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'''Tau functions''' are an important ingredient in the modern mathematical theory of [[integrable system]]s, and have numerous applications in a variety of other domains. They were originally introduced by [[:simple:Ryogo Hirota|Ryogo Hirota]]<ref>{{cite journal | last=Hirota | first=Ryogo | title=Reduction of soliton equations in bilinear form | journal=Physica D: Nonlinear Phenomena | publisher=Elsevier BV | volume=18 | issue=1–3 | year=1986 | issn=0167-2789 | doi=10.1016/0167-2789(86)90173-9 | pages=161–170| bibcode=1986PhyD...18..161H }}</ref> in his ''direct method'' approach to [[soliton]] equations, based on expressing them in an equivalent bilinear form.

The term '''tau function''', or <math> \tau </math>-function, was first used systematically by [[Mikio Sato]]<ref name="Sato">Sato, Mikio, "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds", ''Kokyuroku, RIMS, Kyoto Univ.'', 30–46 (1981).</ref> and his students<ref name="DJKM1">{{cite journal | last1=Date | first1=Etsuro | last2=Jimbo | first2=Michio | last3=Kashiwara | first3=Masaki | last4=Miwa | first4=Tetsuji | author2-link=Michio Jimbo  |author3-link=Masaki Kashiwara |author4-link=Tetsuji Miwa  |title=Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III– | journal=Journal of the Physical Society of Japan | publisher=Physical Society of Japan | volume=50 | issue=11 | year=1981 | issn=0031-9015 | doi=10.1143/jpsj.50.3806 | pages=3806–3812| bibcode=1981JPSJ...50.3806D }}</ref><ref name="DJKM2">{{cite journal | last1=Jimbo | first1=Michio | last2=Miwa | first2=Tetsuji | title=Solitons and infinite-dimensional Lie algebras | journal=Publications of the Research Institute for Mathematical Sciences | publisher=European Mathematical Society Publishing House | volume=19 | issue=3 | year=1983 | issn=0034-5318 | doi=10.2977/prims/1195182017 | pages=943–1001| doi-access=free }}</ref> in the specific context of the [[Kadomtsev–Petviashvili equation|Kadomtsev–Petviashvili (or KP) equation]] and related [[integrable system|integrable hierarchies]]. It is a central ingredient in the theory of [[solitons]]. In this setting, given any <math>\tau</math>-function satisfying a Hirota-type system of bilinear equations (see {{slink||Hirota bilinear residue relation for KP tau functions}} 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 [[Random matrix|matrix model]] [[Partition function (statistical mechanics)|partition functions]] in the spectral theory of [[random matrices]],<ref name="AkBaDiFr">{{cite book | last1=Akemann | first1=G.| last2=Baik | first2= J. | last3=Di Francesco | first3=P. | title=The Oxford Handbook of Random Matrix Theory |year=2011 | publisher=Oxford University Press | location=Oxford | isbn=978-0-19-957400-1}}</ref><ref name="H1">{{cite book | editor-first1= John | editor-last1= Harnad | title=Random Matrices, Random Processes and Integrable Systems | series=  CRM Series in Mathematical Physics | year=2011 | publisher=Springer Verlag | location=New York | isbn=978-1461428770 | doi=10.1007/978-1-4419-9514-8| arxiv= math/0603543 |  bibcode= 2011rmrp.book.....H | s2cid= 117785783 | last1= Dieng | first1= Momar | last2= Tracy | first2= Craig A. }}</ref><ref name="HB1">{{cite book | last1=Harnad | first1= J. |  last2=Balogh | first2= F. | title=Tau functions and Their Applications, Chapts. 11-12 | year=2021 | series =Cambridge Monographs on Mathematical Physics | publisher=Cambridge University Press | location = Cambridge, U.K.
 | isbn= 9781108610902 | doi=10.1017/9781108610902| s2cid= 222379146 }}</ref> and may also serve as [[generating function]]s, in the sense of [[combinatorics]] and [[enumerative geometry]], especially in relation to [[moduli spaces]] of [[Riemann surface]]s, and enumeration of [[branched covering]]s, or so-called [[ELSV formula|Hurwitz numbers]].<ref name="Pa">{{cite journal | last=Pandharipande | first=R. |title=The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere| journal=Letters in Mathematical Physics | publisher=Springer Science and Business Media LLC | volume=53 | issue=1 | year=2000 | issn=0377-9017 | doi=10.1023/a:1026571018707 | pages=59–74| s2cid=17477158 }}</ref><ref name="Ok">{{cite journal | last=Okounkov | first=Andrei | title=Toda equations for Hurwitz numbers | journal=Mathematical Research Letters | publisher=International Press of Boston | volume=7 | issue=4 | year=2000 | issn=1073-2780 | doi=10.4310/mrl.2000.v7.n4.a10 | pages=447–453| arxiv=math/0004128 | s2cid=55141973 }}</ref><ref name="HB2">{{cite book | last1=Harnad | first1= J. |  last2=Balogh | first2= F. | title=Tau functions and Their Applications, Chapts. 13-14 | year=2021 | series =Cambridge Monographs on Mathematical Physics | publisher=Cambridge University Press | location = Cambridge, U.K. | isbn= 9781108610902 | doi=10.1017/9781108610902| s2cid= 222379146 }}</ref>

