Tau function (integrable systems)

A $\backslash tau$-function of isospectral type is defined as a solution of the Hirota bilinear equations (see {{slink||Hirota bilinear residue relation for KP tau functions}} 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 $\backslash tau$-functions for linear systems of Fuchsian type are defined below in {{slink||Fuchsian isomonodromic systems. Schlesinger equations}}. For the more general case of linear ordinary differential equations with rational coefficients, including irregular singularities, they are developed in reference.

{{anchor|Hirota bilinear residue relation}}

A KP (Kadomtsev–Petviashvili) $\backslash tau$-function $\backslash tau(\backslash mathbf\{t\})$

is a function of an infinite collection $\backslash mathbf\{t\}=(t\_1,\; t\_2,\; \backslash dots)$ of variables (called KP flow variables) that satisfies the bilinear formal residue equation

{{NumBlk|::| $\backslash mathrm\{res\}\_\{z=0\}\backslash left(e^\{\backslash sum\_\{i=1\}^\backslash infty\; (\backslash delta\; t\_i)z^i\}\; \backslash tau(\{\backslash bf\; t\}\; -\; [z^\{-1\}])\backslash tau(\{\backslash bf\; s\}\; +\; [z^\{-1\}])\backslash right)dz\; \backslash equiv\; 0,$|{{EquationRef|1}}}}

identically in the $\backslash delta\; t\_j$ variables, where $\backslash mathrm\{res\}\_\{z=0\}$ is the

$z^\{-1\}$ coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in $z$, and

:$$\{\backslash bf\; s\}\; :=\; \{\backslash bf\; t\}\; +\; (\backslash delta\; t\_1,\; \backslash delta\; t\_2,\; \backslash cdots\; ),\; \backslash quad\; [z^\{-1\}]\; :=\; (z^\{-1\},\; \backslash tfrac\{z^\{-2\}\}\{2\},\; \backslash cdots\; \backslash tfrac\{z^\{-j\}\}\{j\},\; \backslash cdots).$$

As explained below in the section {{slink||Formal Baker-Akhiezer function and the KP hierarchy}}, every such $\backslash tau$-function determines a set of solutions to the equations of the KP hierarchy.

If $\backslash tau(t\_1,\; t\_2,\; t\_3,\; \backslash dots\backslash dots)$ is a KP $\backslash tau$-function satisfying

the Hirota residue equation ({{EquationNote|1}}) and we identify the first three flow variables as

::$t\_1\; =x,\; \backslash quad\; t\_2=y,\backslash quad\; t\_3\; =t,$

it follows that the function

::$u(x,y,t):=2\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\backslash log\backslash left(\backslash tau(x,y,t,\; t\_4,\backslash dots)\backslash right)$

satisfies the $2$ (spatial)$+1$ (time) dimensional nonlinear partial differential equation

{{NumBlk|::| $3u\_\{yy\}=\backslash left(4u\_t-6uu\_x-u\_\{xxx\}\backslash right)\_x,$ |{{EquationRef|2}}}}

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 $\backslash tau(t\_1,\; t\_2,\; t\_3,\; \backslash dots\backslash dots)$ 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 $\{\backslash bf\; t\}\; =(t\_1,\; t\_2,\; \backslash dots\; )$. These are collectively known as the KP hierarchy.

If we define the (formal) Baker-Akhiezer function $\backslash psi(z,\; \backslash mathbf\{t\})$

by Sato's formula

:: $\backslash psi(z,\; \backslash mathbf\{t\})\; :=$

e^{\sum_{i=1}^\infty t_i z^i}

\frac{\tau(\mathbf{t} - [z^{-1}])}{\tau(\mathbf{t})}

and expand it as a formal series in the powers of the variable $z$

::$$

\psi(z, \mathbf{t}) = e^{\sum_{i=1}^\infty t_i z^i}

( 1 + \sum_{j=1}^\infty w_j(\mathbf{t}) z^{-j}),

this satisfies an infinite sequence of compatible evolution equations

{{NumBlk|::| $$

\frac{\partial \psi }{\partial t_i} = \mathcal{D}_i \psi, \quad i,j, = 1,2, \dots,

|{{EquationRef|3}}}}

where $\backslash mathcal\{D\}\_i$ is a linear ordinary differential operator of degree $i$

in the variable $x:=\; t\_1$, with coefficients that are functions of the flow variables

$\backslash mathbf\{t\}=(t\_1,\; t\_2,\; \backslash dots)$, defined as follows

::$$

\mathcal{D}_i := \big(\mathcal{L}^i\big)_+

where $\backslash mathcal\{L\}$ is the formal pseudo-differential operator

:: $$

\mathcal{L} = \partial + \sum_{j=1}^\infty u_j(\mathbf{t}) \partial^{-j}

= \mathcal{W} \circ\partial \circ{\mathcal{W}}^{-1}

with $\backslash partial\; :=\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}$,

:: $$

\mathcal{W} := 1 +\sum_{j=1}^\infty w_j(\mathbf{t}) \partial^{-j}

is the wave operator and $\backslash big(\backslash mathcal\{L\}^i\backslash big)\_+$

denotes the projection to the part of $\backslash mathcal\{L\}^i$ containing

purely non-negative powers of $\backslash partial$; i.e. the differential operator part of $\{\backslash mathcal\{L\}\}^i$ .

