Tau function (integrable systems)

There are two notions of ${\displaystyle \tau }$-functions, both introduced by the Sato school. The first is that of isomonodromic ${\displaystyle \tau }$-functions .[5] The second is ${\displaystyle \tau }$-functions of the Sato-Segal-Wilson type [2][6] for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectral deformation equations of Lax type.

A ${\displaystyle \tau }$-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[2] and Segal-Wilson [6] 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.

A KP (Kadomtsev–Petviashvili) ${\displaystyle \tau }$-function ${\displaystyle \tau (\mathbf {t} )}$ is a function of an infinite number of KP flow variables ${\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )}$ that satisfies the following bilinear formal residue equation

${\displaystyle \mathrm {res} _{z=0}\left(e^{\sum _{i=1}^{\infty }(\delta t_{i})z^{i}}\tau ({\bf {t}}-[z^{-1}])\tau ({\bf {s}}+[z^{-1}])\right)dz\equiv 0,}$

(1)

identically in the ${\displaystyle \delta t_{j}}$ variables, where ${\displaystyle \mathrm {res} _{z=0}}$ is the ${\displaystyle z^{-1}}$ coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series' in ${\displaystyle z}$, and

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

If ${\displaystyle \tau (t_{1},t_{2},t_{3},\dots \dots )}$ is a KP ${\displaystyle \tau }$-function satisfying the Hirota residue equation (1) and we identify the first three flow variables as

${\displaystyle t_{1}=x,\quad t_{2}=y,\quad t_{3}=t,\quad }$

it follows that the function

${\displaystyle u(x,y,t):=2{\frac {\partial ^{2}}{\partial x^{2}}}\log \left(\tau (x,y,t,t_{4},\dots )\right)}$

satisfies the ${\displaystyle 2+1}$ dimensional nonlinear partial differential equation

${\displaystyle 3u_{yy}=\left(4u_{t}-6uu_{x}-u_{xxx}\right)_{x},}$

(2)

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 ${\displaystyle \tau (t_{1},t_{2},t_{3},\dots \dots )}$ 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 ${\displaystyle {\bf {t}}=(t_{1},t_{2},\dots )}$. These are collectively known as the KP hierarchy.

If we define the (formal) Baker-Akhiezer function ${\displaystyle \Psi (z,\mathbf {t} )}$ by Sato's formula

${\displaystyle \Psi (z,\mathbf {t} ):=e^{\sum _{i=1}^{\infty }t_{i}z^{i}}{\frac {\tau (\mathbf {t} -[z^{-1}])}{\tau (\mathbf {t} )}}}$

and expand it as a formal series in the powers of the variable ${\displaystyle z}$

${\displaystyle \Psi (z,\mathbf {t} )=e^{\sum _{i=1}^{\infty }t_{i}z^{i}}(1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )z^{-j}),}$

this satisfies an infinite sequence of compatible evolution equations

${\displaystyle {\frac {\partial \Psi }{\partial t_{i}}}={\mathcal {D}}_{i}\Psi ,\quad i,j,=1,2,\dots ,}$

(3)

where ${\displaystyle {\mathcal {D}}_{i}}$ is a linear ordinary differential operator of degree ${\displaystyle i}$ in the variable ${\displaystyle x:=t_{1}}$, with coefficients that are functions of the flow variables ${\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )}$, defined as follows

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

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

${\displaystyle {\mathcal {L}}=\partial +\sum _{j=1}^{\infty }u_{j}(\mathbf {t} )\partial ^{-j}={\mathcal {W}}\circ \partial \circ {\mathcal {W}}^{-1}}$

with ${\displaystyle \partial$ :={\frac {\partial }{\partial x}}} , where

${\displaystyle {\mathcal {W}}:=1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )\partial ^{-j}}$

is the wave operator and ${\displaystyle {\big (}{\mathcal {L}}^{i}{\big )}_{+}}$ denotes the projection to the part of ${\displaystyle {\mathcal {L}}^{i}}$ containing purely non-negative powers of ${\displaystyle \partial }$ ; i.e. to the differential operator part of ${\displaystyle {\mathcal {L}}^{i}}$ .

