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Tau function (integrable systems)
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<h2 id="tau-functions-isospectral-and-isomonodromic">Tau functions: isospectral and isomonodromic</h2> <p>A τ {\displaystyle \tau } -function of <a href="/facts/Isospectral/vvYRzPqT">isospectral</a> type is defined as a solution of the Hirota bilinear equations (see § Hirota bilinear residue relation for KP tau functions below), from which the linear operator undergoing isospectral evolution can be uniquely reconstructed. Geometrically, in the <a href="/facts/Mikio_Sato/LUkFh98o">Sato</a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> and <a href="/facts/Graeme_Segal/M92u6ecE">Segal</a>-Wilson<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> sense, it is the value of the determinant of a <a href="/facts/Fredholm_integral_operator/euHGJbQo">Fredholm integral operator</a>, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) <a href="/facts/Grassmann_manifold/QcmgLRq8">Grassmann manifold</a> onto the <i>origin</i>, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group. It typically arises as a <a href="/facts/Partition_function_(statistical_mechanics)/q65sJrrn">partition function</a>, in the sense of <a href="/facts/Statistical_mechanics/6zMsNYYG">statistical mechanics</a>, many-body <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> or <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theory</a>, as the underlying measure undergoes a linear exponential deformation. </p><p><i><a href="/facts/Isomonodromic_deformation/9S7RmUsU">Isomonodromic</a> τ {\displaystyle \tau } -functions</i> for linear systems of <a href="/facts/Isomonodromic_deformation/9S7RmUsU">Fuchsian</a> type are defined below in § 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.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h3>Hirota bilinear residue relation for KP tau functions</h3> <p>A KP (<a href="/facts/Kadomtsev%E2%80%93Petviashvili_equation/cL7nX4ER">Kadomtsev–Petviashvili</a>) τ {\displaystyle \tau } -function τ ( t ) {\displaystyle \tau (\mathbf {t} )} is a function of an infinite collection t = ( t 1 , t 2 , … ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )} of variables (called <i>KP flow variables</i>) that satisfies the bilinear formal residue equation </p> <table><tbody><tr><td> r e s z = 0 ( e ∑ i = 1 ∞ ( δ t i ) z i τ ( t − [ z − 1 ] ) τ ( s + [ z − 1 ] ) ) d z ≡ 0 , {\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,} </td> <td></td> <td>1</td></tr></tbody></table> <p>identically in the δ t j {\displaystyle \delta t_{j}} variables, where r e s z = 0 {\displaystyle \mathrm {res} _{z=0}} is the z − 1 {\displaystyle z^{-1}} coefficient in the formal Laurent expansion resulting from expanding all factors as Laurent series in z {\displaystyle z} , and </p> s := t + ( δ t 1 , δ t 2 , ⋯ ) , [ z − 1 ] := ( z − 1 , z − 2 2 , ⋯ z − j j , ⋯ ) . {\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 ).} <p>As explained below in the section § Formal Baker-Akhiezer function and the KP hierarchy, every such τ {\displaystyle \tau } -function determines a set of solutions to the equations of the KP hierarchy. </p> <h3>Kadomtsev–Petviashvili equation</h3> <p>If τ ( t 1 , t 2 , t 3 , … … ) {\displaystyle \tau (t_{1},t_{2},t_{3},\dots \dots )} is a KP τ {\displaystyle \tau } -function satisfying the Hirota residue equation (1) and we identify the first three flow variables as </p> t 1 = x , t 2 = y , t 3 = t , {\displaystyle t_{1}=x,\quad t_{2}=y,\quad t_{3}=t,} <p>it follows that the function </p> u ( x , y , t ) := 2 ∂ 2 ∂ x 2 log ( τ ( x , y , t , t 4 , … ) ) {\displaystyle u(x,y,t):=2{\frac {\partial ^{2}}{\partial x^{2}}}\log \left(\tau (x,y,t,t_{4},\dots )\right)} <p>satisfies the 2 {\displaystyle 2} (spatial) + 1 {\displaystyle +1} (time) dimensional nonlinear partial differential equation </p> <table><tbody><tr><td> 3 u y y = ( 4 u t − 6 u u x − u x x x ) x , {\displaystyle 3u_{yy}=\left(4u_{t}-6uu_{x}-u_{xxx}\right)_{x},} </td> <td></td> <td>2</td></tr></tbody></table> <p>known as the <a href="/facts/Kadomtsev%E2%80%93Petviashvili_equation/cL7nX4ER"><i>Kadomtsev-Petviashvili</i> (KP) <i>equation</i></a>. This equation plays a prominent role in plasma physics and in shallow water ocean waves. </p><p>Taking further logarithmic derivatives of τ ( t 1 , t 2 , t 3 , … … ) {\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 t = ( t 1 , t 2 , … ) {\displaystyle {\bf {t}}=(t_{1},t_{2},\dots )} . These are collectively known as the <i>KP hierarchy</i>. </p> <h3>Formal Baker–Akhiezer function and the KP hierarchy</h3> <p>If we define the (formal) Baker-Akhiezer function ψ ( z , t ) {\displaystyle \psi (z,\mathbf {t} )} by Sato's