The pseudodifferential operator $\backslash mathcal\{L\}$ satisfies the infinite system of isospectral deformation equations

{{NumBlk|::| $$

\frac{\partial\mathcal{L} }{\partial t_i} = [\mathcal{D}_i, \mathcal{L} ], \quad i, = 1,2, \dots

|{{EquationRef|4}}}}

and the compatibility conditions for both the system ({{EquationNote|3}}) and

({{EquationNote|4}}) are

{{NumBlk|::|$$

\frac{\partial\mathcal{D}_i}{\partial t_j} - \frac{\partial\mathcal{D}_j}{\partial t_i} + [\mathcal{D}_i, \mathcal{D}_j]=0, \quad i,j, = 1,2, \dots

|{{EquationRef|5}}}}

This is a compatible infinite system of nonlinear partial differential equations, known as the KP (Kadomtsev-Petviashvili) hierarchy, for the functions $\backslash \{u\_j(\backslash mathbf\{t\})\backslash \}\_\{j\backslash in\; \backslash mathbf\{N\}\}$, with respect to the set $\backslash mathbf\{t\}=(t\_1,\; t\_2,\; \backslash dots)$ of independent variables, each of which contains only a finite number of $u\_j$'s, and derivatives only with respect to the three independent variables $(x,\; t\_i,\; t\_j)$. The first nontrivial case of these

is the Kadomtsev-Petviashvili equation ({{EquationNote|2}}).

Thus, every KP $\backslash tau$-function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations.

Consider the overdetermined system of first order matrix partial differential equations

{{NumBlk|:| $\{\backslash partial\; \backslash Psi\; \backslash over\; \backslash partial\; z\}-\; \backslash sum\_\{i=1\}^n\; \{N\_i\; \backslash over\; z\; -\; \backslash alpha\_i\}\; \backslash Psi=0,\; \backslash quad$|{{EquationRef|6}}}}

{{NumBlk|:| $\{\backslash partial\; \backslash Psi\; \backslash over\; \backslash partial\; \backslash alpha\_i\}+\; \{N\_i\; \backslash over\; z\; -\; \backslash alpha\_i\}\; \backslash Psi=0,$ |{{EquationRef|7}}}}

where $\backslash \{N\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$ are a set of $n$ $r\backslash times\; r$ traceless matrices,

$\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$ a set of $n$ complex parameters, $z$ a complex variable, and $\backslash Psi(z,\; \backslash alpha\_1,\; \backslash dots,\; \backslash alpha\_m)$ is an invertible $r\; \backslash times\; r$ matrix valued function of $z$ and $\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$.

These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group

$\backslash pi\_1(\{\backslash bf\; P\}^1\backslash backslash\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\})$ of the Riemann sphere punctured at

the points $\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$ corresponding to the rational covariant derivative operator

:$\{\backslash partial\; \backslash over\; \backslash partial\; z\}-\; \backslash sum\_\{i=1\}^n\; \{N\_i\; \backslash over\; z\; -\; \backslash alpha\_i\}$

to be independent of the parameters $\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations

{{NumBlk|:|$\{\backslash partial\; N\_i\; \backslash over\; \backslash partial\; \backslash alpha\_j\}\; =\; \{[N\_i,\; N\_j]\; \backslash over\; \backslash alpha\_i-\backslash alpha\_j\}\; \backslash quad\; \backslash text\{for\; \}\; i\; \backslash neq\; j,$

\quad {\partial N_i \over \partial \alpha_i} = - \sum_{1\le j \le n, j\neq i}{[N_i, N_j] \over \alpha_i-\alpha_j}. |{{EquationRef|8}}}}

Defining $n$ functions

{{NumBlk|:|$H\_i\; :=\; \backslash frac\{1\}\{2\}\; \backslash sum\_\{1\backslash le\; j\; \backslash le\; n,\; j\backslash neq\; i\}\{\{\backslash rm\; Tr\}(N\_i\; N\_j)\; \backslash over\; \backslash alpha\_i-\backslash alpha\_j\},\; \backslash quad\; i=1,\; \backslash dots\; ,n,$|{{EquationRef|9}}}}

the Schlesinger equations ({{EquationNote|8}}) imply that the differential form

:$\backslash omega\; :=\; \backslash sum\_\{i=1\}^n\; H\_i\; d\backslash alpha\_i$

on the space of parameters is closed:

:$d\backslash omega\; =\; 0$

and hence, locally exact. Therefore, at least locally, there exists a function

$\backslash tau(\backslash alpha\_1,\; \backslash dots,\; \backslash alpha\_n)$

of the parameters, defined within a multiplicative constant, such that

:$\backslash omega\; =\; d\backslash mathrm\{ln\}\backslash tau$

The function $\backslash tau(\backslash alpha\_1,\; \backslash dots,\; \backslash alpha\_n)$ is called the isomonodromic $\backslash tau$-function

associated to the fundamental solution $\backslash Psi$ of the system ({{EquationNote|6}}), ({{EquationNote|7}}).

