Tau function (integrable systems) explained
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.
The term tau function, or
-function, was first used systematically by
Mikio Sato[2] and his students
[3] [4] 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,
[5] [6] [7] 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.
[8] [9] [10] There are two notions of
-functions, both introduced by the
Sato school. The first is
isospectral
-functions of the
Sato–Segal–Wilson type[11] 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.
[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.
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.[12] 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 equationidentically 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
u(x,y,t):=2 | \partial2 |
\partialx2 |
log\left(\tau(x,y,t,t4,...)\right)
satisfies the
(spatial)
(time) dimensional nonlinear partial differential equationknown 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 (formal) Baker-Akhiezer function
by Sato's formula
and expand it as a formal series in the powers of the variable
\psi(z,t)=
(1+
wj(t)z-j),
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 operatorl{L}=\partial+
uj(t)\partial-j=l{W}\circ\partial\circ{l{W}}-1
with
,
is the wave operator and
denotes the projection to the part of
containingpurely non-negative powers of
; i.e. the differential operator part of
.
The pseudodifferential operator
satisfies the infinite system of
isospectral deformation equationsand 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
, 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 theseis 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 equationswhere
are a set of
traceless matrices,
a set of
complex parameters,
a complex variable, and
\Psi(z,\alpha1,...,\alpham)
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 atthe points
corresponding to the rational covariant derivative operator
{\partial\over\partialz}-
{Ni\overz-\alphai}
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[12] Isomonodromic
-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
\tau(\alpha1,...,\alphan)
of the parameters, defined within a multiplicative constant, such that
The function
\tau(\alpha1,...,\alphan)
is called the
isomonodromic
-functionassociated 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:
\{(Ni)ab,(Nj)c,d\}=\deltaij\left((Ni)ad\deltabc-(Ni)cb\deltaad\right)
1\lei,j\len, 1\lea,b,c,d\ler,
and viewing the
functions
defined in as Hamiltonian functions on this Poisson space, the Schlesinger equations may be expressed in Hamiltonian form as
[13] [14] | \partialf(N1,...,Nn) |
\partial\alphai |
=\{f,Hi\}, 1\lei\len
for any differentiable function
.
Reduction of
,
case to
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.
[15] 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.[12] 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.[13] [14] 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 whichnumerous
special solutions are also known.
Fermionic VEV (vacuum expectation value) representations
, is a semi-infinite exterior product space
[16] l{F}=Λinfty/2l{H}= ⊕ n\inl{F}n
defined on a (separable) 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
\psii:=ei\wedge,
:=
, i\inZ,
and satisfy the canonical anti-commutation relations
[\psii,\psik]+=
+=0,
+=\deltaij.
These generate the standard fermionic representation of the Clifford algebra on the direct sum
, corresponding to the scalar product
Q(u+\mu,w+\nu):=\nu(u)+\mu(v), u,v\inl{H}, \mu,\nu\inl{H}*
with the Fock space
as irreducible module.Denote the vacuum state, in the zero fermionic charge sector
, as
|0\rangle:=e-1\wedgee-2\wedge …
,
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
\psi-j|0\rangle=0,
|0\rangle=0, j=0,1,...
The dual fermionic Fock space vacuum state, denoted
, is annihilated by the adjoint operators, acting to the left
\langle0|
=0, \langle0|\psij-1|0=0, j=0,1,...
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
\langle0|:L1, … Lm:|0\rangle=0.
In particular, for a product
of a pair
of linear operators, one has
{:L1L2:}=L1L2-\langle0|L1L2|0\rangle.
The fermionic charge operator
is defined as
The subspace
is the eigenspace of
consisting of all eigenvectors with eigenvalue
C|v;n\rangle=n|v;n\rangle, \forall|v;n\rangle\inl{F}n
.
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
(\alpha1,...\alphar|\beta1,...\betar)
, 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 productof
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.
[2] The case of the null (emptyset) partition
|\emptyset\rangle=|0\rangle
gives the vacuum state, and the dual basis
is defined by
\langle\mu|λ\rangle=\deltaλ,
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 (finite or infinite) 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
l{H}+=span\{e-i\}i\subsetl{H}
of the Hilbert space
.