formula<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> ψ ( z , t ) := e ∑ i = 1 ∞ t i z i τ ( t − [ z − 1 ] ) τ ( t ) {\displaystyle \psi (z,\mathbf {t} ):=e^{\sum _{i=1}^{\infty }t_{i}z^{i}}{\frac {\tau (\mathbf {t} -[z^{-1}])}{\tau (\mathbf {t} )}}} <p>and expand it as a formal series in the powers of the variable z {\displaystyle z} </p> ψ ( z , t ) = e ∑ i = 1 ∞ t i z i ( 1 + ∑ j = 1 ∞ w j ( t ) z − j ) , {\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}),} <p>this satisfies an infinite sequence of compatible evolution equations </p> <table><tbody><tr><td> ∂ ψ ∂ t i = D i ψ , i , j , = 1 , 2 , … , {\displaystyle {\frac {\partial \psi }{\partial t_{i}}}={\mathcal {D}}_{i}\psi ,\quad i,j,=1,2,\dots ,} </td> <td></td> <td>3</td></tr></tbody></table> <p>where D i {\displaystyle {\mathcal {D}}_{i}} is a linear ordinary differential operator of degree i {\displaystyle i} in the variable x := t 1 {\displaystyle x:=t_{1}} , with coefficients that are functions of the flow variables t = ( t 1 , t 2 , … ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )} , defined as follows </p> D i := ( L i ) + {\displaystyle {\mathcal {D}}_{i}:={\big (}{\mathcal {L}}^{i}{\big )}_{+}} <p>where L {\displaystyle {\mathcal {L}}} is the formal <a href="/facts/Pseudo-differential_operator/wEPLVm5c">pseudo-differential operator</a> </p> L = ∂ + ∑ j = 1 ∞ u j ( t ) ∂ − j = W ∘ ∂ ∘ W − 1 {\displaystyle {\mathcal {L}}=\partial +\sum _{j=1}^{\infty }u_{j}(\mathbf {t} )\partial ^{-j}={\mathcal {W}}\circ \partial \circ {\mathcal {W}}^{-1}} <p>with ∂ := ∂ ∂ x {\displaystyle \partial :={\frac {\partial }{\partial x}}} , </p> W := 1 + ∑ j = 1 ∞ w j ( t ) ∂ − j {\displaystyle {\mathcal {W}}:=1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )\partial ^{-j}} <p>is the <i>wave operator</i> and ( L i ) + {\displaystyle {\big (}{\mathcal {L}}^{i}{\big )}_{+}} denotes the projection to the part of L i {\displaystyle {\mathcal {L}}^{i}} containing purely non-negative powers of ∂ {\displaystyle \partial } ; i.e. the differential operator part of L i {\displaystyle {\mathcal {L}}^{i}} . </p><p>The pseudodifferential operator L {\displaystyle {\mathcal {L}}} satisfies the infinite system of <a href="/facts/Lax_pair/dyzB7Ttb">isospectral deformation equations</a> </p> <table><tbody><tr><td> ∂ L ∂ t i = [ D i , L ] , i , = 1 , 2 , … {\displaystyle {\frac {\partial {\mathcal {L}}}{\partial t_{i}}}=[{\mathcal {D}}_{i},{\mathcal {L}}],\quad i,=1,2,\dots } </td> <td></td> <td>4</td></tr></tbody></table> <p>and the <a href="/facts/Frobenius_theorem_(differential_topology)/A8gu0fjt">compatibility conditions</a> for both the system (3) and (4) are </p> <table><tbody><tr><td> ∂ D i ∂ t j − ∂ D j ∂ t i + [ D i , D j ] = 0 , i , j , = 1 , 2 , … {\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 } </td> <td></td> <td>5</td></tr></tbody></table> <p>This is a compatible infinite system of nonlinear partial differential equations, known as the <i>KP (Kadomtsev-Petviashvili) hierarchy</i>, for the functions { u j ( t ) } j ∈ N {\displaystyle \{u_{j}(\mathbf {t} )\}_{j\in \mathbf {N} }} , with respect to the set t = ( t 1 , t 2 , … ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )} of independent variables, each of which contains only a finite number of u j {\displaystyle u_{j}} 's, and derivatives only with respect to the three independent variables ( x , t i , t j ) {\displaystyle (x,t_{i},t_{j})} . The first nontrivial case of these is the <a href="/facts/Kadomtsev-Petviashvili_equation/cL7nX4ER">Kadomtsev-Petviashvili equation</a> (2). </p><p>Thus, every KP τ {\displaystyle \tau } -function provides a solution, at least in the formal sense, of this infinite system of nonlinear partial differential equations. </p> <h2 id="isomonodromic-systems-isomonodromic-tau-functions">Isomonodromic systems. Isomonodromic tau functions</h2> <h3>Fuchsian isomonodromic systems. Schlesinger equations</h3> <p>Consider the <a href="/facts/Overdetermined_system/DMWk4pQR">overdetermined system</a> of first order matrix partial differential equations </p> <table><tbody><tr><td> ∂ Ψ ∂ z − ∑ i = 1 n N i z − α i Ψ = 0 , {\displaystyle {\partial \Psi \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}\Psi =0,\quad } </td> <td></td> <td>6</td></tr></tbody></table> <table><tbody><tr><td> ∂ Ψ ∂ α i + N i z − α i Ψ = 0 , {\displaystyle {\partial \Psi \over \partial \alpha _{i}}+{N_{i} \over z-\alpha _{i}}\Psi =0,} </td> <td></td> <td>7</td></tr></tbody></table> <p>where { N i } i = 1 , … , n {\displaystyle \{N_{i}\}_{i=1,\dots ,n}} are a set of n {\displaystyle n} r × r {\displaystyle r\times r} traceless matrices, { α i } i = 1 , … , n {\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}} a set of n {\displaystyle n} complex parameters, z {\displaystyle z} a complex variable, and