Defining the Lie Poisson brackets on the space of $n$-tuples $\backslash \{N\_i\backslash \}\_\{i=1,\; \backslash dots,\; n\}$ of $r\; \backslash times\; r$ matrices:

:$\backslash \{(N\_i)\_\{ab\},\; (N\_j)\_\{c,d\}\backslash \}\; =$

\delta_{ij}\left((N_i)_{ad}\delta_{bc} - (N_i)_{cb}\delta_{ad}\right)

:$1\; \backslash le\; i,j\; \backslash le\; n,\; \backslash quad\; 1\backslash le\; a,b,c,d\; \backslash le\; r,$

and viewing the $n$ functions $\backslash \{H\_i\backslash \}\_\{i=1,\; \backslash dots,n\}$ defined in ({{EquationNote|9}}) as Hamiltonian functions on this Poisson space, the Schlesinger equations ({{EquationNote|8}})

may be expressed in Hamiltonian form as {{cite journal | last1=Harnad | first1=J. | title=Dual Isomonodromic Deformations and Moment Maps into Loop Algebras | journal=Communications in Mathematical Physics | publisher=Springer | volume=166| issue=11 | year=1994 | doi=10.1007/BF02112319 | pages=337–365 | arxiv=hep-th/9301076 | bibcode=1994CMaPh.166..337H | s2cid=14665305 }}

{{cite journal | last1=Bertola | first1=M. | last2=Harnad| first2=J. | last3=Hurtubise | first3=J. | title=Hamiltonian structure of rational isomonodromic deformation systems | journal=Journal of Mathematical Physics | publisher=American Institute of Physics | volume=64| year=2023 | issue=8 | doi=10.1063/5.0142532 | pages=083502 | arxiv=2212.06880 | bibcode=2023JMP....64h3502B }}

:$\backslash frac\{\backslash partial\; f(N\_1,\; \backslash dots,\; N\_n)\}\{\backslash partial\; \backslash alpha\_i\}\; =\; \backslash \{f,\; H\_i\backslash \},\; \backslash quad\; 1\backslash le\; i\; \backslash le\; n$

for any differentiable function $f(N\_1,\; \backslash dots,\; N\_n)$.

The simplest nontrivial case of the Schlesinger equations is when $r=2$ and $n=3$. By applying a Möbius transformation to the variable $z$,

two of the finite poles may be chosen to be at $0$ and $1$, and the third viewed as the independent variable.

Setting the sum $\backslash sum\_\{i=1\}^3\; N\_i$ of the matrices appearing in

({{EquationNote|6}}), which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under $Gl(2)$ conjugation, we obtain a system equivalent to the most generic case $P\_\{VI\}$ of the six Painlevé transcendent equations, for which many detailed classes of explicit solutions are known.{{Citation |last1=Fokas | first1=Athanassios S. | last2=Its | first2=Alexander R. | last3=Kapaev | first3=Andrei A. | last4=Novokshenov | first4=Victor Yu. | title=Painlevé transcendents: The Riemann–Hilbert approach | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3651-4 | mr=2264522 | year=2006 | volume=128}}{{Citation | last1=Conte | first1=R.| last2=Musette | first2=M. | title=The Painlevé handbook, second edition | publisher=Springer Nature | location=Switzerland | series=Mathematical physics studies | isbn=978-3-030-53339-7 | year=2020 }}{{cite journal | last1=Lisovyy | first1=Oleg | last2=Tykhyy | first2=Yuriy | title=Algebraic solutions of the sixth Painlevé equation | journal = Journal of Geometry and Physics | volume=85 | pages=124–163 | year=2014 | doi=10.1016/j.geomphys.2014.05.010 | bibcode=2014JGP....85..124L | s2cid=50552982 | doi-access=free | arxiv=0809.4873 }}

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 $\backslash tau$-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 $z$ and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.

Taking all possible confluences of the poles appearing in ({{EquationNote|6}}) for the $r=2$ and $n=3$ case, including the one at $z=\backslash infty$, and making the corresponding reductions, we obtain all other instances $P\_\{I\}\; \backslash cdots\; P\_V$

of the Painlevé transcendents, for which

numerous special solutions are also known.

The fermionic Fock space $\backslash mathcal\{F\}$, is a semi-infinite exterior product space

{{cite journal | last1=Kac | first1=V. | last2=Peterson | first2=D.H. | title=Spin and wedge representations of infinite-dimensional Lie Algebras and groups | journal=Proc. Natl. Acad. Sci. U.S.A. | volume=58 | issue=6 | year=1981 | doi=10.1073/pnas.78.6.3308 | pages=3308–3312 | pmid=16593029 | pmc=319557 | bibcode=1981PNAS...78.3308K | doi-access=free }}

:$\backslash mathcal\{F\}\; =\; \backslash Lambda^\{\backslash infty/2\}\backslash mathcal\{H\}\; =\; \backslash oplus\_\{n\backslash in\; \backslash mathbf\{Z\}\}\backslash mathcal\{F\}\_n$

defined on a (separable) Hilbert space $\backslash mathcal\{H\}$ with basis elements

$\backslash \{e\_i\backslash \}\_\{i\backslash in\; \backslash mathbf\{Z\}\}$ and dual basis elements

$\backslash \{e^i\backslash \}\_\{i\backslash in\; \backslash mathbf\{Z\}\}$ for $\backslash mathcal\{H\}^*$.