There are two notions of <math>\tau </math>-functions, both introduced by the [[Mikio Sato|Sato]] school. The first is ''[[isospectral]] <math>\tau </math>-functions'' of the ''[[Mikio Sato|Sato]]–[[Graeme Segal|Segal]]–Wilson type''<ref name="Sato" /><ref name="SW">{{cite journal | last1=Segal | first1=Graeme | last2=Wilson | first2=George | title=Loop groups and equations of KdV type | journal=Publications mathématiques de l'IHÉS | publisher=Springer Science and Business Media LLC | volume=61 | issue=1 | year=1985 | issn=0073-8301 | doi=10.1007/bf02698802 | pages=5–65| s2cid=54967353 | url=http://www.numdam.org/item/PMIHES_1985__61__5_0/ }}</ref> for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying [[isospectral]] [[Lax pair|deformation equations]] of [[Lax pair|Lax]] type. The second is ''[[Isomonodromic deformation|isomonodromic]] <math>\tau </math>-functions''.<ref name="JMU">{{cite journal | last1=Jimbo | first1=Michio | last2=Miwa | first2=Tetsuji | last3=Ueno | first3=Kimio | title=Monodromy preserving deformation of linear ordinary differential equations with rational coefficients | journal=Physica D: Nonlinear Phenomena | publisher=Elsevier BV | volume=2 | issue=2 | year=1981 | issn=0167-2789 | doi=10.1016/0167-2789(81)90013-0 | pages=306–352}}</ref>

Depending on the specific application, a <math>\tau  </math>-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 equation|''Hamilton–Jacobi'']] approach to [[Integrable system|''Liouville integrable Hamiltonian systems'']], ''[[Hamilton's principal function]]'', evaluated on the level surfaces of a complete set of [[Poisson bracket|Poisson]] commuting invariants, plays a role similar to the <math>\tau </math>-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 <math> \tau </math>-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 [[Mikio Sato|Sato]]<ref name="Sato" /> and [[Graeme Segal|Segal]]-Wilson<ref name="SW" /> 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 (statistical mechanics)|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 deformation|Isomonodromic]] <math>\tau</math>-functions'' for linear systems of [[Isomonodromic deformation|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.<ref name="JMU"/>

===Hirota bilinear residue relation for KP tau functions===
{{anchor|Hirota bilinear residue relation}}

A KP ([[Kadomtsev–Petviashvili equation|Kadomtsev–Petviashvili]]) <math>\tau</math>-function <math>\tau(\mathbf{t})</math> 
is a function of an infinite collection <math>\mathbf{t}=(t_1, t_2, \dots)</math> of variables (called ''KP flow variables'') that satisfies the bilinear formal residue equation
{{NumBlk|::| <math> \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, </math>|{{EquationRef|1}}}}
identically in the <math> \delta t_j </math> variables, where <math>\mathrm{res}_{z=0}</math> is the 
<math>z^{-1}</math> coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in <math>z</math>, and

:<math display=block> {\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).  </math>

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

===Kadomtsev–Petviashvili equation===

If <math>\tau(t_1, t_2, t_3, \dots\dots) </math> is a KP <math>\tau </math>-function satisfying 
the Hirota residue equation ({{EquationNote|1}}) and we identify the first three flow variables as
::<math>t_1 =x, \quad t_2=y,\quad t_3 =t, </math>

it follows that the function
::<math>u(x,y,t):=2\frac{\partial^2}{\partial x^2}\log\left(\tau(x,y,t, t_4,\dots)\right) </math>

satisfies the <math>2 </math> (spatial)<math>  +1</math> (time) dimensional nonlinear partial differential equation
{{NumBlk|::| <math>3u_{yy}=\left(4u_t-6uu_x-u_{xxx}\right)_x, </math> |{{EquationRef|2}}}}
known as the [[Kadomtsev–Petviashvili equation|''Kadomtsev-Petviashvili'' (KP) ''equation'']]. This equation plays a prominent role in plasma physics and in shallow water ocean waves.