This corresponds, under the Bose-Fermi correspondence, to a decomposable element
|\tauw\rangle=\sumλ\piλ(w)|λ\rangle
of the Fock space
which, up to projectivization, is the image of the Grassmannian element
under the
Plücker mapl{Pl}:span(w1,w2,...)\longrightarrow[w1\wedgew2\wedge … ]=[|\tauw\rangle],
where
is a basis for the subspace
and
denotes projectivization ofan element of
.
The Plücker coordinates
satisfy an infinite set of bilinearrelations, 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 (VEV):
\tauw(t)=\langle0|\hat{\gamma}+(t)\hat{g}|0\rangle,
where
\Gamma+=\{\hat{\gamma}+(t)=
\}\subsetGl(l{H})
is the abelian subgroup of
that generates the KP flows, and
Ji:=\sumj\in\psij
, i=1,2...
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 (at least formally) 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 (or finite) Grassmann manifold. In fact, the simplest class of (polynomial) 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
\{\alphak,\betak,\gammak\}k=1,
with
's all distinct,
, and define the functions
yk({\bft}):=
+\gammak
k=1,...,N,
we arrive at the Wronskian determinant formula
| (N) |
\tau | |
| \vec\alpha,\vec\beta,\vec\gamma |
({\bft}):=
\begin{vmatrix}
y1({\bft})&y2({\bft})& … &yN({\bft})\\
y1'({\bft})&y2'({\bft})& … &yN'({\bft})\\
\vdots&\vdots&\ddots&\vdots\
({\bft})&
({\bft})& … &
({\bft})\\
\end{vmatrix},
which gives the general
-soliton
-function.
[3] [4] [17] 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
ai\circaj=bi\circbj=0, ai\circbj=\deltaij, 1\leqi,j\leqg.
Let
be a basis for the space
of
holomorphic differentials satisfying the standard normalization conditions
\omegaj=\deltaij,
\omegaj=Bij,
where
is the
Riemann matrix of periods. The matrix
belongs to the
Siegel upper half space Sg=\left\{B\inMatg x (C) \colon BT=B, Im(B)ispositivedefinite\right\}.
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:
is a pole of order
at
with vanishing residue.
around
is
\Omegak=d(\zeta-k)+
Qij\zetajd\zeta
.
is normalized to have vanishing
-cycles:
Denote by
the vector of
-cycles of
:
Denote the image of
under the
Abel map
E:=l{A}(l{D})\inCg, Ej=l{A}j(l{D}):=
\omegaj
with arbitrary base point
.
Then the following is a KP
-function:
[18] \tau(X,,pinfty,\zeta)}(t):=e-{1\over\sumijQijtitj}
\theta\left(E
tkUk|B\right)
.
Matrix model partition functions as KP
-functions
Let
be the Lebesgue measure on the
dimensional space
of
complex Hermitian matrices.Let
be a conjugation invariant integrable density function
\rho(UMU\dagger)=\rho(M), U\inU(N).
Define a deformation family of measures
d\muN,\rho(t):=
\rho(M)d\mu0(M)
for small
and let
\tauN,\rho({\bft}):=
}d\muN,\rho({\bft}).
be the
partition function for this
random matrix model.
[19] [5] Then
satisfies the bilinear Hirota residue equation, and hence is a
-function of the KP hierarchy.
[20]
-functions of hypergeometric type. Generating function for Hurwitz numbers
Let
be a (doubly) 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 vaiables, viewed as (normalized) power sums
of the infinite sequence of auxiliary variables
and
,
defined by:
,
the functionis a double KP
-function, both in the
and the
variables, known as a
-function of
hypergeometric type.
[21] In particular, choosing
for some small parameter
, denoting the corresponding content product coefficient as
and setting
,
the resulting
-function can be equivalently expanded aswhere
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
pλ(t)=
(t), with pi(t):=
=iti
is the power sum symmetric function. Equation thus shows that the (formal) KP hypergeometric
-function corresponding to the content product coefficients
is a generating function, in the combinatorial sense, for simple Hurwitz numbers.
[8] [9] [10] References
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