The free fermionic creation and annihilation operators

$\backslash \{\backslash psi\_j,\; \backslash psi^\{\backslash dagger\}\_j\backslash \}\_\{j\; \backslash in\; \backslash mathbf\{Z\}\}$ act as endomorphisms on

$\backslash mathcal\{F\}$ via exterior and interior multiplication by the basis elements

: $\backslash psi\_i\; :=\; e\_i\; \backslash wedge,\; \backslash quad\; \backslash psi^\backslash dagger\_i\; :=\; i\_\{e^i\},\; \backslash quad\; i\; \backslash in\; \backslash mathbf\{Z\},$

and satisfy the canonical anti-commutation relations

:$[\backslash psi\_i,\backslash psi\_k]\_+\; =\; [\backslash psi^\backslash dagger\_i,\backslash psi^\backslash dagger\_k]\_+=\; 0,\; \backslash quad\; [\backslash psi\_i,\backslash psi^\backslash dagger\_k]\_+=\; \backslash delta\_\{ij\}.$

These generate the standard fermionic representation of the Clifford algebra

on the direct sum $\backslash mathcal\{H\}\; +\backslash mathcal\{H\}^*$, corresponding to the scalar product

:$Q(u\; +\; \backslash mu,\; w\; +\; \backslash nu)\; :=\; \backslash nu(u)\; +\; \backslash mu(v),\; \backslash quad\; u,v\; \backslash in\; \backslash mathcal\{H\},\backslash \; \backslash mu,\; \backslash nu\; \backslash in\; \backslash mathcal\{H\}^*$

with the Fock space $\backslash mathcal\{F\}$ as irreducible module.

Denote the vacuum state, in the zero fermionic charge sector $\backslash mathcal\{F\}\_0$, as

: $|0\backslash rangle\; :=\; e\_\{-1\}\backslash wedge\; e\_\{-2\}\; \backslash wedge\; \backslash cdots$,

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

:$\backslash psi\_\{-j\}|0\; \backslash rangle\; =\; 0,\; \backslash quad\; \backslash psi^\{\backslash dagger\}\_\{j-1\}|0\; \backslash rangle\; =\; 0,\; \backslash quad\; j=0,\; 1,\; \backslash dots$

The dual fermionic Fock space vacuum state, denoted $\backslash langle\; 0\; |$, is annihilated by the adjoint operators, acting to the left

:$\backslash langle\; 0|\; \backslash psi^\backslash dagger\_\{-j\}\; =\; 0,\; \backslash quad\; \backslash langle\; 0\; |\; \backslash psi\_\{j-1\}|0\; =\; 0,\; \backslash quad\; j=0,\; 1,\; \backslash dots$

Normal ordering $:\; L\_1,\; \backslash cdots\; L\_m:$ 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

:$\backslash langle\; 0\; |:\; L\_1,\; \backslash cdots\; L\_m:|0\; \backslash rangle\; =0.$

In particular, for a product $L\_1\; L\_2$ of a pair $(L\_1,\; L\_2)$ of linear operators, one has

:$\{:L\_1\; L\_2:\}\; =\; L\_1\; L\_2\; -\; \backslash langle\; 0\; |\; L\_1\; L\_2|0\; \backslash rangle.$

The fermionic charge operator $C$ is defined as

:$C\; =\; \backslash sum\_\{i\backslash in\; \backslash mathbf\{Z\}\}\; :\backslash psi\_i\; \backslash psi^\backslash dagger\_i:$

The subspace $\backslash mathcal\{F\}\_n\; \backslash subset\; \backslash mathcal\{F\}$ is the eigenspace of $C$

consisting of all eigenvectors with eigenvalue $n$

: $C\; |\; v;\; n\backslash rangle\; =\; n\; |\; v;\; n\backslash rangle,\; \backslash quad\; \backslash forall\; |\; v;\; n\backslash rangle\; \backslash in\; \backslash mathcal\{F\}\_n$.

The standard orthonormal basis $\backslash \{|\backslash lambda\backslash rangle\backslash \}$ for the zero fermionic charge sector $\backslash mathcal\{F\}\_0$ is labelled by integer partitions

$\backslash lambda\; =\; (\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_\{\backslash ell(\backslash lambda)\})$,

where $\backslash lambda\_1\backslash ge\; \backslash cdots\; \backslash ge\; \backslash lambda\_\{\backslash ell(\backslash lambda)\}$

is a weakly decreasing sequence of $\backslash ell(\backslash lambda)$ positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition

$(5,\; 4,\; 1)$.