Taking further logarithmic derivatives of <math>\tau(t_1, t_2, t_3, \dots\dots)</math> 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 <math>{\bf t} =(t_1, t_2, \dots )</math>. These are collectively known as the ''KP hierarchy''.

=== Formal Baker–Akhiezer function and the KP hierarchy ===

If we define the (formal) Baker-Akhiezer function <math>\psi(z, \mathbf{t})</math>
by Sato's formula<ref name="Sato" /><ref name="DJKM1" /> 
:: <math> \psi(z, \mathbf{t}) :=
e^{\sum_{i=1}^\infty t_i z^i} 
\frac{\tau(\mathbf{t} - [z^{-1}])}{\tau(\mathbf{t})}</math>

and expand it as a formal series in the powers of the variable <math>z</math>
::<math>
\psi(z, \mathbf{t}) = e^{\sum_{i=1}^\infty t_i z^i} 
( 1 + \sum_{j=1}^\infty w_j(\mathbf{t}) z^{-j}), </math>

this satisfies an infinite sequence of compatible evolution equations
{{NumBlk|::| <math>
\frac{\partial \psi }{\partial t_i} =  \mathcal{D}_i \psi,  \quad i,j, = 1,2, \dots,
</math>|{{EquationRef|3}}}}

where <math> \mathcal{D}_i</math> is a linear ordinary differential operator of degree <math>i</math> 
in the variable <math>x:= t_1</math>, with coefficients that are functions of the flow variables 
<math> \mathbf{t}=(t_1, t_2, \dots)</math>, defined as follows
::<math>
\mathcal{D}_i := \big(\mathcal{L}^i\big)_+
</math>

where <math>\mathcal{L}</math> is the formal [[pseudo-differential operator]]
:: <math>
\mathcal{L} = \partial + \sum_{j=1}^\infty  u_j(\mathbf{t}) \partial^{-j}
= \mathcal{W} \circ\partial \circ{\mathcal{W}}^{-1}
</math>

with <math> \partial := \frac{\partial}{\partial x} </math>,
:: <math>
\mathcal{W} := 1 +\sum_{j=1}^\infty w_j(\mathbf{t}) \partial^{-j}
</math>

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

The pseudodifferential operator <math>\mathcal{L}</math> satisfies the infinite system of [[Lax pair|isospectral deformation equations]]
{{NumBlk|::| <math>
\frac{\partial\mathcal{L} }{\partial t_i} = [\mathcal{D}_i, \mathcal{L} ], \quad i, = 1,2, \dots
</math>|{{EquationRef|4}}}}

and the [[Frobenius theorem (differential topology)|compatibility conditions]] for both the system ({{EquationNote|3}}) and 
({{EquationNote|4}}) are
{{NumBlk|::|<math>
\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
</math> |{{EquationRef|5}}}}

This is a compatible infinite system of nonlinear partial differential equations, known as the ''KP (Kadomtsev-Petviashvili) hierarchy'', for the functions <math>\{u_j(\mathbf{t})\}_{j\in \mathbf{N}}</math>, with respect to the set <math>\mathbf{t}=(t_1, t_2, \dots)</math> of independent variables, each of which contains only a finite number of <math>u_j</math>'s, and derivatives only with respect to the three independent variables <math>(x, t_i, t_j)</math>. The first nontrivial case of these
is the [[Kadomtsev-Petviashvili equation]] ({{EquationNote|2}}).