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

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial t_{i}}}=[{\mathcal {D}}_{i},{\mathcal {L}}],\quad i,=1,2,\dots }$

(4)

and the compatibility conditions for both the system (3) and (4) are

${\displaystyle {\frac {\partial {\mathcal {D}}_{i}}{\partial t_{j}}}-{\frac {\partial {\mathcal {D}}_{j}}{\partial t_{i}}}+[{\mathcal {D}}_{i},{\mathcal {D}}_{j}]=0,\quad i,j,=1,2,\dots }$

This is a compatible infinite system of nonlinear partial differential equations, known as the KP (Kadomtsev-Petviashvili) hierarchy, for the functions ${\displaystyle \{u_{j}(\mathbf {t} )\}_{j\in \mathbf {N} }}$, with respect to the set ${\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )}$ of independent variables, each of which contains only a finite number of ${\displaystyle u_{j}}$'s, and derivatives only with respect to the three independent variables ${\displaystyle (x,t_{i},t_{j})}$. The first nontrivial case of these is the Kadomtsev-Petviashvili equation (2).

Thus, every KP ${\displaystyle \tau }$ 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

${\displaystyle {\partial \Psi \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}\Psi =0,\quad }$

(5)

${\displaystyle {\partial \Psi \over \partial \alpha _{i}}+{N_{i} \over z-\alpha _{i}}\Psi =0,}$

(6)

where ${\displaystyle \{N_{i}\}_{i=1,\dots ,n}}$ are a set of ${\displaystyle n}$ ${\displaystyle r\times r}$ traceless matrices, ${\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}}$ a set of ${\displaystyle n}$ complex parameters and ${\displaystyle z}$ a complex variable, and ${\displaystyle \Psi (z,\alpha _{1},\dots ,\alpha _{m})}$ is an invertible ${\displaystyle r\times r}$ matrix valued function of ${\displaystyle z}$ and ${\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}}$. These are the necessary and sufficient conditions for the based monodromy representation of the fundamental group ${\displaystyle \pi _{0}({\bf {P}}^{1}\backslash \{\alpha _{i}\}_{i=1,\dots ,n})}$ of the Riemann sphere punctured at the points ${\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}}$ corresponding to the rational covariant derivative operator

${\displaystyle {\partial \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}}$

to be independent of the parameters ${\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}}$; i.e. that changes in these parameters induce an isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations [5]

${\displaystyle {\partial N_{i} \over \partial \alpha _{j}}={[N_{i},N_{j}] \over \alpha _{i}-\alpha _{j}}\quad {\text{ for }}i\neq j,}$

${\displaystyle {\partial N_{i} \over \partial \alpha _{i}}=-\sum _{1\leq j\leq n,j\neq i}{[N_{i},N_{j}] \over \alpha _{i}-\alpha _{j}}.}$

Defining the ${\displaystyle n}$ functions

${\displaystyle H_{i}={\frac {1}{2}}\sum _{1\leq j\leq n,j\neq i}{{\rm {Tr}}(N_{i}N_{j}) \over \alpha _{i}-\alpha _{j}},\quad i=1,\dots ,n,}$

the Schlesinger equations imply that the differential form

${\displaystyle \omega$ :=\sum _{i=1}^{n}H_{i}d\alpha _{i}}

on the space of parameters is closed:

${\displaystyle d\omega =0}$

and hence, locally exact. Therefore, at least locally, there exists a function ${\displaystyle \tau (\alpha _{1},\dots ,\alpha _{n})}$ of the parameters, defined within a multiplicative constant, such that

${\displaystyle \omega =d\mathrm {ln} \tau }$

The function ${\displaystyle \tau (\alpha _{1},\dots ,\alpha _{n})}$ is called the isomonodromic ${\displaystyle \tau }$-function associated to the fundamental solution ${\displaystyle \Psi }$ of the system (5), (6). For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes parameters and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic ${\displaystyle \tau }$-functions may be defined in a similar way, using differentials on the extended parameter space.[5]