:Image:Young diagram for 541 partition.svg

An alternative notation for a partition $\backslash lambda$ consists of the

Frobenius indices

$(\backslash alpha\_1,\; \backslash dots\; \backslash alpha\_r\; |\; \backslash beta\_1,\; \backslash dots\; \backslash beta\; \_r)$, where $\backslash alpha\_i$

denotes the arm length; i.e. the number $\backslash lambda\_i\; -i$ of boxes in the Young diagram to the right of the $i$'th diagonal box, $\backslash beta\_i$ denotes the leg length, i.e. the number of boxes in the Young diagram below the $i$'th diagonal box, for $i=1,\; \backslash dots,\; r$, where $r$ is the Frobenius rank, which is the number of elements along the principal diagonal.

The basis element $|\backslash lambda\backslash rangle$ is then given by acting on the vacuum with a product

of $r$ pairs of creation and annihilation operators, labelled by the Frobenius indices

: $|\backslash lambda\backslash rangle\; =\; (-1)^\{\backslash sum\_\{j=1\}^r\; \backslash beta\_j\}$

\prod_{k=1}^r \big(\psi_{\alpha_k} \psi^\dagger_{-\beta_k-1}\big)| 0 \rangle.

The integers $\backslash \{\backslash alpha\_i\backslash \}\_\{i=1,\; \backslash dots,\; r\}$ indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while

$\backslash \{-\backslash beta\_i-1\backslash \}\_\{i=1,\; \backslash dots,\; r\}$ 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 $|\backslash emptyset\backslash rangle\; =\; |0\; \backslash rangle$ gives the vacuum state, and the dual basis $\backslash \{\backslash langle\; \backslash mu|\backslash \}$ is defined by

: $\backslash langle\; \backslash mu|\backslash lambda\backslash rangle\; =\; \backslash delta\_\{\backslash lambda,\; \backslash mu\}$

Any KP $\backslash tau$-function can be expressed as a sum

{{NumBlk|:| $\backslash tau\_w(\backslash mathbf\{t\})\; =\; \backslash sum\_\{\backslash lambda\}\; \backslash pi\_\backslash lambda(w)\; s\_\backslash lambda(\backslash mathbf\{t\}),$ | {{EquationRef|10}}}}

where $\backslash mathbf\{t\}\; =\; (t\_1,\; t\_2,\; \backslash dots,\; \backslash dots)$ are the KP flow variables,

$s\_\backslash lambda(\backslash mathbf\{t\})$ is the Schur function

corresponding to the partition $\backslash lambda$, viewed as a function of the normalized power sum variables

:$t\_i\; :=\; [\backslash mathbf\{x\}]\_i\; :=\; \backslash frac\{1\}\{i\}\; \backslash sum\_\{a=1\}^n\; x\_a^i\; \backslash quad\; i\; =\; 1,2,\; \backslash dots$

in terms of an auxiliary (finite or infinite) sequence of variables

$\backslash mathbf\{x\}:=(x\_1,\; \backslash dots,\; x\_N)$ and the constant coefficients

$\backslash pi\_\backslash lambda(w)$ may be viewed as the Plücker coordinates of an

element $w\backslash in\; \backslash mathrm\{Gr\}\_\{\backslash mathcal\{H\}\_+\}(\backslash mathcal\{H\})$

of the infinite dimensional Grassmannian consisting of the orbit, under the action of

the general linear group $\backslash mathrm\{Gl\}(\backslash mathcal\{H\})$, of the subspace

$\backslash mathcal\{H\}\_+\; =\; \backslash mathrm\{span\}\backslash \{e\_\{-i\}\backslash \}\_\{i\; \backslash in\; \backslash mathbf\{N\}\}\; \backslash subset\; \backslash mathcal\{H\}$

of the Hilbert space $\backslash mathcal\{H\}$.

This corresponds, under the Bose-Fermi correspondence, to a decomposable element

:$|\backslash tau\_w\backslash rangle\; =\; \backslash sum\_\{\backslash lambda\}\; \backslash pi\_\{\backslash lambda\}(w)\; |\backslash lambda\; \backslash rangle$

of the Fock space $\backslash mathcal\{F\}\_0$ which, up to projectivization, is the image of the Grassmannian element $w\backslash in\; \backslash mathrm\{Gr\}\_\{\backslash mathcal\{H\}\_+\}(\backslash mathcal\{H\})$ under the

Plücker map

:$\backslash mathcal\{Pl\}:\; \backslash mathrm\{span\}(w\_1,\; w\_2,\; \backslash dots\; )$

\longrightarrow [w_1 \wedge w_2 \wedge \cdots ]= [|\tau_w\rangle],

where $(w\_1,\; w\_2,\; \backslash dots\; )$ is a basis for the subspace

$w\backslash subset\; \backslash mathcal\{H\}$ and $[\; \backslash cdots]$ denotes projectivization of

an element of $\backslash mathcal\{F\}$.

The Plücker coordinates $\backslash \{\backslash pi\_\backslash lambda(w)\backslash \}$ satisfy an infinite set of bilinear

relations, the Plücker relations, defining the image of the Plücker embedding

into the projectivization $\backslash mathbf\{P\}(\backslash mathcal\{F\})$ of the fermionic Fock space,

which are equivalent to the Hirota bilinear residue relation ({{EquationNote|1}}).