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

==Isomonodromic systems. Isomonodromic tau functions==

===Fuchsian isomonodromic systems. Schlesinger equations===

Consider the [[overdetermined system]] of first order matrix partial differential equations
{{NumBlk|:| <math>{\partial \Psi \over \partial z}- \sum_{i=1}^n {N_i \over z - \alpha_i} \Psi=0, \quad </math>|{{EquationRef|6}}}}
{{NumBlk|:| <math> {\partial \Psi \over \partial \alpha_i}+ {N_i \over z - \alpha_i} \Psi=0, </math> |{{EquationRef|7}}}}
where <math> \{N_i\}_{i=1, \dots, n}</math> are a set of <math>n</math> <math>r\times r</math> traceless matrices, 
<math>\{\alpha_i\}_{i=1, \dots, n}</math> a set of <math>n</math> complex parameters, <math>z</math> a complex variable, and <math>\Psi(z, \alpha_1, \dots, \alpha_m)</math> is an invertible <math>r \times r</math> matrix valued function of <math>z</math> and <math>\{\alpha_i\}_{i=1, \dots, n}</math>.
These are the necessary and sufficient conditions for the based [[monodromy]] representation of the fundamental group
<math>\pi_1({\bf P}^1\backslash\{\alpha_i\}_{i=1, \dots, n})</math> of the Riemann sphere punctured at
the points <math>\{\alpha_i\}_{i=1, \dots, n}</math> corresponding to the rational covariant derivative operator 
:<math>{\partial  \over \partial z}- \sum_{i=1}^n {N_i \over z - \alpha_i}</math> 
to be independent of the parameters <math>\{\alpha_i\}_{i=1, \dots, n}</math>; i.e. that changes in these parameters induce an [[isomonodromic deformation]]. The [[Frobenius theorem (differential topology)|compatibility conditions]] for this system are the [[Schlesinger equations]]<ref name="JMU"/>
{{NumBlk|:|<math> {\partial N_i \over \partial \alpha_j} = {[N_i, N_j] \over \alpha_i-\alpha_j} \quad \text{for } i \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}. </math> |{{EquationRef|8}}}}

===Isomonodromic <math>\tau</math>-function===

Defining <math>n</math> functions
{{NumBlk|:|<math> H_i := \frac{1}{2} \sum_{1\le j \le n, j\neq i}{{\rm Tr}(N_i N_j) \over \alpha_i-\alpha_j}, \quad i=1, \dots ,n, </math>|{{EquationRef|9}}}} 
the [[Isomonodromic deformation|Schlesinger equations]] ({{EquationNote|8}}) imply that the differential form
:<math>\omega := \sum_{i=1}^n H_i d\alpha_i</math>
on the space of parameters is closed:
:<math>d\omega = 0</math>
and hence, locally exact. Therefore, at least locally, there exists a function 
<math>\tau(\alpha_1, \dots, \alpha_n)</math>
of the parameters, defined within a multiplicative constant, such that
:<math>\omega = d\mathrm{ln}\tau </math>
The function <math>\tau(\alpha_1, \dots, \alpha_n)</math> is called the ''isomonodromic <math>\tau</math>-function''
associated to the fundamental solution <math>\Psi</math> of the system ({{EquationNote|6}}), ({{EquationNote|7}}).

===Hamiltonian structure of the Schlesinger equations===
Defining the Lie [[Poisson bracket]]s on the space of <math>n</math>-tuples <math>\{N_i\}_{i=1, \dots, n}</math> of <math>r \times r </math> matrices:
:<math>\{(N_i)_{ab}, (N_j)_{c,d}\} = 
\delta_{ij}\left((N_i)_{ad}\delta_{bc} - (N_i)_{cb}\delta_{ad}\right)</math>
:<math> 1 \le i,j \le n, \quad 1\le a,b,c,d \le r, </math>
and viewing the <math>n</math> functions <math>\{H_i\}_{i=1, \dots,n}</math> defined in ({{EquationNote|9}}) as Hamiltonian functions on this Poisson space, the Schlesinger equations ({{EquationNote|8}})
may be expressed in Hamiltonian form as <ref name="H2">{{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 }}</ref>
<ref name="BHH">{{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 }}</ref>
:<math> \frac{\partial f(N_1, \dots, N_n)}{\partial \alpha_i} = \{f, H_i\}, \quad 1\le i \le n </math>
for any differentiable function <math>f(N_1, \dots, N_n)</math>.

=== Reduction of <math>r=2</math>, <math>n=3</math> case to [[Painleve equations|<math>P_{VI}</math>]]===

The simplest nontrivial case of the Schlesinger equations is when <math>r=2</math> and <math>n=3</math>. By applying a [[Mobius transformation|Möbius transformation]] to the variable <math>z</math>,
two of the finite poles may be chosen to be at <math>0</math> and <math>1</math>, and the third viewed as the independent variable.
Setting the sum <math> \sum_{i=1}^3 N_i</math> 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 <math>Gl(2)</math> conjugation, we obtain a system equivalent to the most generic case <math>P_{VI}</math> of the six [[Painleve equations|Painlevé transcendent equations]], for which many detailed classes of [[Painleve equations|explicit solutions]] are known.<ref name="FIKN">{{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}}</ref><ref name="CM">{{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 }}</ref><ref name="LT">{{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 }}</ref>

===Non-Fuchsian isomonodromic systems===

For non-Fuchsian systems, with higher order poles, the ''generalized'' monodromy data include ''[[Isomonodromic deformation#Extended monodromy data|Stokes matrices and connection matrices]]'', and there are further [[isomonodromic deformation]] parameters associated with the local asymptotics, but the ''isomonodromic <math> \tau </math>-functions'' may be defined in a similar way, using differentials on the extended parameter space.<ref name="JMU"/>
There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter <math>z</math> and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.<ref name="H2"/><ref name="BHH"/>