Ψ ( z , α 1 , … , α m ) {\displaystyle \Psi (z,\alpha _{1},\dots ,\alpha _{m})} is an invertible r × r {\displaystyle r\times r} matrix valued function of z {\displaystyle z} and { α i } i = 1 , … , n {\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}} . These are the necessary and sufficient conditions for the based <a href="/facts/Monodromy/ldkpxExM">monodromy</a> representation of the fundamental group π 1 ( P 1 ∖ { α i } i = 1 , … , n ) {\displaystyle \pi _{1}({\bf {P}}^{1}\backslash \{\alpha _{i}\}_{i=1,\dots ,n})} of the Riemann sphere punctured at the points { α i } i = 1 , … , n {\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}} corresponding to the rational covariant derivative operator </p> ∂ ∂ z − ∑ i = 1 n N i z − α i {\displaystyle {\partial \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}} <p>to be independent of the parameters { α i } i = 1 , … , n {\displaystyle \{\alpha _{i}\}_{i=1,\dots ,n}} ; i.e. that changes in these parameters induce an <a href="/facts/Isomonodromic_deformation/9S7RmUsU">isomonodromic deformation</a>. The <a href="/facts/Frobenius_theorem_(differential_topology)/A8gu0fjt">compatibility conditions</a> for this system are the <a href="/facts/Schlesinger_equations/9S7RmUsU">Schlesinger equations</a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p> <table><tbody><tr><td> ∂ N i ∂ α j = [ N i , N j ] α i − α j for i ≠ j , ∂ N i ∂ α i = − ∑ 1 ≤ j ≤ n , j ≠ i [ N i , N j ] α i − α j . {\displaystyle {\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\leq j\leq n,j\neq i}{[N_{i},N_{j}] \over \alpha _{i}-\alpha _{j}}.} </td> <td></td> <td>8</td></tr></tbody></table> <h3>Isomonodromic τ {\displaystyle \tau } -function</h3> <p>Defining n {\displaystyle n} functions </p> <table><tbody><tr><td> H i := 1 2 ∑ 1 ≤ j ≤ n , j ≠ i T r ( N i N j ) α i − α j , i = 1 , … , n , {\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,} </td> <td></td> <td>9</td></tr></tbody></table> <p>the <a href="/facts/Isomonodromic_deformation/9S7RmUsU">Schlesinger equations</a> (8) imply that the differential form </p> ω := ∑ i = 1 n H i d α i {\displaystyle \omega :=\sum _{i=1}^{n}H_{i}d\alpha _{i}} <p>on the space of parameters is closed: </p> d ω = 0 {\displaystyle d\omega =0} <p>and hence, locally exact. Therefore, at least locally, there exists a function τ ( α 1 , … , α n ) {\displaystyle \tau (\alpha _{1},\dots ,\alpha _{n})} of the parameters, defined within a multiplicative constant, such that </p> ω = d l n τ {\displaystyle \omega =d\mathrm {ln} \tau } <p>The function τ ( α 1 , … , α n ) {\displaystyle \tau (\alpha _{1},\dots ,\alpha _{n})} is called the <i>isomonodromic τ {\displaystyle \tau } -function</i> associated to the fundamental solution Ψ {\displaystyle \Psi } of the system (6), (7). </p> <h3>Hamiltonian structure of the Schlesinger equations</h3> <p>Defining the Lie <a href="/facts/Poisson_bracket/7WUH7VXV">Poisson brackets</a> on the space of n {\displaystyle n} -tuples { N i } i = 1 , … , n {\displaystyle \{N_{i}\}_{i=1,\dots ,n}} of r × r {\displaystyle r\times r} matrices: </p> { ( N i ) a b , ( N j ) c , d } = δ i j ( ( N i ) a d δ b c − ( N i ) c b δ a d ) {\displaystyle \{(N_{i})_{ab},(N_{j})_{c,d}\}=\delta _{ij}\left((N_{i})_{ad}\delta _{bc}-(N_{i})_{cb}\delta _{ad}\right)} 1 ≤ i , j ≤ n , 1 ≤ a , b , c , d ≤ r , {\displaystyle 1\leq i,j\leq n,\quad 1\leq a,b,c,d\leq r,} <p>and viewing the n {\displaystyle n} functions { H i } i = 1 , … , n {\displaystyle \{H_{i}\}_{i=1,\dots ,n}} defined in (9) as Hamiltonian functions on this Poisson space, the Schlesinger equations (8) may be expressed in Hamiltonian form as <a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> <a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> ∂ f ( N 1 , … , N n ) ∂ α i = { f , H i } , 1 ≤ i ≤ n {\displaystyle {\frac {\partial f(N_{1},\dots ,N_{n})}{\partial \alpha _{i}}}=\{f,H_{i}\},\quad 1\leq i\leq n} <p>for any differentiable function f ( N 1 , … , N n ) {\displaystyle f(N_{1},\dots ,N_{n})} . </p> <h3>Reduction of r = 2 {\displaystyle r=2} , n = 3 {\displaystyle n=3} case to <a href="/facts/Painleve_equations/wIkH4Y1G"> P V I {\displaystyle P_{VI}} </a></h3> <p>The simplest nontrivial case of the Schlesinger equations is when r = 2 {\displaystyle r=2} and n = 3 {\displaystyle n=3} . By applying a <a href="/facts/Mobius_transformation/YEkTTLfK">Möbius transformation</a> to the variable z {\displaystyle z} , two of the finite poles may be chosen to be at 0 {\displaystyle 0} and 1 {\displaystyle 1} , and the third viewed as the independent variable. Setting the sum ∑ i = 1 3 N i {\displaystyle \sum _{i=1}^{3}N_{i}} of the matrices appearing in (6), which is an invariant of the Schlesinger equations, equal to a constant, and quotienting by its stabilizer under G l ( 2 ) {\displaystyle Gl(2)} conjugation, we obtain a system equivalent to the most generic case P V I {\displaystyle P_{VI}} of the six <a href="/facts/Painleve_equations/wIkH4Y1G">Painlevé transcendent equations</a>, for which many detailed classes of <a href="/facts/Painleve_equations/wIkH4Y1G">explicit solutions</a> are known.