If $w\; =\; g(\backslash mathcal\{H\}\_+)$ for a group element $g\; \backslash in\; \backslash mathrm\{Gl\}(\backslash mathcal\{H\})$

with fermionic representation $\backslash hat\{g\}$, then the $\backslash tau$-function $\backslash tau\_w(\backslash mathbf\{t\})$ can be expressed as the fermionic vacuum state expectation value (VEV):

:$\backslash tau\_w(\backslash mathbf\{t\})\; =\; \backslash langle\; 0\; |\; \backslash hat\{\backslash gamma\}\_+(\backslash mathbf\{t\})\; \backslash hat\{g\}\; |\; 0\; \backslash rangle,$

where

:$\backslash Gamma\_+\; =\backslash \{\backslash hat\{\backslash gamma\}\_+(\backslash mathbf\{t\})\; =\; e^\{\backslash sum\_\{i=1\}^\backslash infty\; t\_i\; J\_i\}\backslash \}$

\subset \mathrm{Gl}(\mathcal{H})

is the abelian subgroup of $\backslash mathrm\{Gl\}(\backslash mathcal\{H\})$ that generates the KP flows, and

:$J\_i\; :=\; \backslash sum\_\{j\backslash in\; \backslash mathbf\{Z\}\}\; \backslash psi\_j\; \backslash psi^\backslash dagger\_\{j+i\},\; \backslash quad\; i=1,2\; \backslash dots$

are the ""current"" components.

As seen in equation ({{EquationNote|9}}), every KP $\backslash tau$-function can be represented (at least formally) as a linear combination of Schur functions, in which the coefficients $\backslash pi\_\backslash lambda(w)$ satisfy the bilinear set of Plucker relations corresponding to an element $w$ of an infinite (or finite) Grassmann manifold. In fact, the simplest class of (polynomial) tau functions consists of the Schur functions $s\_\backslash lambda(\backslash mathbf\{t\})$ themselves, which correspond to the special element of the Grassmann manifold whose image under the Plücker map is $|\backslash lambda>$.

If we choose $3N$ complex constants

$\backslash \{\backslash alpha\_k,\; \backslash beta\_k,\; \backslash gamma\_k\backslash \}\_\{k=1,\; \backslash dots,\; N\}$

with $\backslash alpha\_k,\; \backslash beta\_k$'s all distinct, $\backslash gamma\_k\; \backslash ne\; 0$, and define the functions

:$$

y_k({\bf t}) := e^{\sum_{i=1}^\infty t_i \alpha_k^i} +\gamma_k e^{\sum_{i=1}^\infty t_i \beta_k^i} \quad k=1,\dots, N,

we arrive at the Wronskian determinant formula

:$$

\tau^{(N)}_{\vec\alpha, \vec\beta, \vec\gamma}({\bf t}):=

\begin{vmatrix}

y_1({\bf t})& y_2({\bf t}) &\cdots& y_N({\bf t})\\

y_1'({\bf t})& y_2'({\bf t}) &\cdots& y_N'({\bf t})\\

\vdots & \vdots &\ddots &\vdots\\

y_1^{(N-1)}({\bf t})& y_2^{(N-1)}({\bf t}) &\cdots& y_N^{(N-1)}({\bf t})\\

\end{vmatrix},

which gives the general $N$-soliton $\backslash tau$-function.{{cite book | last1=Harnad | first1= J. | last2=Balogh | first2= F. | title=Tau functions and Their Applications, Chapt. 3 | year=2021 | series =Cambridge Monographs on Mathematical Physics | publisher=Cambridge University Press | location = Cambridge, U.K. | isbn= 9781108610902 | doi=10.1017/9781108610902| s2cid= 222379146 }}

Let $X$ be a compact Riemann surface of genus $g$ and fix a canonical homology basis $a\_1,\; \backslash dots,\; a\_g,\; b\_1,\; \backslash dots,\; b\_g$

of $H\_1(X,\backslash mathbf\{Z\})$ with intersection numbers

:$$

a_i \circ a_j = b_i \circ b_j =0, \quad a_i\circ b_j =\delta_{ij},\quad 1\leq i,j \leq g.

Let $\backslash \{\backslash omega\_i\backslash \}\_\{i=1,\; \backslash dots,\; g\}$ be a basis for the space $H^1(X)$ of holomorphic differentials satisfying the standard normalization conditions

:$$

\oint_{a_i} \omega_j =\delta_{ij}, \quad \oint_{b_j }\omega_j = B_{ij},

where $B$ is the Riemann matrix of periods.

The matrix $B$ belongs to the Siegel upper half space

:$$

\mathbf{S}_g=\left\{B \in \mathrm{Mat}_{g\times g}(\mathbf{C})\ \colon\ B^T = B,\ \text{Im}(B)

\text{ is positive definite}\right\}.

The Riemann $\backslash theta$ function on $\backslash mathbf\{C\}^g$ corresponding to the period matrix $B$ is defined to be

:$\backslash theta(Z\; |\; B)\; :=\; \backslash sum\_\{N\backslash in\; \backslash Z^g\}\; e^\{i\backslash pi\; (N,\; B\; N)\; +\; 2i\backslash pi\; (N,\; Z)\}.$

Choose a point $p\_\backslash infty\; \backslash in\; X$, a local parameter $\backslash zeta$ in a neighbourhood of $p\_\{\backslash infty\}$ with $\backslash zeta(p\_\backslash infty)=0$ and

a positive divisor of degree $g$

:$$

\mathcal{D}:= \sum_{i=1}^g p_i,\quad p_i \in X.