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

${\displaystyle {\mathcal {F}}=\Lambda ^{\infty /2}{\mathcal {H}}=\oplus _{n\in \mathbf {Z} }{\mathcal {F}}_{n}}$

defined on a (separable) Hilbert space ${\displaystyle {\mathcal {H}}}$ with basis elements ${\displaystyle \{e_{i}\}_{i\in \mathbf {Z} }\}}$ and dual basis elements ${\displaystyle \{e^{i}\}_{i\in \mathbf {Z} }\}}$ for ${\displaystyle {\mathcal {H}}^{*}}$.

The free fermionic creation and annihilation operators ${\displaystyle \{\psi _{j},\psi _{j}^{\dagger }\}_{j\in \mathbf {Z} }}$ act as endomorphisms on ${\displaystyle {\mathcal {F}}}$ via exterior and interior multiplication by the basis elements

${\displaystyle \psi _{i}:=e_{i}\wedge ,\quad \psi _{i}^{\dagger }:=i_{e^{i}},\quad i\in \mathbf {Z} ,}$

and satisfy the canonical anti-commutation relations

${\displaystyle [\psi _{i},\psi _{k}]_{+}=[\psi _{i}^{\dagger },\psi _{k}^{\dagger }]_{+}=0,\quad [\psi _{i},\psi _{k}^{\dagger }]_{+}=\delta _{ij}.}$

These generate the standard fermionic representation of the Clifford algebra on the direct sum ${\displaystyle {\mathcal {H}}+{\mathcal {H}}^{*}}$, corresponding to the scalar product

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

with the Fock space ${\displaystyle {\mathcal {F}}}$ as irreducible module. Denote the vacuum state, in the zero fermionic charge sector ${\displaystyle {\mathcal {F}}_{0}}$, as

${\displaystyle |0\rangle$ :=e_{-1}\wedge e_{-2}\wedge \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

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

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

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

Normal ordering ${\displaystyle :L_{1},\cdots L_{m}:}$ 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

${\displaystyle \langle 0|:L_{1},\cdots L_{m}:|0\rangle =0.}$

In particular, for a product ${\displaystyle L_{1}L_{2}}$ of a pair ${\displaystyle (L_{1},L_{2})}$ of linear operators

${\displaystyle :L_{1}L_{2}:=L_{1}L_{2}-\langle 0|L_{1}L_{2}|0\rangle .}$

The fermionic charge operator ${\displaystyle C}$ is defined as

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

The subspace ${\displaystyle {\mathcal {F}}_{n}\subset {\mathcal {F}}}$ is the eigenspace of ${\displaystyle C}$ consisting of all eigenvectors with eigenvalue ${\displaystyle n}$

${\displaystyle C|v;n\rangle =n|v;n\rangle ,\quad \forall |v;n\rangle \in {\mathcal {F}}_{n}}$.

The standard orthonormal basis ${\displaystyle \{|\lambda \rangle \}}$ for the zero fermionic charge sector ${\displaystyle {\mathcal {F}}_{0}}$ is labelled by integer partitions ${\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})}$, where ${\displaystyle \lambda _{1}\geq \cdots \geq \lambda _{\ell (\lambda )}}$ is a weakly decreasing sequence of ${\displaystyle \ell (\lambda )}$ positive integers, which can equivalently be represented by a Young diagram, as depicted here for the partition ${\displaystyle (5,4,1)}$.

Young diagram of the partition (5, 4, 1)

An alternative notation for a partition ${\displaystyle \lambda }$ consists of the Frobenius indices ${\displaystyle (\alpha _{1},\dots \alpha _{r}|\beta _{1},\dots \beta _{r})}$, where ${\displaystyle \alpha _{i}}$ denotes the arm length; i.e. the number ${\displaystyle \lambda _{i}-i}$ of boxes in the Young diagram to the right of the ${\displaystyle i}$'th diagonal box, ${\displaystyle \beta _{i}}$ denotes the leg length, i.e. the number of boxes in the Young diagram below the ${\displaystyle i}$'th diagonal box, for ${\displaystyle i=1,\dots ,r}$, where ${\displaystyle r}$ is the Frobenius rank, which is the number of diagonal elements.