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> </p> <h3>Non-Fuchsian isomonodromic systems</h3> <p>For non-Fuchsian systems, with higher order poles, the <i>generalized</i> monodromy data include <i><a href="/facts/Isomonodromic_deformation/9S7RmUsU">Stokes matrices and connection matrices</a></i>, and there are further <a href="/facts/Isomonodromic_deformation/9S7RmUsU">isomonodromic deformation</a> parameters associated with the local asymptotics, but the <i>isomonodromic τ {\displaystyle \tau } -functions</i> may be defined in a similar way, using differentials on the extended parameter space.<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a> There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter z {\displaystyle z} and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> </p><p>Taking all possible confluences of the poles appearing in (6) for the r = 2 {\displaystyle r=2} and n = 3 {\displaystyle n=3} case, including the one at z = ∞ {\displaystyle z=\infty } , and making the corresponding reductions, we obtain all other instances P I ⋯ P V {\displaystyle P_{I}\cdots P_{V}} of the <a href="/facts/Painleve_equations/wIkH4Y1G">Painlevé transcendents</a>, for which numerous <a href="/facts/Painleve_equations/wIkH4Y1G">special solutions</a> are also known.<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> </p> <h2 id="fermionic-vev-vacuum-expectation-value-representations">Fermionic VEV (vacuum expectation value) representations</h2> <p>The <a href="/facts/Fock_space/I2PYLlhl">fermionic Fock space</a> F {\displaystyle {\mathcal {F}}} , is a semi-infinite exterior product space <a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> </p> F = Λ ∞ / 2 H = ⊕ n ∈ Z F n {\displaystyle {\mathcal {F}}=\Lambda ^{\infty /2}{\mathcal {H}}=\oplus _{n\in \mathbf {Z} }{\mathcal {F}}_{n}} <p>defined on a (separable) Hilbert space H {\displaystyle {\mathcal {H}}} with basis elements { e i } i ∈ Z {\displaystyle \{e_{i}\}_{i\in \mathbf {Z} }} and dual basis elements { e i } i ∈ Z {\displaystyle \{e^{i}\}_{i\in \mathbf {Z} }} for H ∗ {\displaystyle {\mathcal {H}}^{*}} . </p><p>The free fermionic creation and annihilation operators { ψ j , ψ j † } j ∈ Z {\displaystyle \{\psi _{j},\psi _{j}^{\dagger }\}_{j\in \mathbf {Z} }} act as endomorphisms on F {\displaystyle {\mathcal {F}}} via exterior and interior multiplication by the basis elements </p> ψ i := e i ∧ , ψ i † := i e i , i ∈ Z , {\displaystyle \psi _{i}:=e_{i}\wedge ,\quad \psi _{i}^{\dagger }:=i_{e^{i}},\quad i\in \mathbf {Z} ,} <p>and satisfy the canonical anti-commutation relations </p> [ ψ i , ψ k ] + = [ ψ i † , ψ k † ] + = 0 , [ ψ i , ψ k † ] + = δ i j . {\displaystyle [\psi _{i},\psi _{k}]_{+}=[\psi _{i}^{\dagger },\psi _{k}^{\dagger }]_{+}=0,\quad [\psi _{i},\psi _{k}^{\dagger }]_{+}=\delta _{ij}.} <p>These generate the standard fermionic representation of the Clifford algebra on the direct sum H + H ∗ {\displaystyle {\mathcal {H}}+{\mathcal {H}}^{*}} , corresponding to the scalar product </p> Q ( u + μ , w + ν ) := ν ( u ) + μ ( v ) , u , v ∈ H , μ , ν ∈ H ∗ {\displaystyle Q(u+\mu ,w+\nu ):=\nu (u)+\mu (v),\quad u,v\in {\mathcal {H}},\ \mu ,\nu \in {\mathcal {H}}^{*}} <p>with the Fock space F {\displaystyle {\mathcal {F}}} as irreducible module. Denote the vacuum state, in the zero fermionic charge sector F 0 {\displaystyle {\mathcal {F}}_{0}} , as </p> | 0 ⟩ := e − 1 ∧ e − 2 ∧ ⋯ {\displaystyle |0\rangle :=e_{-1}\wedge e_{-2}\wedge \cdots } , <p>which corresponds to the <i><a href="/facts/Dirac_sea/x40KMcu3">Dirac sea</a></i> of states along the real integer lattice in which all negative integer locations are occupied and all non-negative ones are empty. </p><p>This is annihilated by the following operators </p> ψ − j | 0 ⟩ = 0 , ψ j − 1 † | 0 ⟩ = 0 , j = 0 , 1 , … {\displaystyle \psi _{-j}|0\rangle =0,\quad \psi _{j-1}^{\dagger }|0\rangle =0,\quad j=0,1,\dots } <p>The dual fermionic Fock space vacuum state, denoted ⟨ 0 | {\displaystyle \langle 0|} , is annihilated by the adjoint operators, acting to the left </p> ⟨ 0 | ψ − j † = 0 , ⟨ 0 | ψ j − 1 | 0 = 0 , j = 0 , 1 , … {\displaystyle \langle 0|\psi _{-j}^{\dagger }=0,\quad \langle 0|\psi _{j-1}|0=0,\quad j=0,1,\dots } <p><a href="/facts/Normal_ordering/M4qQg6Bu">Normal ordering</a> : L 1 , ⋯ L m : {\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 </p> ⟨ 0 | : L 1 , ⋯ L m : | 0 ⟩ = 0. {\displaystyle \langle 0|:L_{1},\cdots L_{m}:|0\rangle =0.