Taking all possible confluences of the poles appearing in ({{EquationNote|6}}) for the <math>r=2</math> and <math>n=3</math> case, including the one at <math>z=\infty</math>, and making the corresponding reductions, we obtain all other instances <math>P_{I} \cdots P_V</math>
of the [[Painleve equations|Painlevé transcendents]], for which
numerous [[Painleve equations|special solutions]] are also known.<ref name="FIKN"/><ref name="CM"/>

==Fermionic VEV (vacuum expectation value) representations ==

The [[Fock space|fermionic Fock space]] <math>\mathcal{F}</math>, is a semi-infinite exterior product space
<ref name="KP">{{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 }}</ref>

:<math>\mathcal{F} = \Lambda^{\infty/2}\mathcal{H} = \oplus_{n\in \mathbf{Z}}\mathcal{F}_n </math>
defined on a (separable) Hilbert space <math>\mathcal{H} </math> with basis elements 
<math>\{e_i\}_{i\in \mathbf{Z}}</math> and dual basis elements 
<math>\{e^i\}_{i\in \mathbf{Z}}</math> for <math>\mathcal{H}^* </math>.

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

: <math>\psi_i := e_i \wedge, \quad \psi^\dagger_i := i_{e^i}, \quad i \in \mathbf{Z},</math>
and satisfy the canonical anti-commutation relations

:<math> [\psi_i,\psi_k]_+ = [\psi^\dagger_i,\psi^\dagger_k]_+= 0, \quad  [\psi_i,\psi^\dagger_k]_+= \delta_{ij}.</math>

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

:<math>Q(u + \mu, w + \nu) := \nu(u) + \mu(v), \quad u,v \in \mathcal{H},\ \mu, \nu \in \mathcal{H}^* </math>

with the Fock space <math>\mathcal{F}</math> as irreducible module.
Denote the vacuum state, in the zero fermionic charge sector <math>\mathcal{F}_0</math>, as

: <math>|0\rangle := e_{-1}\wedge e_{-2} \wedge \cdots</math>,

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

:<math> \psi_{-j}|0 \rangle = 0, \quad \psi^{\dagger}_{j-1}|0 \rangle = 0, \quad j=0, 1, \dots </math>

The dual fermionic Fock space vacuum state, denoted <math>\langle 0 |</math>, is annihilated by the adjoint operators, acting to the left

:<math> \langle 0| \psi^\dagger_{-j} = 0, \quad  \langle 0 | \psi_{j-1}|0 = 0, \quad j=0, 1, \dots </math>

[[Normal ordering]] <math>: L_1, \cdots L_m:</math> 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
:<math> \langle 0 |: L_1, \cdots L_m:|0 \rangle =0. </math>
In particular, for a product <math>L_1 L_2</math> of a pair <math>(L_1, L_2)</math> of linear operators, one has

:<math> {:L_1 L_2:} = L_1 L_2 - \langle 0 | L_1 L_2|0 \rangle. </math>

The ''fermionic charge'' operator <math> C </math> is defined as

:<math> C = \sum_{i\in \mathbf{Z}} :\psi_i \psi^\dagger_i: </math>

The subspace <math>\mathcal{F}_n \subset \mathcal{F}</math> is the eigenspace of <math> C</math>
consisting of all eigenvectors with eigenvalue <math>n</math>

: <math> C | v; n\rangle = n | v; n\rangle, \quad \forall | v; n\rangle \in \mathcal{F}_n  </math>.

The standard orthonormal basis <math>\{|\lambda\rangle\}</math> for the zero fermionic charge sector <math>\mathcal{F}_0</math> is labelled by [[integer partition]]s
<math> \lambda = (\lambda_1, \dots, \lambda_{\ell(\lambda)})</math>,
where <math>\lambda_1\ge \cdots \ge \lambda_{\ell(\lambda)}</math>
is a weakly decreasing sequence of <math>\ell(\lambda)</math> positive integers, which can equivalently be represented by a [[Young diagram]], as depicted here for the partition 
<math>(5, 4, 1)</math>.