The basis element ${\displaystyle |\lambda \rangle }$ is then given by acting on the vacuum with a product of ${\displaystyle r}$ pairs of creation and annihilation operators, labelled by the Frobenius indices

${\displaystyle |\lambda \rangle =(-1)^{\sum _{j=1}^{r}\beta _{j}}\prod _{k=1}^{r}{\big (}\psi _{\alpha _{k}}\psi _{-\beta _{k}-1}^{\dagger }{\big )}|0\rangle .}$

The integers ${\displaystyle \{\alpha _{i}\}_{i=1,\dots ,r}}$ indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while ${\displaystyle \{-\beta _{i}-1\}_{i=1,\dots ,r}}$ 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.[2]

The case of the null (emptyset) partition ${\displaystyle |\emptyset \rangle =|0\rangle }$ gives the vacuum state, and the dual basis ${\displaystyle \langle \mu |\}}$ is defined by

${\displaystyle \langle \mu |\lambda \rangle =\delta _{\lambda ,\mu }}$

Then any KP ${\displaystyle \tau }$-function can be expressed as a sum

${\displaystyle \tau _{w}(\mathbf {t} )=\sum _{\lambda }\pi _{\lambda }(w)s_{\lambda }(\mathbf {t} )}$

where ${\displaystyle \mathbf {t} =(t_{1},t_{2},\dots ,\dots )}$ are the KP flow variables, ${\displaystyle s_{\lambda }(\mathbf {t} )}$ is the Schur function corresponding to the partition ${\displaystyle \lambda }$, viewed as a function of the normalized power sum variables

${\displaystyle t_{i}:=[\mathbf {x} ]_{i}:={\tfrac {1}{i}}\sum _{a=1}^{n}x_{a}^{i}\quad i=1,2,\dots }$

in terms of an auxiliary (finite or infinite) sequence of variables ${\displaystyle \mathbf {x}$ :=(x_{1},\dots ,x_{N})} and the constant coefficients ${\displaystyle \pi _{\lambda }(w)}$ may be viewed as the Plucker coordinates of an element ${\displaystyle w\in \mathrm {Gr} _{{\mathcal {H}}_{+}}({\mathcal {H}})}$ of the infinite dimensional Grassmannian consisting of the orbit, under the action of the general linear group ${\displaystyle \mathrm {Gl} ({\mathcal {H}})}$, of the subspace ${\displaystyle {\mathcal {H}}_{+}=\mathrm {span} \{e_{-i}\}_{i\in \mathbf {N} }\subset {\mathcal {H}}}$ of the Hilbert space ${\displaystyle {\mathcal {H}}}$.

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

${\displaystyle |\tau _{w}\rangle =\sum _{\lambda }\pi _{\lambda }(w)|\lambda \rangle }$

of the Fock space ${\displaystyle {\mathcal {F}}_{0}}$ which, up to projectivization is the image of the Grassmannian element ${\displaystyle w\in \mathrm {Gr} _{{\mathcal {H}}_{+}}({\mathcal {H}})}$ under the Plucker map

${\displaystyle {\mathcal {Pl}}:\mathrm {span} (w_{1},w_{2},\dots )\longrightarrow [w_{1}\wedge w_{2}\wedge \cdots ]=[|\tau _{w}\rangle ],}$

where ${\displaystyle (w_{1},w_{2},\dots )}$ is a basis for the subspace ${\displaystyle w\subset {\mathcal {H}}}$ and ${\displaystyle [\cdots ]}$ denotes projectivization of an element of ${\displaystyle {\mathcal {F}}}$.

The Plucker coordinates ${\displaystyle \{\pi _{\lambda }(w)\}}$ satisfy an infinite set of bilinear relations, the Plucker relations, defining the Plücker embedding into the projecivization ${\displaystyle \mathbf {P} ({\mathcal {F}})}$ of the fermionic Fock space, which are equivalent to the Hirota bilinear residue relation (1).