} <p>In particular, for a product L 1 L 2 {\displaystyle L_{1}L_{2}} of a pair ( L 1 , L 2 ) {\displaystyle (L_{1},L_{2})} of linear operators, one has </p> : L 1 L 2 : = L 1 L 2 − ⟨ 0 | L 1 L 2 | 0 ⟩ . {\displaystyle {:L_{1}L_{2}:}=L_{1}L_{2}-\langle 0|L_{1}L_{2}|0\rangle .} <p>The <i>fermionic charge</i> operator C {\displaystyle C} is defined as </p> C = ∑ i ∈ Z : ψ i ψ i † : {\displaystyle C=\sum _{i\in \mathbf {Z} }:\psi _{i}\psi _{i}^{\dagger }:} <p>The subspace F n ⊂ F {\displaystyle {\mathcal {F}}_{n}\subset {\mathcal {F}}} is the eigenspace of C {\displaystyle C} consisting of all eigenvectors with eigenvalue n {\displaystyle n} </p> C | v ; n ⟩ = n | v ; n ⟩ , ∀ | v ; n ⟩ ∈ F n {\displaystyle C|v;n\rangle =n|v;n\rangle ,\quad \forall |v;n\rangle \in {\mathcal {F}}_{n}} . <p>The standard orthonormal basis { | λ ⟩ } {\displaystyle \{|\lambda \rangle \}} for the zero fermionic charge sector F 0 {\displaystyle {\mathcal {F}}_{0}} is labelled by <a href="/facts/Integer_partition/LtyXWqwH">integer partitions</a> λ = ( λ 1 , … , λ ℓ ( λ ) ) {\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})} , where λ 1 ≥ ⋯ ≥ λ ℓ ( λ ) {\displaystyle \lambda _{1}\geq \cdots \geq \lambda _{\ell (\lambda )}} is a weakly decreasing sequence of ℓ ( λ ) {\displaystyle \ell (\lambda )} positive integers, which can equivalently be represented by a <a href="/facts/Young_diagram/e5lI4bck">Young diagram</a>, as depicted here for the partition ( 5 , 4 , 1 ) {\displaystyle (5,4,1)} . </p> <p>An alternative notation for a partition λ {\displaystyle \lambda } consists of the <a href="/facts/Ferdinand_Georg_Frobenius/4cXXQyu5">Frobenius</a> indices ( α 1 , … α r | β 1 , … β r ) {\displaystyle (\alpha _{1},\dots \alpha _{r}|\beta _{1},\dots \beta _{r})} , where α i {\displaystyle \alpha _{i}} denotes the <i>arm length</i>; i.e. the number λ i − i {\displaystyle \lambda _{i}-i} of boxes in the Young diagram to the right of the i {\displaystyle i} 'th diagonal box, β i {\displaystyle \beta _{i}} denotes the <i>leg length</i>, i.e. the number of boxes in the Young diagram below the i {\displaystyle i} 'th diagonal box, for i = 1 , … , r {\displaystyle i=1,\dots ,r} , where r {\displaystyle r} is the <i>Frobenius rank</i>, which is the number of elements along the principal diagonal. </p><p>The basis element | λ ⟩ {\displaystyle |\lambda \rangle } is then given by acting on the vacuum with a product of r {\displaystyle r} pairs of creation and annihilation operators, labelled by the Frobenius indices </p> | λ ⟩ = ( − 1 ) ∑ j = 1 r β j ∏ k = 1 r ( ψ α k ψ − β k − 1 † ) | 0 ⟩ . {\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 .} <p>The integers { α i } i = 1 , … , r {\displaystyle \{\alpha _{i}\}_{i=1,\dots ,r}} indicate, relative to the Dirac sea, the occupied non-negative sites on the integer lattice while { − β i − 1 } i = 1 , … , r {\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 <i>Maya diagram</i>.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a> </p><p>The case of the null (emptyset) partition | ∅ ⟩ = | 0 ⟩ {\displaystyle |\emptyset \rangle =|0\rangle } gives the vacuum state, and the dual basis { ⟨ μ | } {\displaystyle \{\langle \mu |\}} is defined by </p> ⟨ μ | λ ⟩ = δ λ , μ {\displaystyle \langle \mu |\lambda \rangle =\delta _{\lambda ,\mu }} <p>Any KP τ {\displaystyle \tau } -function can be expressed as a sum </p> <table><tbody><tr><td> τ w ( t ) = ∑ λ π λ ( w ) s λ ( t ) , {\displaystyle \tau _{w}(\mathbf {t} )=\sum _{\lambda }\pi _{\lambda }(w)s_{\lambda }(\mathbf {t} ),} </td> <td></td> <td>10</td></tr></tbody></table> <p>where t = ( t 1 , t 2 , … , … ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\dots ,\dots )} are the KP flow variables, s λ ( t ) {\displaystyle s_{\lambda }(\mathbf {t} )} is the <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur function</a> corresponding to the partition λ {\displaystyle \lambda } , viewed as a function of the normalized power sum variables </p> t i := [ x ] i := 1 i ∑ a = 1 n x a i i = 1 , 2 , … {\displaystyle t_{i}:=[\mathbf {x} ]_{i}:={\frac {1}{i}}\sum _{a=1}^{n}x_{a}^{i}\quad i=1,2,\dots } <p>in terms of an auxiliary (finite or infinite) sequence of variables x := ( x 1 , … , x N ) {\displaystyle \mathbf {x} :=(x_{1},\dots ,x_{N})} and the constant coefficients π λ ( w ) {\displaystyle \pi _{\lambda }(w)} may be viewed as the <a href="/facts/Pl%C3%BCcker_coordinates/SvB7rfeQ">Plücker coordinates</a> of an element w ∈ G r H + ( H ) {\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 G l ( H ) {\displaystyle \mathrm {Gl} ({\mathcal {H}})} , of the subspace H + = s p a n { e − i } i ∈ N ⊂ H {\displaystyle {\mathcal {H}}_{+}=\mathrm {span} \{e_{-i}\}_{i\in \mathbf {N} }\subset {\mathcal {H}}} of the Hilbert space H {\displaystyle {\mathcal {H}}} . </p><p>This corresponds, under the <i>Bose-Fermi correspondence</i>, to a <a href="/facts/Exterior_algebra/rdN4TvpR">decomposable</a> element </p> | τ w ⟩ = ∑ λ π λ ( w ) | λ ⟩ {\displaystyle |\tau _{w}\rangle =\sum _{\lambda }\pi _{\lambda }(w)|\lambda \rangle } <p>of the Fock space F 0 {\displaystyle {\mathcal {F}}_{0}} which, up to projectivization, is the image of the Grassmannian element w ∈ G r H + ( H ) {\displaystyle w\in \mathrm {Gr} _{{\mathcal {H}}_{+}}({\mathcal {H}})} under the <a href="/facts/Plucker_coordinates/SvB7rfeQ">Plücker map</a> </p> P l : s p a n ( w 1 , w 2 , … ) ⟶ [ w 1 ∧ w 2 ∧ ⋯ ] = [ | τ w ⟩ ] , {\displaystyle {\mathcal {Pl}}:\mathrm {span} (w_{1},w_{2},\dots )\longrightarrow [w_{1}\wedge w_{2}\wedge \cdots ]=[|\tau _{w}\rangle ],} <p>where ( w 1 , w 2 , … ) {\displaystyle (w_{1},w_{2},\dots )} is a basis for the subspace w ⊂ H {\displaystyle w\subset {\mathcal {H}}} and [ ⋯ ] {\displaystyle [\cdots ]} denotes projectivization of an element of F {\displaystyle {\mathcal {F}}} . </p><p>The Plücker coordinates { π λ ( w ) } {\displaystyle \{\pi _{\lambda }(w)\}} satisfy an infinite set of bilinear relations, the <a href="/facts/Pl%C3%BCcker_relations/85rFMBKE">Plücker relations</a>, defining the image of the <a href="/facts/Pl%C3%BCcker_embedding/85rFMBKE">Plücker embedding</a> into the projectivization P ( F ) {\displaystyle \mathbf {P} ({\mathcal {F}})} of the fermionic Fock space, which are equivalent to the Hirota bilinear residue relation (1). </p><p>If w = g ( H + ) {\displaystyle w=g({\mathcal {H}}_{+})} for a group element g ∈ G l ( H ) {\displaystyle g\in \mathrm {Gl} ({\mathcal {H}})} with fermionic representation g ^ {\displaystyle {\hat {g}}} , then the τ {\displaystyle \tau } -function τ w ( t ) {\displaystyle \tau _{w}(\mathbf {t} )} can be expressed as the fermionic vacuum state expectation value (VEV): </p> τ w ( t ) = ⟨ 0 | γ ^ + ( t ) g ^ | 0 ⟩ , {\displaystyle \tau _{w}(\mathbf {t} )=\langle 0|{\hat {\gamma }}_{+}(\mathbf {t} ){\hat {g}}|0\rangle ,} <p>where </p> Γ + = { γ ^ + ( t ) = e ∑ i = 1 ∞ t i J i } ⊂ G l ( H ) {\displaystyle \Gamma _{+}=\{{\hat {\gamma }}_{+}(\mathbf {t} )=e^{\sum _{i=1}^{\infty }t_{i}J_{i}}\}\subset \mathrm {Gl} ({\mathcal {H}})} <p>is the abelian subgroup of G l ( H ) {\displaystyle \mathrm {Gl} ({\mathcal {H}})} that generates the KP flows, and </p> J i := ∑ j ∈ Z ψ j ψ j + i † , i = 1 , 2 … {\displaystyle J_{i}:=\sum _{j\in \mathbf {Z} }\psi _{j}\psi _{j+i}^{\dagger },\quad i=1,2\dots } <p>are the ""current"" components. </p> <h2 id="examples-of-solutions-to-the-equations-of-the-kp-hierarchy">Examples of solutions to the equations of the KP hierarchy</h2> <h3>Schur functions</h3> <p>As seen in equation (9), every KP τ {\displaystyle \tau } -function can be represented (at least formally) as a linear combination of <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur functions</a>, in which the coefficients π λ ( w ) {\displaystyle \pi _{\lambda }(w)} satisfy the bilinear set of <a href="/facts/Plucker_relations/85rFMBKE">Plucker relations</a> corresponding to an element w {\displaystyle w} of an infinite (or finite) Grassmann manifold. In fact, the simplest class of (polynomial) tau functions consists of the <a href="/facts/Schur_polynomial/mBzF2Mdr">Schur functions</a> s λ ( t ) {\displaystyle s_{\lambda }(\mathbf {t} )} themselves, which correspond to the special element of the Grassmann manifold whose image under the <a href="/facts/Plucker_relations/85rFMBKE">Plücker map</a> is | λ > {\displaystyle |\lambda >} . </p> <h3>Multisoliton solutions</h3> <p>If we choose 3 N {\displaystyle 3N} complex constants { α k , β k , γ k } k = 1 , … , N {\displaystyle \{\alpha _{k},\beta _{k},\gamma _{k}\}_{k=1,\dots ,N}} with α k , β k {\displaystyle \alpha _{k},\beta _{k}} 's all distinct, γ k ≠ 0 {\displaystyle \gamma _{k}\neq 0} , and define the functions </p> y k ( t ) := e ∑ i = 1 ∞ t i α k i + γ k e ∑ i = 1 ∞ t i β k i k = 1 , … , N , {\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,} <p>we arrive at the Wronskian determinant formula </p> τ α → , β → , γ → ( N ) ( t ) := | y 1 ( t ) y 2 ( t ) ⋯ y N ( t ) y 1 ′ ( t ) y 2 ′ ( t ) ⋯ y N ′ ( t ) ⋮ ⋮ ⋱ ⋮ y 1 ( N − 1 ) ( t ) y 2 ( N − 1 ) ( t ) ⋯ y N ( N − 1 ) ( t ) | , {\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}},} <p>which gives the general <a href="/facts/Solitons/5rC19BSU"> N {\displaystyle N} -soliton</a> τ {\displaystyle \tau } -function.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a><a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p> <h3>Theta function solutions associated to algebraic curves</h3> <p>Let X {\displaystyle X} be a compact Riemann surface of genus g {\displaystyle g} and fix a canonical homology basis a 1 , … , a g , b 1 , … , b g {\displaystyle a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}} of H 1 ( X , Z ) {\displaystyle H_{1}(X,\mathbf {Z} )} with intersection numbers </p> a i ∘ a j = b i ∘ b j = 0 , a i ∘ b j = δ i j , 1 ≤ i , j ≤ g . {\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.