:[[Image:Young diagram for 541 partition.svg|thumb|center|150px|Young diagram of the partition (5, 4, 1)]]

An alternative notation for a partition <math>\lambda</math> consists of the 
[[Ferdinand Georg Frobenius|Frobenius]] indices
<math>(\alpha_1, \dots \alpha_r | \beta_1, \dots \beta _r)</math>, where <math>\alpha_i</math>
denotes the ''arm length''; i.e. the number <math>\lambda_i -i</math> of boxes in the Young diagram to the right of the <math>i</math>'th diagonal box, <math>\beta_i</math> denotes the ''leg length'', i.e. the number of boxes in the Young diagram below the <math>i</math>'th diagonal box, for <math>i=1, \dots, r</math>, where <math>r</math> is the ''Frobenius rank'', which is the number of elements along the principal diagonal.

The basis element <math>|\lambda\rangle</math> is then given by acting on the vacuum with a product
of <math>r</math> pairs of creation and annihilation operators, labelled by the Frobenius indices

: <math> |\lambda\rangle = (-1)^{\sum_{j=1}^r \beta_j} 
\prod_{k=1}^r \big(\psi_{\alpha_k} \psi^\dagger_{-\beta_k-1}\big)| 0 \rangle. </math>

The integers <math>\{\alpha_i\}_{i=1, \dots, r}</math> indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while 
<math>\{-\beta_i-1\}_{i=1, \dots, r}</math> 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''.<ref name="Sato"/>

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

: <math>\langle \mu|\lambda\rangle = \delta_{\lambda, \mu} </math>

Any KP <math>\tau</math>-function can be expressed as a sum
{{NumBlk|:| <math>\tau_w(\mathbf{t}) = \sum_{\lambda} \pi_\lambda(w) s_\lambda(\mathbf{t}), </math> | {{EquationRef|10}}}}

where <math>\mathbf{t} = (t_1, t_2, \dots, \dots)</math> are the KP flow variables,
<math>s_\lambda(\mathbf{t})</math> is the [[Schur polynomial|Schur function]] 
corresponding to the partition <math>\lambda</math>, viewed as a function of the normalized power sum variables

:<math> t_i := [\mathbf{x}]_i := \frac{1}{i} \sum_{a=1}^n x_a^i \quad i = 1,2, \dots </math>

in terms of an auxiliary (finite or infinite) sequence of variables 
<math>\mathbf{x}:=(x_1, \dots, x_N)</math> and the constant coefficients 
<math>\pi_\lambda(w)</math> may be viewed as the [[Plücker coordinates]] of an 
element <math>w\in \mathrm{Gr}_{\mathcal{H}_+}(\mathcal{H}) </math> 
of the infinite dimensional Grassmannian consisting of the orbit, under the action of 
the general linear group <math>\mathrm{Gl}(\mathcal{H})</math>, of the subspace 
<math>\mathcal{H}_+ = \mathrm{span}\{e_{-i}\}_{i \in \mathbf{N}} \subset \mathcal{H} </math> 
of the Hilbert space <math>\mathcal{H}</math>.

This corresponds, under the ''Bose-Fermi correspondence'', to a [[Exterior algebra|decomposable]] element

:<math> |\tau_w\rangle = \sum_{\lambda} \pi_{\lambda}(w) |\lambda \rangle </math>

of the Fock space <math>\mathcal{F}_0</math> which, up to projectivization, is the image of the Grassmannian element <math>w\in \mathrm{Gr}_{\mathcal{H}_+}(\mathcal{H}) </math> under the 
[[Plucker coordinates|Plücker map]]

:<math> \mathcal{Pl}: \mathrm{span}(w_1, w_2, \dots ) 
\longrightarrow [w_1 \wedge w_2 \wedge \cdots ]= [|\tau_w\rangle], </math>

where <math>(w_1, w_2, \dots )</math> is a basis for the subspace 
<math>w\subset \mathcal{H}</math> and <math>[ \cdots]</math> denotes projectivization of
an element of <math>\mathcal{F}</math>.

The Plücker coordinates <math>\{\pi_\lambda(w)\}</math> satisfy an infinite set of bilinear
relations, the [[Plücker relations]], defining the image of the [[Plücker embedding]] 
into the projectivization <math>\mathbf{P}(\mathcal{F})</math> of the fermionic Fock space,
which are equivalent to the Hirota bilinear residue relation ({{EquationNote|1}}).

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

:<math>\tau_w(\mathbf{t}) = \langle 0 | \hat{\gamma}_+(\mathbf{t}) \hat{g} | 0 \rangle,   </math>

where

:<math>\Gamma_+ =\{\hat{\gamma}_+(\mathbf{t}) = e^{\sum_{i=1}^\infty t_i J_i}\} 
\subset \mathrm{Gl}(\mathcal{H}) </math>

is the abelian subgroup of <math>\mathrm{Gl}(\mathcal{H})</math> that generates the KP flows, and

:<math> J_i := \sum_{j\in \mathbf{Z}} \psi_j \psi^\dagger_{j+i}, \quad i=1,2 \dots </math>

are the ""current"" components.

