Tau function (integrable systems)

Tau functions are an important ingredient in the modern 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 formula_1-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. 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.
Definition of formula_2-functions.
There are two notions of formula_3-functions, both introduced by the Sato school. The first is that of "isomonodromic formula_3-functions" 
. The second is "formula_3-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.
A formula_1-function of isospectral type is 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 (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.
Hirota bilinear residue relation for KP formula_2-functions.
A KP (Kadomtsev–Petviashvili) formula_2-function formula_9 
is a function of an infinite number of KP flow variables formula_10 that satisfies the
following bilinear formal residue equation
identically in the formula_11 variables, where formula_12 is the 
formula_13 coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series' in formula_14, and
Kadomtsev-Petviashvili equation.
If formula_16 is a KP formula_3-function satisfying 
the Hirota residue equation () and we identify the first three flow variables as
it follows that the function
satisfies the formula_20 dimensional nonlinear partial differential equation
known as the "Kadomtsev-Petviashvili" (KP) "equation", which plays a prominent role in plasma physics and in shallow water ocean waves.
Taking further logarithmic derivatives of formula_21 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 formula_22. These are collectively known as the "KP hierarchy".
Formal Baker-Akhiezer function and the KP hierarchy.
If we define the (formal) Baker-Akhiezer function formula_23
by Sato's formula
and expand it as a formal series in the powers of the variable formula_14
this satisfies an infinite sequence of compatible evolution equations
where formula_27 is a linear ordinary differential operator of degree formula_28 
in the variable formula_29, with coefficients that are functions of the flow variables 
formula_30, defined as follows
where formula_32 is the formal pseudo-differential operator
with formula_34,
where
is the "wave operator" and formula_36
denotes the projection to the part of formula_37 containing
purely non-negative powers of formula_38; i.e. to the differential operator part of
formula_39 .
The pseudodifferential operator formula_32 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 (Kadomtsev-Petviashvili) hierarchy", for the functions
formula_42, with respect to the set 
formula_10 of independent variables, each of which contains 
only a finite number of formula_44's, and derivatives only with respect to the three independent variables formula_45. The first nontrivial case of these
is the Kadomtsev-Petviashvili equation ().
Thus, every KP formula_2 function provides a solution, at least in the formal sense,
of this infinite system of nonlinear partial differential equations.
Fuchsian isomonodromic systems: Isomonodromic formula_1-functions.
Consider the overdetermined system of first order matrix partial differential equations
where formula_48 are a set of formula_49 formula_50 traceless matrices, 
formula_51 a set of formula_49 complex parameters and formula_14 a complex variable, and formula_54 is an invertible formula_55 matrix valued function of formula_14 and formula_51.
These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group
formula_58 of the Riemann sphere punctured at
the points formula_51 corresponding to the rational covariant derivative operator 
to be independent of the parameters formula_51; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations 
Defining the formula_49 functions
defined on a (separable) Hilbert space formula_66 with basis elements 
formula_67 and dual basis elements 
formula_68 for formula_69.
The free fermionic creation and annihilation operators 
formula_70 act as endomorphisms on
formula_71 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 formula_74,
corresponding to the scalar product
with the Fock space formula_71 as irreducible module.
Denote the vacuum state, in the zero fermionic charge sector formula_77, 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 formula_80, is annihilated by the adjoint operators, acting to the left
Normal ordering formula_82 of a product of 
linear operators (i.e., finite or infinite linear combinations of creation and annihilation operators) is defined so that its vacuum expectation value (VEV) vanishes
In particular, for a product formula_84 of a pair formula_85
of linear operators
The "fermionic charge" operator formula_87 is defined as
The subspace formula_89 is the eigenspace of formula_90
consisting of all eigenvectors with eigenvalue formula_49
The standard orthonormal basis formula_93 for the zero fermionic charge sector formula_77 is labelled by integer partitions
formula_95,
where formula_96
is a weakly decreasing sequence of formula_97 positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition 
formula_98.
An alternative notation for a partition formula_99 consists of the 
Frobenius indices
formula_100, where formula_101
denotes the "arm length"; i.e. the number formula_102 of boxes in the Young diagram to the right of the formula_28'th diagonal box, formula_104 denotes the "leg length", i.e. the number of boxes in the Young diagram below the formula_28'th diagonal box, for formula_106, where formula_107 is the "Frobenius rank", which is the number of diagonal elements.