} <p>Let { ω i } i = 1 , … , g {\displaystyle \{\omega _{i}\}_{i=1,\dots ,g}} be a basis for the space H 1 ( X ) {\displaystyle H^{1}(X)} of <a href="/facts/Differential_of_the_first_kind/yNpPfRVV">holomorphic differentials</a> satisfying the standard normalization conditions </p> ∮ a i ω j = δ i j , ∮ b j ω j = B i j , {\displaystyle \oint _{a_{i}}\omega _{j}=\delta _{ij},\quad \oint _{b_{j}}\omega _{j}=B_{ij},} <p>where B {\displaystyle B} is the <i><a href="/facts/Schottky_problem/kvYIAndF">Riemann matrix</a></i> of periods. The matrix B {\displaystyle B} belongs to the <i>Siegel upper half space</i> </p> S g = { B ∈ M a t g × g ( C ) : B T = B , Im ( B ) is positive definite } . {\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\}.} <p>The <a href="/facts/Theta_function/g09xvoQQ">Riemann θ {\displaystyle \theta } function</a> on C g {\displaystyle \mathbf {C} ^{g}} corresponding to the <a href="/facts/Period_mapping/9alkmCtU">period matrix</a> B {\displaystyle B} is defined to be </p> θ ( Z | B ) := ∑ N ∈ Z g e i π ( N , B N ) + 2 i π ( N , Z ) . {\displaystyle \theta (Z|B):=\sum _{N\in \mathbb {Z} ^{g}}e^{i\pi (N,BN)+2i\pi (N,Z)}.} <p>Choose a point p ∞ ∈ X {\displaystyle p_{\infty }\in X} , a local parameter ζ {\displaystyle \zeta } in a neighbourhood of p ∞ {\displaystyle p_{\infty }} with ζ ( p ∞ ) = 0 {\displaystyle \zeta (p_{\infty })=0} and a positive <a href="/facts/Divisor/DKNa2GvC">divisor</a> of degree g {\displaystyle g} </p> D := ∑ i = 1 g p i , p i ∈ X . {\displaystyle {\mathcal {D}}:=\sum _{i=1}^{g}p_{i},\quad p_{i}\in X.} <p>For any positive integer k ∈ N + {\displaystyle k\in \mathbf {N} ^{+}} let Ω k {\displaystyle \Omega _{k}} be the unique <a href="/facts/Differential_of_the_first_kind/yNpPfRVV">meromorphic differential</a> of the second kind characterized by the following conditions: </p> <ul><li>The only singularity of Ω k {\displaystyle \Omega _{k}} is a pole of order k + 1 {\displaystyle k+1} at p = p ∞ {\displaystyle p=p_{\infty }} with vanishing residue.</li> <li>The expansion of Ω k {\displaystyle \Omega _{k}} around p = p ∞ {\displaystyle p=p_{\infty }} is Ω k = d ( ζ − k ) + ∑ j = 1 ∞ Q i j ζ j d ζ {\displaystyle \Omega _{k}=d(\zeta ^{-k})+\sum _{j=1}^{\infty }Q_{ij}\zeta ^{j}d\zeta } .</li> <li> Ω k {\displaystyle \Omega _{k}} is normalized to have vanishing a {\displaystyle a} -cycles: ∮ a i Ω j = 0. {\displaystyle \oint _{a_{i}}\Omega _{j}=0.} </li></ul> <p>Denote by U k ∈ C g {\displaystyle \mathbf {U} _{k}\in \mathbf {C} ^{g}} the vector of b {\displaystyle b} -cycles of Ω k {\displaystyle \Omega _{k}} : </p> ( U k ) j := ∮ b j Ω k . {\displaystyle (\mathbf {U} _{k})_{j}:=\oint _{b_{j}}\Omega _{k}.} <p>Denote the image of D {\displaystyle {\mathcal {D}}} under the <a href="/facts/Abel%E2%80%93Jacobi_map/W1f1S827">Abel</a> map A : S g ( X ) → C g {\displaystyle {\mathcal {A}}:{\mathcal {S}}^{g}(X)\to \mathbf {C} ^{g}} </p> E := A ( D ) ∈ C g , E j = A j ( D ) := ∑ j = 1 g ∫ p 0 p i ω j {\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}} <p>with arbitrary base point p 0 {\displaystyle p_{0}} . </p><p>Then the following is a KP τ {\displaystyle \tau } -function:<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> </p> τ ( X , D , p ∞ , ζ ) ( t ) := e − 1 2 ∑ i j Q i j t i t j θ ( E + ∑ k = 1 ∞ t k U k | B ) {\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)} . <h3>Matrix model partition functions as KP τ {\displaystyle \tau } -functions</h3> <p>Let d μ 0 ( M ) {\displaystyle d\mu _{0}(M)} be the Lebesgue measure on the N 2 {\displaystyle N^{2}} dimensional space H N × N {\displaystyle {\mathbf {H} }^{N\times N}} of N × N {\displaystyle N\times N} complex Hermitian matrices. Let ρ ( M ) {\displaystyle \rho (M)} be a conjugation invariant integrable density function </p> ρ ( U M U † ) = ρ ( M ) , U ∈ U ( N ) . {\displaystyle \rho (UMU^{\dagger })=\rho (M),\quad U\in U(N).} <p>Define a deformation family of measures </p> d μ N , ρ ( t ) := e Tr ( ∑ i = 1 ∞ t i M i ) ρ ( M ) d μ 0 ( M ) {\displaystyle d\mu _{N,\rho }(\mathbf {t} ):=e^{{\text{ Tr }}(\sum _{i=1}^{\infty }t_{i}M^{i})}\rho (M)d\mu _{0}(M)} <p>for small t = ( t 1 , t 2 , ⋯ ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\cdots )} and let </p> τ N , ρ ( t ) := ∫ H N × N d μ N , ρ ( t ) . {\displaystyle \tau _{N,\rho }({\bf {t}}):=\int _{{\mathbf {H} }^{N\times N}}d\mu _{N,\rho }({\bf {t}}).} <p>be the <a href="/facts/Partition_function_(statistical_mechanics)/q65sJrrn">partition function</a> for this <a href="/facts/Random_matrix/Yqo1lwFW">random matrix model</a>.