== Examples of solutions to the equations of the KP hierarchy ==

=== Schur functions ===

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

=== Multisoliton solutions===

If we choose <math>3N</math> complex constants
<math>\{\alpha_k, \beta_k, \gamma_k\}_{k=1, \dots, N}</math>
with <math>\alpha_k, \beta_k</math>'s all distinct, <math>\gamma_k \ne 0</math>, and define the functions
:<math>
 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,
</math>
we arrive at the Wronskian determinant formula
:<math>
\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},
</math>
which gives the general [[Solitons|<math>N</math>-soliton]] <math>\tau</math>-function.<ref name="DJKM1"/><ref name="DJKM2"/><ref name="HB3">{{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 }}</ref>

=== Theta function solutions associated to algebraic curves ===

Let <math>X</math> be a compact Riemann surface of genus <math>g</math> and fix a canonical homology basis <math>a_1, \dots, a_g, b_1, \dots, b_g</math>
of <math>H_1(X,\mathbf{Z})</math> with intersection numbers 
:<math>
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.
</math>
Let <math>\{\omega_i\}_{i=1, \dots, g}</math> be a basis for the space <math>H^1(X)</math> of [[Differential of the first kind|holomorphic differentials]] satisfying the standard normalization conditions
:<math>
\oint_{a_i} \omega_j =\delta_{ij}, \quad \oint_{b_j }\omega_j = B_{ij},
</math>

where <math>B</math> is the ''[[Schottky problem|Riemann matrix]]'' of periods. 
The matrix <math>B</math> belongs to the ''Siegel upper half space'' 
:<math>
\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\}.
</math>

The [[Theta function|Riemann <math>\theta</math> function]] on <math>\mathbf{C}^g</math> corresponding to the [[Period mapping|period matrix]] <math>B</math> is defined to be
:<math> \theta(Z | B) := \sum_{N\in \Z^g} e^{i\pi (N, B N) + 2i\pi (N, Z)}.
</math>

Choose a point <math>p_\infty \in X</math>, a local parameter <math>\zeta</math> in a neighbourhood of <math>p_{\infty}</math> with <math>\zeta(p_\infty)=0</math> and 
a positive [[divisor]] of degree <math>g</math> 
:<math>
\mathcal{D}:= \sum_{i=1}^g p_i,\quad p_i \in X.
</math>
For any positive integer <math>k\in \mathbf{N}^+</math> let <math>\Omega_k</math> be the unique [[Differential of the first kind|meromorphic differential]] of the second kind characterized by the following conditions:

* The only singularity of <math>\Omega_k</math> is a pole of order <math>k+1</math> at <math>p=p_\infty</math> with vanishing residue.
* The expansion of <math>\Omega_k</math> around <math>p=p_{\infty}</math> is
*:<math>\Omega_k = d(\zeta^{-k} ) + \sum_{j=1}^\infty Q_{ij} \zeta^j d\zeta</math>.
* <math>\Omega_k</math> is normalized to have vanishing <math>a</math>-cycles:
*:<math> \oint_{a_i }\Omega_j =0. </math>

Denote by <math>\mathbf{U}_k \in \mathbf{C}^g</math> the vector of <math>b</math>-cycles of <math>\Omega_k</math>:
:<math>(\mathbf{U}_k)_j := \oint_{b_j} \Omega_k.</math>

Denote the image of <math>{\mathcal D}</math> under the [[Abel–Jacobi map|Abel]] map 
<math>\mathcal{A}: \mathcal{S}^g(X) \to \mathbf{C}^g </math>  
:<math> \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 </math>
with arbitrary base point <math>p_0</math>.

Then the following is a KP <math>\tau</math>-function:<ref name="Dub">{{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  }}</ref> 
:<math> \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) </math>.