If ${\displaystyle w=g({\mathcal {H}}_{+})}$ for a group element ${\displaystyle g\in \mathrm {Gl} ({\mathcal {H}})}$ with fermionic representation ${\displaystyle {\hat {g}}}$, then the ${\displaystyle \tau }$-function ${\displaystyle \tau _{w}(\mathbf {t} )}$ can be expressed as the fermionic vacuum state expectation value (VEV):

${\displaystyle \tau _{w}(\mathbf {t} )=\langle 0|{\hat {\gamma }}_{+}(\mathbf {t} ){\hat {g}}|0\rangle ,}$

where

${\displaystyle \Gamma _{+}=\{{\hat {\gamma }}_{+}(\mathbf {t} )=e^{\sum _{i=1}^{\infty }t_{i}J_{i}}\}\subset \mathrm {Gl} ({\mathcal {H}})}$

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

${\displaystyle J_{i}:=\sum _{j\in \mathbf {Z} }\psi _{j}\psi _{j+i}^{\dagger },\quad i=1,2\dots }$

are the ""current"" components.

If we choose ${\displaystyle 3N}$ complex constants ${\displaystyle \{\alpha _{k},\beta _{k},\gamma _{k}\}_{k=1,\dots ,N}}$ with ${\displaystyle \alpha _{k},\beta _{k}}$'s all distinct, ${\displaystyle \gamma _{k}\neq 0}$, and define the functions

${\displaystyle y_{k}({\bf {t}}):=e^{\sum _{i=1}^{\infty }t_{i}\alpha _{k}^{i}}+\gamma _{k}e^{\sum _{i=1}^{\infty }t_{i}\beta _{k}^{i}}\quad k=1,\dots ,N,}$

we arrive at the Wronskian determinant formula

${\displaystyle \tau _{{\vec {\alpha }},{\vec {\beta }},{\vec {\gamma }}}^{(N)}({\bf {t}}):={\begin{vmatrix}y_{1}({\bf {t}})&y_{2}({\bf {t}})&\cdots &y_{N}({\bf {t}})\\y_{1}'({\bf {t}})&y_{2}'({\bf {t}})&\cdots &y_{N}'({\bf {t}})\\\vdots &\vdots &\ddots &\vdots \\y_{1}^{(N-1)}({\bf {t}})&y_{2}^{(N-1)}({\bf {t}})&\cdots &y_{N}^{(N-1)}({\bf {t}})\\\end{vmatrix}}.}$

which gives the general ${\displaystyle N}$-soliton solution.[3][4]

Let ${\displaystyle X}$ be a compact Riemann surface of genus ${\displaystyle g}$ and fix a canonical homology basis ${\displaystyle a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}}$ of ${\displaystyle H_{1}(X,\mathbf {Z} )}$ with intersection numbers

${\displaystyle a_{i}\circ a_{j}=b_{i}\circ b_{j}=0,\quad a_{i}\circ b_{j}=\delta _{ij},\quad 1\leq i,j\leq g.}$

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

${\displaystyle \oint _{a_{i}}\omega _{j}=\delta _{ij},\quad \oint _{b_{j}}\omega _{j}=B_{ij},}$

where ${\displaystyle B}$ is the Riemann matrix of periods. The matrix ${\displaystyle B}$ belongs to the Siegel upper half space

${\displaystyle \mathbf {S} _{g}=\left\{B\in \mathrm {Mat} _{g\times g}(\mathbf {C} )\ \colon \ B^{T}=B,\ {\text{Im}}(B){\text{ is positive definite}}\right\}.}$

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

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

Choose a point ${\displaystyle p_{\infty }\in X}$, a local parameter ${\displaystyle \zeta }$ in a neighbourhood of ${\displaystyle p_{\infty }}$ with ${\displaystyle \zeta (p_{\infty })=0}$ and a positive divisor of degree ${\displaystyle g}$

${\displaystyle {\mathcal {D}}:=\sum _{i=1}^{g}p_{i},\quad p_{i}\in X.}$

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

- The only singularity of ${\displaystyle \Omega _{k}}$ is a pole of order ${\displaystyle k+1}$ at ${\displaystyle p=p_{\infty }}$ with vanishing residue.

