Tau functions are an important ingredient in the modern mathematical theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in his direct method approach to soliton equations, based on expressing them in an equivalent bilinear form. The term tau function, or τ {\displaystyle \tau } -function, was first used systematically by Mikio Sato and his students in the specific context of the Kadomtsev–Petviashvili (or KP) equation and related integrable hierarchies. It is a central ingredient in the theory of solitons. In this setting, given any τ {\displaystyle \tau } -function satisfying a Hirota-type system of bilinear equations (see § 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 matrix model partition functions in the spectral theory of random matrices, and may also serve as generating functions, in the sense of combinatorics and enumerative geometry, especially in relation to moduli spaces of Riemann surfaces, and enumeration of branched coverings, or so-called Hurwitz numbers. There are two notions of τ {\displaystyle \tau } -functions, both introduced by the Sato school. The first is isospectral τ {\displaystyle \tau } -functions of the Sato–Segal–Wilson type for integrable hierarchies, such as the KP hierarchy, which are parametrized by linear operators satisfying isospectral deformation equations of Lax type. The second is isomonodromic τ {\displaystyle \tau } -functions. Depending on the specific application, a τ {\displaystyle \tau } -function may either be: 1) an analytic function of a finite or infinite number of independent, commuting flow variables, or deformation parameters; 2) a discrete function of a finite or infinite number of denumerable variables; 3) a formal power series expansion in a finite or infinite number of expansion variables, which need have no convergence domain, but serves as generating function for certain enumerative invariants appearing as the coefficients of the series; 4) a finite or infinite (Fredholm) determinant whose entries are either specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite or infinite dimensional) with entries similarly of polynomial or quasi-polynomial type. Examples of all these types are given below. In the Hamilton–Jacobi approach to Liouville integrable Hamiltonian systems, Hamilton's principal function, evaluated on the level surfaces of a complete set of Poisson commuting invariants, plays a role similar to the τ {\displaystyle \tau } -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 τ {\displaystyle \tau } -function of isospectral 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 Sato and Segal-Wilson sense, it is the value of the determinant of a Fredholm integral operator, interpreted as the orthogonal projection of an element of a suitably defined (infinite dimensional) Grassmann manifold onto the origin, as that element evolves under the linear exponential action of a maximal abelian subgroup of the general linear group. It typically arises as a partition function, in the sense of statistical mechanics, many-body quantum mechanics or quantum field theory, as the underlying measure undergoes a linear exponential deformation. Isomonodromic τ {\displaystyle \tau } -functions for linear systems of Fuchsian 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.
Hirota bilinear residue relation for KP tau functions
A KP (Kadomtsev–Petviashvili) 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 KP flow variables) that satisfies the bilinear formal residue equation
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
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 ).}
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.
Kadomtsev–Petviashvili equation 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
t 1 = x , t 2 = y , t 3 = t , {\displaystyle t_{1}=x,\quad t_{2}=y,\quad t_{3}=t,}
it follows that the function
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)}
satisfies the 2 {\displaystyle 2} (spatial) + 1 {\displaystyle +1} (time) dimensional nonlinear partial differential equation
known as the Kadomtsev-Petviashvili (KP) equation. This equation plays a prominent role in plasma physics and in shallow water ocean waves. Taking further logarithmic derivatives of 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 KP hierarchy.
Formal Baker–Akhiezer function and the KP hierarchy If we define the (formal) Baker-Akhiezer function ψ ( z , t ) {\displaystyle \psi (z,\mathbf {t} )}
by Sato's formula
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} )}}}
and expand it as a formal series in the powers of the variable z {\displaystyle z}
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}),}
this satisfies an infinite sequence of compatible evolution equations
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
D i := ( L i ) + {\displaystyle {\mathcal {D}}_{i}:={\big (}{\mathcal {L}}^{i}{\big )}_{+}}
where L {\displaystyle {\mathcal {L}}} is the formal pseudo-differential operator
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}}
with ∂ := ∂ ∂ x {\displaystyle \partial :={\frac {\partial }{\partial x}}} ,
W := 1 + ∑ j = 1 ∞ w j ( t ) ∂ − j {\displaystyle {\mathcal {W}}:=1+\sum _{j=1}^{\infty }w_{j}(\mathbf {t} )\partial ^{-j}}
is the wave operator 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}} . The pseudodifferential operator L {\displaystyle {\mathcal {L}}} satisfies the infinite system of isospectral deformation equations
and the compatibility conditions for both the system (3) and (4) are
This is a compatible infinite system of nonlinear partial differential equations, known as the KP (Kadomtsev-Petviashvili) hierarchy, 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 Kadomtsev-Petviashvili equation (2). 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.