<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a><a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> Then τ N , ρ ( t ) {\displaystyle \tau _{N,\rho }(\mathbf {t} )} satisfies the bilinear Hirota residue equation (1), and hence is a τ {\displaystyle \tau } -function of the KP hierarchy.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a> </p> <h3> τ {\displaystyle \tau } -functions of hypergeometric type. Generating function for Hurwitz numbers</h3> <p>Let { r i } i ∈ Z {\displaystyle \{r_{i}\}_{i\in \mathbf {Z} }} be a (doubly) infinite sequence of complex numbers. For any integer partition λ = ( λ 1 , … , λ ℓ ( λ ) ) {\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{\ell (\lambda )})} define the <i>content product</i> coefficient </p> r λ := ∏ ( i , j ) ∈ λ r j − i {\displaystyle r_{\lambda }:=\prod _{(i,j)\in \lambda }r_{j-i}} , <p>where the product is over all pairs ( i , j ) {\displaystyle (i,j)} of positive integers that correspond to boxes of the Young diagram of the partition λ {\displaystyle \lambda } , viewed as positions of matrix elements of the corresponding ℓ ( λ ) × λ 1 {\displaystyle \ell (\lambda )\times \lambda _{1}} matrix. Then, for every pair of infinite sequences t = ( t 1 , t 2 , … ) {\displaystyle \mathbf {t} =(t_{1},t_{2},\dots )} and s = ( s 1 , s 2 , … ) {\displaystyle \mathbf {s} =(s_{1},s_{2},\dots )} of complex variables, viewed as (normalized) power sums t = [ x ] , s = [ y ] {\displaystyle \mathbf {t} =[\mathbf {x} ],\ \mathbf {s} =[\mathbf {y} ]} of the infinite sequence of auxiliary variables </p> x = ( x 1 , x 2 , … ) {\displaystyle \mathbf {x} =(x_{1},x_{2},\dots )} and y = ( y 1 , y 2 , … ) {\displaystyle \mathbf {y} =(y_{1},y_{2},\dots )} , <p>defined by: </p> t j := 1 j ∑ a = 1 ∞ x a j , s j := 1 j ∑ j = 1 ∞ y a j {\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}} , <p>the function </p> <table><tbody><tr><td> τ r ( t , s ) := ∑ λ r λ s λ ( t ) s λ ( s ) {\displaystyle \tau ^{r}(\mathbf {t} ,\mathbf {s} ):=\sum _{\lambda }r_{\lambda }s_{\lambda }(\mathbf {t} )s_{\lambda }(\mathbf {s} )} </td> <td></td> <td>11</td></tr></tbody></table> <p>is a <i>double</i> KP τ {\displaystyle \tau } -function, both in the t {\displaystyle \mathbf {t} } and the s {\displaystyle \mathbf {s} } variables, known as a τ {\displaystyle \tau } -function of <i>hypergeometric type</i>.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a> </p><p>In particular, choosing </p> r j = r j β := e j β {\displaystyle r_{j}=r_{j}^{\beta }:=e^{j\beta }} <p>for some small parameter β {\displaystyle \beta } , denoting the corresponding content product coefficient as r λ β {\displaystyle r_{\lambda }^{\beta }} and setting </p> s = ( 1 , 0 , … ) =: t 0 {\displaystyle \mathbf {s} =(1,0,\dots )=:\mathbf {t} _{0}} , <p>the resulting τ {\displaystyle \tau } -function can be equivalently expanded as </p> <table><tbody><tr><td> τ r β ( t , t 0 ) = ∑ λ ∑ d = 0 ∞ β d d ! H d ( λ ) p λ ( t ) , {\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} ),} </td> <td></td> <td>12</td></tr></tbody></table> <p>where { H d ( λ ) } {\displaystyle \{H_{d}(\lambda )\}} are the <i>simple Hurwitz numbers</i>, which are 1 n ! {\displaystyle {\frac {1}{n!}}} times the number of ways in which an element k λ ∈ S n {\displaystyle k_{\lambda }\in {\mathcal {S}}_{n}} of the symmetric group S n {\displaystyle {\mathcal {S}}_{n}} in n = | λ | {\displaystyle n=|\lambda |} elements, with cycle lengths equal to the parts of the partition λ {\displaystyle \lambda } , can be factorized as a product of d {\displaystyle d} 2 {\displaystyle 2} -cycles </p> k λ = ( a 1 b 1 ) … ( a d b d ) {\displaystyle k_{\lambda }=(a_{1}b_{1})\dots (a_{d}b_{d})} , <p>and </p> p λ ( t ) = ∏ i = 1 ℓ ( λ ) p λ i ( t ) , with p i ( t ) := ∑ a = 1 ∞ x a i = i t i {\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}} <p>is the power sum symmetric function. Equation (12) thus shows that the (formal) KP hypergeometric τ {\displaystyle \tau } -function (11) corresponding to the content product coefficients r λ β {\displaystyle r_{\lambda }^{\beta }} is a generating function, in the combinatorial sense, for simple Hurwitz numbers.<a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a> </p> <h2 id="bibliography">Bibliography</h2> <ul><li>Dickey, L.A. (2003), <i>Soliton Equations and Hamiltonian Systems</i>, Advanced Series in Mathematical Physics, vol. 26, World Scientific Publishing Co., Inc., River Edge, NJ, 2nd Ed., <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1142%2F5108">10.1142/5108</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9789810202156</li> <li><a href="/facts/John_Harnad/aaYARDAo">Harnad, J.</a>; Balogh, F. 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