For any positive integer $k\backslash in\; \backslash mathbf\{N\}^+$ let $\backslash Omega\_k$ be the unique meromorphic differential of the second kind characterized by the following conditions:

• The only singularity of $\backslash Omega\_k$ is a pole of order $k+1$ at $p=p\_\backslash infty$ with vanishing residue.
• The expansion of $\backslash Omega\_k$ around $p=p\_\{\backslash infty\}$ is
• :$\backslash Omega\_k\; =\; d(\backslash zeta^\{-k\}\; )\; +\; \backslash sum\_\{j=1\}^\backslash infty\; Q\_\{ij\}\; \backslash zeta^j\; d\backslash zeta$.
• $\backslash Omega\_k$ is normalized to have vanishing $a$-cycles:
• :$\backslash oint\_\{a\_i\; \}\backslash Omega\_j\; =0.$

Denote by $\backslash mathbf\{U\}\_k\; \backslash in\; \backslash mathbf\{C\}^g$ the vector of $b$-cycles of $\backslash Omega\_k$:

:$(\backslash mathbf\{U\}\_k)\_j\; :=\; \backslash oint\_\{b\_j\}\; \backslash Omega\_k.$

Denote the image of $\{\backslash mathcal\; D\}$ under the Abel map

$\backslash mathcal\{A\}:\; \backslash mathcal\{S\}^g(X)\; \backslash to\; \backslash mathbf\{C\}^g$

:$\backslash mathbf\{E\}\; :=\; \backslash mathcal\{A\}(\backslash mathcal\{D\})\; \backslash in\; \backslash mathbf\{C\}^g,\; \backslash quad\; \backslash mathbf\{E\}\_j$

= \mathcal{A}_j (\mathcal{D}) := \sum_{j=1}^g \int_{p_0}^{p_i}\omega_j

with arbitrary base point $p\_0$.

Then the following is a KP $\backslash tau$-function:{{cite journal | last=Dubrovin | first=B.A. | title=Theta Functions and Nonlinear Equations | journal=Russ. Math. Surv. | volume=36 | issue=1 | year=1981 | pages=11–92 | doi= 10.1070/RM1981v036n02ABEH002596 | bibcode=1981RuMaS..36...11D |url=https://iopscience.iop.org/article/10.1070/RM1981v036n02ABEH002596 | s2cid=54967353 }}

:$\backslash tau\_\{(X,\; \backslash mathcal\{D\},\; p\_\backslash infty,\; \backslash zeta)\}(\backslash mathbf\{t\}):=\; e^\{-\{1\backslash over\; 2\}\; \backslash sum\_\{ij\}\; Q\_\{ij\}t\; \_i\; t\_j\}$

\theta\left(\mathbf{E} +\sum_{k=1}^\infty t_k \mathbf{U}_k \Big|B\right) .

Let $d\backslash mu\_0(M)$ be the Lebesgue measure on the $N^2$ dimensional space $\{\backslash mathbf\; H\}^\{N\backslash times\; N\}$ of $N\backslash times\; N$ complex Hermitian matrices.

Let $\backslash rho(M)$ be a conjugation invariant integrable density function

:$$

\rho(U M U^{\dagger}) = \rho(M), \quad U\in U(N).

Define a deformation family of measures

:$$

d\mu_{N,\rho}(\mathbf{t}) := e^{\text{ Tr }(\sum_{i=1}^\infty t_i M^i)} \rho(M) d\mu_0 (M)

for small $\backslash mathbf\{t\}=\; (t\_1,\; t\_2,\; \backslash cdots)$ and let

:$$

\tau_{N,\rho}({\bf t}):= \int_{{\mathbf H}^{N\times N} }d\mu_{N,\rho}({\bf t}).

be the partition function for this random matrix model.M.L. Mehta, "Random Matrices", 3rd ed., vol. 142 of Pure and Applied Mathematics, Elsevier, Academic Press, {{ISBN|9780120884094}} (2004)

Then $\backslash tau\_\{N,\backslash rho\}(\backslash mathbf\{t\})$ satisfies the bilinear Hirota residue equation ({{EquationNote|1}}), and hence is a $\backslash tau$-function of the KP hierarchy.{{cite journal | last1=Kharchev | first1=S. | last2=Marshakov | first2=A. | last3=Mironov | first3=A. | last4=Orlov | first4=A. | last5=Zabrodin | first5=A. | title=Matrix models among integrable theories: Forced hierarchies and operator formalism | journal=Nuclear Physics B | publisher=Elsevier BV | volume=366 | issue=3 | year=1991 | issn=0550-3213 | doi=10.1016/0550-3213(91)90030-2 | pages=569–601| bibcode=1991NuPhB.366..569K }}

Let $\backslash \{r\_i\backslash \}\_\{i\backslash in\; \backslash mathbf\{Z\}\}$ be a (doubly) infinite sequence of complex numbers.