- The expansion of ${\displaystyle \Omega _{k}}$ around ${\displaystyle p=p_{\infty }}$ is

${\displaystyle \Omega _{k}=d(\zeta ^{-k})+\sum _{j=1}^{\infty }Q_{ij}\zeta ^{j}d\zeta }$.

- ${\displaystyle \Omega _{k}}$ is normalized to have vanishing ${\displaystyle a}$-cycles:

${\displaystyle \oint _{a_{i}}\Omega _{j}=0.}$

Denote by ${\displaystyle \mathbf {U} _{k}\in \mathbf {C} ^{g}}$ the vector of ${\displaystyle b}$-cycles of ${\displaystyle \Omega _{k}}$:

${\displaystyle (\mathbf {U} _{k})_{j}:=\oint _{b_{j}}\Omega _{k}.}$

Denote the image of ${\displaystyle {\mathcal {D}}}$ under the Abel map ${\displaystyle {\mathcal {A}}:{\mathcal {S}}^{g}(X)\to \mathbf {C} ^{g}}$

${\displaystyle \mathbf {E}$ :={\mathcal {A}}({\mathcal {D}})\in \mathbf {C} ^{g},\quad \mathbf {E} _{j}={\mathcal {A}}_{j}({\mathcal {D}}):=\sum _{j=1}^{g}\int _{p_{0}}^{p_{i}}\omega _{j}}

with arbitrary base point ${\displaystyle p_{0}}$.

Then the following is a KP ${\displaystyle \tau }$-function:

${\displaystyle \tau _{(X,{\mathcal {D}},p_{\infty },\zeta )}(\mathbf {t} ):=e^{-{1 \over 2}\sum _{ij}Q_{ij}t_{i}t_{j}}\theta \left(\mathbf {E} +\sum _{k=1}^{\infty }t_{k}\mathbf {U} _{k}{\Big |}B\right).}$

Let ${\displaystyle d\mu _{0}(M)}$ be the Lebesgue measure on the ${\displaystyle N^{2}}$ dimensional space ${\displaystyle {\mathbf {H} }^{N\times N}}$ of ${\displaystyle N\times N}$ complex Hermitian matrices. Let ${\displaystyle \rho (M)}$ be a conjugation invariant integrable density function

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

Define a deformation family of measures

${\displaystyle d\mu _{N,\rho }(\mathbf {t} ):=e^{{\text{ Tr }}(\sum _{i=1}^{\infty }t_{i}M^{i})}\rho (M)d\mu _{0}(M)}$

for small ${\displaystyle \mathbf {t} =(t_{1},t_{2},\cdots )}$ and let

${\displaystyle \tau _{N,\rho }({\bf {t}}):=\int _{{\mathbf {H} }^{N\times N}}d\mu _{N,\rho }({\bf {t}}).}$

be the partition function for this random matrix model.[7] Then ${\displaystyle \tau _{N,\rho }(\mathbf {t} )}$ satisfies the bilinear Hirota residue equation (1), and hence is a ${\displaystyle \tau }$-function of the KP hierarchy.[8]

Let ${\displaystyle \{r_{i}\}_{i\in \mathbf {Z} }}$ be a (doubly) infinite sequence of complex numbers. For any integer partition ${\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})}$ define the content product coefficient

${\displaystyle r_{\lambda }:=\prod _{(i,j)\in \lambda }r_{j-i}}$

where the product is over all pairs ${\displaystyle (i,j)}$ of positive integers that correspond to boxes of the Young diagram of the partition ${\displaystyle \lambda }$, viewed as positions of matrix elements of the corresponding ${\displaystyle \ell (\lambda )\times \lambda _{1}}$ matrix. Then, for every pair of infinite sequences ${\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )}$ and ${\displaystyle \mathbf {s} =(s_{1},s_{2},\dots )}$ of complex vaiables, viewed as (normalized) power sums ${\displaystyle \mathbf {t} =[\mathbf {x} ],\ \mathbf {s} =[\mathbf {y} ]}$ of the infinite sequence of auxiliary variables ${\displaystyle \mathbf {x} =(x_{1},x_{2},\dots )}$ and ${\displaystyle \mathbf {y} =(y_{1},xy2,\dots )}$, defined by

