomorphisms on
via exterior and interior multiplication by the basis elements
and satisfy the canonical anti-commutation relations
These generate the standard fermionic representation of the Clifford algebra
on the direct sum, corresponding to the scalar product
with the Fock space as irreducible module.
Denote the vacuum state, in the zero fermionic charge sector, as
which corresponds to the Dirac sea of states along the real integer lattice in which all negative integer locations are occupied and all non-negative ones are empty.
This is annihilated by the following operators
The dual fermionic Fock space vacuum state, denoted, is annihilated by the adjoint operators, acting to the left
Normal ordering of a product of
linear operators is defined so that its vacuum expectation value vanishes
In particular, for a product of a pair of linear operators, one has
The fermionic charge operator is defined as
The subspace is the eigenspace of
consisting of all eigenvectors with eigenvalue
The standard orthonormal basis for the zero fermionic charge sector is labelled by integer partitions
where
is a weakly decreasing sequence of positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition
An alternative notation for a partition consists of the
Frobenius indices
, where
denotes the arm length; i.e. the number of boxes in the Young diagram to the right of the 'th diagonal box, denotes the leg length, i.e. the number of boxes in the Young diagram below the 'th diagonal box, for, where is the Frobenius rank, which is the number of elements along the principal diagonal.
The basis element is then given by acting on the vacuum with a product
of pairs of creation and annihilation operators, labelled by the Frobenius indices
The integers indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while
indicate the unoccupied negative integer sites.
The corresponding diagram, consisting of infinitely many occupied and unoccupied sites on the integer lattice that are a finite perturbation of the Dirac sea are referred to as a Maya diagram.
The case of the null partition gives the vacuum state, and the dual basis is defined by
Any KP -function can be expressed as a sum
where are the KP flow variables,
is the Schur function
corresponding to the partition, viewed as a function of the normalized power sum variables
in terms of an auxiliary sequence of variables
and the constant coefficients
may be viewed as the Plücker coordinates of an
element
of the infinite dimensional Grassmannian consisting of the orbit, under the action of
the general linear group, of the subspace
of the Hilbert space.
This corresponds, under the Bose-Fermi correspondence, to a decomposable element
of the Fock space which, up to projectivization, is the image of the Grassmannian element under the
Plücker map
where is a basis for the subspace
and denotes projectivization of
an element of.
The Plücker coordinates satisfy an infinite set of bilinear
relations, the Plücker relations, defining the image of the Plücker embedding
into the projectivization of the fermionic Fock space,
which are equivalent to the Hirota bilinear residue relation.
If for a group element
with fermionic representation, then the -function can be expressed as the fermionic vacuum state expectation value :
where
is the abelian subgroup of that generates the KP flows, and
are the ""current"" components.

Examples of solutions to the equations of the KP hierarchy

Schur functions

As seen in equation, every KP -function can be represented as a linear combination of Schur functions, in which the coefficients satisfy the bilinear set of Plucker relations corresponding to an element of an infinite Grassmann manifold. In fact, the simplest class of tau functions consists of the Schur functions themselves, which correspond to the special element of the Grassmann manifold whose image under the Plücker map is.

Multisoliton solutions

If we choose complex constants
with 's all distinct,, and define the functions
we arrive at the Wronskian determinant formula
which gives the general -soliton -function.

Theta function solutions associated to algebraic curves

Let be a compact Riemann surface of genus and fix a canonical homology basis
of with intersection numbers
Let be a basis for the space of holomorphic differentials satisfying the standard normalization conditions
where is the Riemann matrix of periods.
The matrix belongs to the Siegel upper half space
The Riemann function on corresponding to the period matrix is defined to be
Choose a point, a local parameter in a neighbourhood of with and
a positive divisor of degree
For any positive integer let be the unique meromorphic differential of the second kind characterized by the following conditions:

The only singularity of is a pole of order at with vanishing residue.
The expansion of around is
:.
is normalized to have vanishing -cycles:
:

Denote by the vector of -cycles of :
Denote the image of under the Abel map

with arbitrary base point.
Then the following is a KP -function:

Matrix model partition functions as KP $\tau$ -functions

Let be the Lebesgue measure on the dimensional space of complex Hermitian matrices.
Let be a conjugation invariant integrable density function
Define a deformation family of measures
for small and let
be the partition function for this random matrix model.
Then satisfies the bilinear Hirota residue equation, and hence is a -function of the KP hierarchy.

$\tau$ -functions of hypergeometric type. Generating function for Hurwitz numbers

Let be a infinite sequence of complex numbers.
For any integer partition define the content product coefficient
where the product is over all pairs of positive integers that correspond to boxes of the Young diagram of the partition, viewed as positions of matrix elements of the corresponding
matrix.
Then, for every pair of infinite sequences and of complex variables, viewed as power sums
of the infinite sequence of auxiliary variables
defined by:
the function
is a double KP -function, both in the and the variables, known as a -function of hypergeometric type.
In particular, choosing
for some small parameter, denoting the corresponding content product coefficient as
and setting
the resulting -function can be equivalently expanded as
where are the simple Hurwitz numbers, which are
times the number of ways in which an element
of the symmetric group in elements, with cycle lengths equal to the parts of the partition, can be factorized as a product of -cycles
and
is the power sum symmetric function. Equation thus shows that the KP hypergeometric -function corresponding to the content product coefficients is a generating function, in the combinatorial sense, for simple Hurwitz numbers.