Isomonodromic systems. Isomonodromic tau functions
Fuchsian isomonodromic systems. Schlesinger equations Consider the overdetermined system of first order matrix partial differential equations
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 monodromy 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
∂ ∂ z − ∑ i = 1 n N i z − α i {\displaystyle {\partial \over \partial z}-\sum _{i=1}^{n}{N_{i} \over z-\alpha _{i}}}
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 isomonodromic deformation. The compatibility conditions for this system are the Schlesinger equations
Isomonodromic τ {\displaystyle \tau } -function Defining n {\displaystyle n} functions
the Schlesinger equations (8) imply that the differential form
ω := ∑ i = 1 n H i d α i {\displaystyle \omega :=\sum _{i=1}^{n}H_{i}d\alpha _{i}}
on the space of parameters is closed:
d ω = 0 {\displaystyle d\omega =0}
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
ω = d l n τ {\displaystyle \omega =d\mathrm {ln} \tau }
The function τ ( α 1 , … , α n ) {\displaystyle \tau (\alpha _{1},\dots ,\alpha _{n})} is called the isomonodromic τ {\displaystyle \tau } -function associated to the fundamental solution Ψ {\displaystyle \Psi } of the system (6), (7).
Hamiltonian structure of the Schlesinger equations Defining the Lie Poisson brackets 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:
{ ( 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,}
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
∂ 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}
for any differentiable function f ( N 1 , … , N n ) {\displaystyle f(N_{1},\dots ,N_{n})} .
Reduction of r = 2 {\displaystyle r=2} , n = 3 {\displaystyle n=3} case to P V I {\displaystyle P_{VI}}
The simplest nontrivial case of the Schlesinger equations is when r = 2 {\displaystyle r=2} and n = 3 {\displaystyle n=3} . By applying a Möbius transformation 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 Painlevé transcendent equations, for which many detailed classes of explicit solutions are known.
Non-Fuchsian isomonodromic systems For non-Fuchsian systems, with higher order poles, the generalized monodromy data include Stokes matrices and connection matrices, and there are further isomonodromic deformation parameters associated with the local asymptotics, but the isomonodromic τ {\displaystyle \tau } -functions may be defined in a similar way, using differentials on the extended parameter space. There is similarly a Poisson bracket structure on the space of rational matrix valued functions of the spectral parameter z {\displaystyle z} and corresponding spectral invariant Hamiltonians that generate the isomonodromic deformation dynamics. 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 Painlevé transcendents, for which numerous special solutions are also known.
Fermionic VEV (vacuum expectation value) representations The fermionic Fock space F {\displaystyle {\mathcal {F}}} , is a semi-infinite exterior product space
F = Λ ∞ / 2 H = ⊕ n ∈ Z F n {\displaystyle {\mathcal {F}}=\Lambda ^{\infty /2}{\mathcal {H}}=\oplus _{n\in \mathbf {Z} }{\mathcal {F}}_{n}}
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}}^{*}} . 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
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} ,}
and satisfy the canonical anti-commutation relations
[ 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}.}
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
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}}^{*}}
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
| 0 ⟩ := e − 1 ∧ e − 2 ∧ ⋯ {\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
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 }
The dual fermionic Fock space vacuum state, denoted ⟨ 0 | {\displaystyle \langle 0|} , is annihilated by the adjoint operators,