The basis element formula_108 is then given by acting on the vacuum with a product
of formula_107 pairs of creation and annihilation operators, labelled by the Frobenius indices
The integers formula_111 indicate, relative to the Dirac sea,
the occupied non-negative sites on the integer lattice while 
formula_112 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 (emptyset) partition formula_113
gives the vacuum state, and the dual basis formula_114 is defined by
Then any KP formula_2-function can be expressed as a sum
where formula_118 are the KP flow variables,
formula_119 is the Schur function 
corresponding to the partition formula_99, viewed as a function of the normalized power sum variables
in terms of an auxiliary (finite or infinite) sequence of variables 
formula_122 and the constant coefficients 
formula_123 may be viewed as the Plucker coordinates of an 
element formula_124 
of the infinite dimensional Grassmannian consisting of the orbit, under the action of 
the general linear group formula_125, of the subspace 
formula_126 
of the Hilbert space formula_127.
This corresponds, under the "Bose-Fermi correspondence", to a decomposable element
of the Fock space formula_77 which, up to projectivization is the image
of the Grassmannian element formula_124 under the 
Plucker map
where formula_132 is a basis for the subspace 
formula_133 and formula_134 denotes projectivization of
an element of formula_71.
The Plucker coordinates formula_136 satisfy an infinite set of bilinear
relations, the Plucker relations, defining the Plücker embedding 
into the projecivization formula_137 of the fermionic Fock space,
which are equivalent to the Hirota bilinear residue relation ().
If formula_138 for a group element formula_139
with fermionic representation formula_140, then the formula_2-function
formula_142 can be expressed as the fermionic vacuum state expectation value (VEV):
where
is the abelian subgroup of formula_125 that generates the KP flows, and
are the "current" components.
Multisoliton solutions.
If we choose formula_147 complex constants
formula_148
with formula_149's all distinct, formula_150, and define the functions
we arrive at the Wronskian determinant formula
which gives the general formula_153-soliton solution.
Theta function solutions associated to algebraic curves.
Let formula_154 be a compact Riemann surface of genus formula_155 and fix a canonical homology basis formula_156
of formula_157 with intersection numbers 
Let formula_159 be a basis for the space formula_160 of holomorphic differentials satisfying the standard normalization conditions
where formula_162 is the "Riemann matrix" of periods. 
The matrix formula_162 belongs to the "Siegel upper half space" 
The Riemann formula_165 function on formula_166 corresponding to the period matrix formula_162 is defined to be
Choose a point formula_169, a local parameter formula_170 in a neighbourhood of formula_171 with formula_172 and 
a positive divisor of degree formula_155 
For any positive integer formula_175 let formula_176 be the unique meromorphic differential of the second kind characterized by the following conditions:
Denote by formula_186 the vector of formula_187-cycles of formula_176:
Denote the image of formula_190 under the Abel map 
formula_191 
with arbitrary base point formula_193.
Then the following is a KP formula_2-function:
Matrix model partition functions as KP formula_2-functions.
Let formula_197 be the Lebesgue measure on the formula_198 dimensional space formula_199 of formula_200 complex Hermitian matrices.
Let formula_201 be a conjugation invariant integrable density function
Define a deformation family of measures
for small formula_204 and let
For any integer partition formula_95
define the "content product" coefficient
where the product is over all pairs formula_208 of positive integers that
correspond to boxes of the Young diagram of the partition formula_209,
viewed as positions of matrix elements of the corresponding 
formula_210 matrix.
Then, for every pair of infinite sequences formula_211
and formula_212 of complex vaiables, viewed
as (normalized) power sums formula_213
of the infinite sequence of auxiliary variables formula_214 and
formula_215, defined by
the function
is a "double" KP formula_2-function, both in the formula_219
and the formula_220 variables, known as a formula_2 function of "hypergeometric type".
In particular, choosing
for some small parameter formula_223, denoting the corresponding content product coefficient as formula_224
and setting formula_225, the resulting 
formula_1-function can be equivalently expanded as
where formula_227 are the "simple Hurwitz numbers", which are 
formula_228 times the number of ways in which an element 
formula_229 of the symmetric group formula_230 in formula_231 elements, with cycle lengths 
equal to the parts of the partition formula_99, can be factorized as a product of formula_233 formula_234-cycles
and
is the power sum symmetric function. Equation () thus shows that
the (formal) KP hypergeometric formula_2-function corresponding to the content
product coefficients formula_224 is a generating
function, in the combinatorial sense, for simple Hurwitz numbers.