=== Matrix model partition functions as KP <math>\tau</math>-functions ===

Let <math>d\mu_0(M)</math> be the Lebesgue measure on the <math>N^2</math> dimensional space <math>{\mathbf H}^{N\times N}</math> of <math>N\times N</math> complex Hermitian matrices.
Let <math>\rho(M)</math> be a conjugation invariant integrable density function
:<math>
\rho(U M U^{\dagger}) = \rho(M), \quad U\in U(N).
</math>
Define a deformation family of measures
:<math>
d\mu_{N,\rho}(\mathbf{t}) := e^{\text{ Tr }(\sum_{i=1}^\infty t_i M^i)} \rho(M) d\mu_0 (M)
</math>
for small <math>\mathbf{t}= (t_1, t_2, \cdots)</math> and let
:<math>
\tau_{N,\rho}({\bf t}):= \int_{{\mathbf H}^{N\times N} }d\mu_{N,\rho}({\bf t}).
</math>
be the [[Partition function (statistical mechanics)|partition function]] for this [[Random matrix|random matrix model]].<ref>M.L. Mehta, "Random Matrices",  3rd ed., vol. 142 of ''Pure and Applied Mathematics'', Elsevier, Academic Press, {{ISBN|9780120884094}} (2004)</ref><ref name="AkBaDiFr"/>
Then <math>\tau_{N,\rho}(\mathbf{t})</math> satisfies the bilinear Hirota residue equation ({{EquationNote|1}}), and hence is a <math>\tau </math>-function of the KP hierarchy.<ref>{{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 }}</ref>

=== <math>\tau</math>-functions of hypergeometric type. Generating function for Hurwitz numbers ===

Let <math>\{r_i\}_{i\in \mathbf{Z}}</math> be a (doubly) infinite sequence of complex numbers.
For any integer partition <math> \lambda = (\lambda_1, \dots, \lambda_{\ell(\lambda)})</math> define the ''content product'' coefficient
:<math>r_{\lambda} := \prod_{(i,j)\in \lambda} r_{j-i}</math>,
where the product is over all pairs <math>(i,j)</math> of positive integers that correspond to boxes of the Young diagram of the partition <math> \lambda </math>, viewed as positions of matrix elements of the corresponding 
<math>\ell(\lambda) \times \lambda_1</math> matrix.
Then, for every pair of infinite sequences <math> \mathbf{t} = (t_1, t_2, \dots )</math> and <math> \mathbf{s} = (s_1, s_2, \dots )</math> of complex variables, viewed as (normalized) power sums <math> \mathbf{t} = [\mathbf{x}], \ \mathbf{s} = [\mathbf{y}]</math>
of the infinite sequence of auxiliary variables 
::<math> \mathbf{x} = (x_1, x_2, \dots )</math> and <math> \mathbf{y} = (y_1, y_2, \dots )</math>,

defined by: 
::<math> 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</math>,

the function
{{NumBlk|:|<math> \tau^r(\mathbf{t},\mathbf{s}) := \sum_{\lambda}r_\lambda s_\lambda(\mathbf{t})s_\lambda(\mathbf{s}) </math>|{{EquationRef|11}}}}
is a ''double'' KP <math>\tau </math>-function, both in the <math> \mathbf{t}</math> and the <math> \mathbf{s}</math> variables, known as a <math>\tau </math>-function of ''hypergeometric type''.<ref>{{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 }}</ref>

In particular, choosing
:<math>r_j = r^{\beta}_j := e^{j\beta}</math>
for some small parameter <math>\beta</math>, denoting the corresponding content product coefficient as <math> r_\lambda^\beta</math>
and setting 
::<math>\mathbf{s} = (1, 0, \dots)=: \mathbf{t}_0</math>,

the resulting <math> \tau </math>-function can be equivalently expanded as
{{NumBlk|::|<math>
\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}),
</math>|{{EquationRef|12}}}}
where <math>\{H_d(\lambda)\}</math> are the ''simple Hurwitz numbers'', which are 
<math> \frac{1}{n!}</math> times the number of ways in which an element 
<math>k_\lambda \in \mathcal{S}_{n}</math> of the symmetric group <math>\mathcal{S}_{n}</math> in <math>n=|\lambda|</math> elements, with cycle lengths equal to the parts of the partition <math>\lambda</math>, can be factorized as a product of <math>d</math> <math>2</math>-cycles
::<math> k_\lambda = (a_1 b_1)\dots (a_d b_d)</math>,
and 
::<math>  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^i_a = i t_i  </math> 
is the power sum symmetric function. Equation ({{EquationNote|12}}) thus shows that the (formal) KP hypergeometric <math>\tau </math>-function ({{EquationNote|11}}) corresponding to the content product coefficients  <math> r_\lambda^\beta</math> is a generating function, in the combinatorial sense, for simple Hurwitz numbers.<ref name ="Pa"/><ref name = "Ok"/><ref name = "HB2"/>

==References==

<references />

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[[Category:Dynamical systems]]
[[Category:Mathematical physics]]
[[Category:Integrable systems]]
[[Category:Solitons]]
[[Category:Special functions]]
[[Category:Generating functions]]
[[Category:Partition functions]]
[[Category:Random matrices]]
[[Category:Combinatorics]]