${\displaystyle t_{j}:={\tfrac {1}{j}}\sum _{a=1}^{\infty }x_{a}^{j},\quad s_{j}:={\tfrac {1}{j}}\sum _{j=1}^{\infty }y_{a}^{j},}$

the function

${\displaystyle \tau ^{r}(\mathbf {t} ,\mathbf {s} ):=\sum _{\lambda }r_{\lambda }s_{\lambda }(\mathbf {t} )s_{\lambda }(\mathbf {s} )}$

is a double KP ${\displaystyle \tau }$-function, both in the ${\displaystyle \mathbf {t} }$ and the ${\displaystyle \mathbf {s} }$ variables, known as a ${\displaystyle \tau }$ function of hypergeometric type.[9]

In particular, choosing

${\displaystyle r_{j}=r_{j}^{\beta }:=e^{j\beta }}$

for some small parameter ${\displaystyle \beta }$, denoting the corresponding content product coefficient as ${\displaystyle r_{\lambda }^{\beta }}$ and setting ${\displaystyle \mathbf {s} =(1,0,\dots )=:\mathbf {t} _{0}}$, the resulting ${\displaystyle \tau }$-function can be equivalently expanded as

${\displaystyle \tau ^{r^{\beta }}(\mathbf {t} ,\mathbf {t} _{0})=\sum _{\lambda }\sum _{d=0}^{\infty }{\frac {\beta ^{d}}{d!}}H_{d}(\lambda )p_{\lambda }(\mathbf {t} ),}$

(7)

where ${\displaystyle H_{d}(\lambda )}$ are the simple Hurwitz numbers, which are ${\displaystyle {\frac {1}{n!}}}$ times the number of ways in which an element ${\displaystyle k_{\lambda }\in {\mathcal {S}}_{n}}$ of the symmetric group ${\displaystyle {\mathcal {S}}_{n}}$ in ${\displaystyle n=|\lambda |}$ elements, with cycle lengths equal to the parts of the partition ${\displaystyle \lambda }$, can be factorized as a product of ${\displaystyle d}$ ${\displaystyle 2}$-cycles

${\displaystyle h_{\lambda }=(a_{1}b_{1})\dots (a_{d}b_{d}),}$

and

${\displaystyle p_{\lambda }(\mathbf {t} )=\prod _{i=1}^{\ell (\lambda )}p_{\lambda _{i}}(\mathbf {t} ),{\text{ with }}p_{i}(\mathbf {t} ):=\sum _{a=1}^{\infty }x_{a}^{i}=it_{i}}$

is the power sum symmetric function. Equation (7) thus shows that the (formal) KP hypergeometric ${\displaystyle \tau }$-function corresponding to the content product coefficients ${\displaystyle r_{\lambda }^{\beta }}$ is a generating function, in the combinatorial sense, for simple Hurwitz numbers. [10] [11]

• Dickey, L.A. (2003), "Soliton Equations and Hamiltonian Systems", vol. 26 of Advanced Series in Mathematical Physics. World Scientific Publishing Co., Inc., River Edge, NJ, 2nd ed.
• Harnad, J.; Balogh, F. (2021), "Tau functions and Their Applications", Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K.
• Hirota, R. (2004), "The Direct Method in Soliton Theory", Cambridge University Press, Cambridge , U.K.
• Jimbo, M.; Miwa, T. (1999), "Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras", Cambridge University Press, Cambridge , U.K., Cambridge Tracts in Mathematics, 135
• Kodama, Y. (2017), KP Solitons and the Grassmannians: Combinatorics and Geometry of Two-Dimensional Wave Patterns, Springer Briefs in Mathematical Physics, Springer Nature