For any integer partition $\backslash lambda\; =\; (\backslash lambda\_1,\; \backslash dots,\; \backslash lambda\_\{\backslash ell(\backslash lambda)\})$ define the content product coefficient

:$r\_\{\backslash lambda\}\; :=\; \backslash prod\_\{(i,j)\backslash in\; \backslash lambda\}\; r\_\{j-i\}$,

where the product is over all pairs $(i,j)$ of positive integers that correspond to boxes of the Young diagram of the partition $\backslash lambda$, viewed as positions of matrix elements of the corresponding

$\backslash ell(\backslash lambda)\; \backslash times\; \backslash lambda\_1$ matrix.

Then, for every pair of infinite sequences $\backslash mathbf\{t\}\; =\; (t\_1,\; t\_2,\; \backslash dots\; )$ and $\backslash mathbf\{s\}\; =\; (s\_1,\; s\_2,\; \backslash dots\; )$ of complex variables, viewed as (normalized) power sums $\backslash mathbf\{t\}\; =\; [\backslash mathbf\{x\}],\; \backslash \; \backslash mathbf\{s\}\; =\; [\backslash mathbf\{y\}]$

of the infinite sequence of auxiliary variables

::$\backslash mathbf\{x\}\; =\; (x\_1,\; x\_2,\; \backslash dots\; )$ and $\backslash mathbf\{y\}\; =\; (y\_1,\; y\_2,\; \backslash dots\; )$,

defined by:

::$t\_j\; :=\; \backslash tfrac\{1\}\{j\}\backslash sum\_\{a=1\}^\backslash infty\; x\_a^j,\; \backslash quad\; s\_j\; :=\; \backslash tfrac\{1\}\{j\}\; \backslash sum\_\{j=1\}^\backslash infty\; y\_a^j$,

the function

{{NumBlk|:|$\backslash tau^r(\backslash mathbf\{t\},\backslash mathbf\{s\})\; :=\; \backslash sum\_\{\backslash lambda\}r\_\backslash lambda\; s\_\backslash lambda(\backslash mathbf\{t\})s\_\backslash lambda(\backslash mathbf\{s\})$|{{EquationRef|11}}}}

is a double KP $\backslash tau$-function, both in the $\backslash mathbf\{t\}$ and the $\backslash mathbf\{s\}$ variables, known as a $\backslash tau$-function of hypergeometric type.{{cite journal | last=Orlov | first=A. Yu. | title=Hypergeometric Functions as Infinite-Soliton Tau Functions | journal=Theoretical and Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=146 | issue=2 | year=2006 | issn=0040-5779 | doi=10.1007/s11232-006-0018-4 | pages=183–206| bibcode=2006TMP...146..183O | s2cid=122017484 }}

In particular, choosing

:$r\_j\; =\; r^\{\backslash beta\}\_j\; :=\; e^\{j\backslash beta\}$

for some small parameter $\backslash beta$, denoting the corresponding content product coefficient as $r\_\backslash lambda^\backslash beta$

and setting

::$\backslash mathbf\{s\}\; =\; (1,\; 0,\; \backslash dots)=:\; \backslash mathbf\{t\}\_0$,

the resulting $\backslash tau$-function can be equivalently expanded as

{{NumBlk|::|$$

\tau^{r^\beta}(\mathbf{t},\mathbf{t}_0)

= \sum_{\lambda}\sum_{d=0}^\infty \frac{\beta^d}{d!} H_d(\lambda)p_\lambda(\mathbf{t}),

|{{EquationRef|12}}}}

where $\backslash \{H\_d(\backslash lambda)\backslash \}$ are the simple Hurwitz numbers, which are

$\backslash frac\{1\}\{n!\}$ times the number of ways in which an element

$k\_\backslash lambda\; \backslash in\; \backslash mathcal\{S\}\_\{n\}$ of the symmetric group $\backslash mathcal\{S\}\_\{n\}$ in $n=|\backslash lambda|$ elements, with cycle lengths equal to the parts of the partition $\backslash lambda$, can be factorized as a product of $d$ $2$-cycles

::$k\_\backslash lambda\; =\; (a\_1\; b\_1)\backslash dots\; (a\_d\; b\_d)$,

and

::$p\_\{\backslash lambda\}(\backslash mathbf\{t\})\; =\; \backslash prod\_\{i=1\}^\{\backslash ell(\backslash lambda)\}\; p\_\{\backslash lambda\_i\}(\backslash mathbf\{t\}),\; \backslash \; \backslash text\{with\}\backslash \; p\_i(\backslash mathbf\{t\})\; :=\; \backslash sum\_\{a=1\}^\backslash infty\; x^i\_a\; =\; i\; t\_i$

is the power sum symmetric function. Equation ({{EquationNote|12}}) thus shows that the (formal) KP hypergeometric $\backslash tau$-function ({{EquationNote|11}}) corresponding to the content product coefficients $r\_\backslash lambda^\backslash beta$ is a generating function, in the combinatorial sense, for simple Hurwitz numbers.

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