On the Bilinear Equations for Fredholm Determinants Appearing in Random Matrices^{†}

J. Harnad

Department of Mathematics and Statistics, Concordia University 7141 Sherbrooke W., Montréal, Qué., Canada H4B 1R6, and Centre de recherches mathématiques, Université de Montréal C. P. 6128, succ. centre ville, Montréal, Qué., Canada H3C 3J7 e-mail: harnad@crm.umontreal.ca

Abstract

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic τ-functions.

1. Differential equations for Fredholm determinants in random matrices

In the theory of random matrices, it is known that in suitably defined double scaling limits the generating functions for spectral distributions are given by Fredholm determinants of certain integral operators [M,TW1–3]. For example, in the universality class of the Gaussian Unitary Ensemble (GUE), in the bulk of the spectrum, the probability of having exactly {m_{1}, ⋯ m_{n}} scaled eigenvalues in the sequence of disjoint intervals {([a_{1}, a_{2}], ⋯ [a_{2n−1}, a_{2n}]} is

E(m_{1}, …, m_{n}) = \frac{(−1)m̄}{m_{1}! ⋯ m_{n}!} \frac{∂m̄τ^{S}}{∂λ_{1}^{m_{1}} ⋯ ∂λ_{n}^{m_{n}}} |_{λ_{1}=···=λ_{m}=1},  m̄ = ∑_{j} m_{j},  (1.1)

where τ^{S} is the Fredholm determinant

τ^{S} := det(1 − K̂^{S})  (1.2)

of the integral operator K̂_{s} : L^{2}(R, C) → L^{2}(R, C) with the sine kernel

(K̂^{S}v)(x) = ∑_{j=1}^{n} λ_{j} ∫_{a_{2j−1}}^{a_{2j}} \frac{sin(π(x − y))}{π(x − y)}v(y)dy.  (1.3)

† Research at MSRI is supported in part by NSF grant DMS-9701755. Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Fonds FCAR du Québec.

J. Harnad

Rescaling at the edge of the spectrum, the corresponding quantity is given by the Fredholm determinant

τ^{A} := det(1 − K̂^{A})  (1.4)

of the operator with the Airy kernel [TW2]

(K̂^{A}v)(x) = ∑_{j=1}^{n} λ_{j} ∫_{a_{2j−1}}^{a_{2j}} \frac{Ai(x))Ai′(y) − Ai(y)Ai′(x)}{x − y}v(y)dy,  (1.5)

where Ai(x) is the Airy function. If the measure is taken to be the one associated with either the Laguerre or Bessel orthogonal polynomials, rescaling at the edge leads to the Fredholm determinant

τ_{α}^{B} := det(1 − K̂_{α}^{B})  (1.6)

of the operator with Bessel kernel [TW3]

(K̂_{α}^{B}v)(x) = ∑_{j=1}^{n} λ_{j} ∫_{a_{2j−1}}^{a_{2j}} \frac{J_{α}(\sqrt{x})\sqrt{y}J′_{α}(\sqrt{y}) − J_{α}(\sqrt{y})\sqrt{x}J′_{α}(\sqrt{x})}{2(x − y)}v(y)dy,  (1.7)

where J_{α}(x) is the Bessel function with index α.

It was shown by Tracy and Widom [TW1–3], extending earlier results of the Kyoto school [JMMS], that all these Fredholm determinants can be computed by quadratures in terms of solutions of certain associated nonautonomous Hamiltonian systems in which the end points {a_{j}} play the rôle of multi-time deformation variables. Moreover, these Fredholm determinants may be interpreted as isomonodromic τ-functions [HTW, HI] in the sense of [JMU, JM].

More recently, Adler, Shiota and van Moerbeke [ASV1, ASV2] have shown that the Fredholm determinants τ^{A}, τ_{α}^{B} satisfy hierarchies of bilinear differential equations with respect to the endpoint parameters. These follow from combining Virasoro constraints satisfied by certain associated KP τ-functions with the bilinear equations they also satisfy with repect to the KP flow parameters {t_{1}, t_{2}, …}, evaluated at the zero values of these parameters. The approach of [ASV1, ASV2] was based on the application of vertex operators, integrated over the intervals {[a_{2j−1}, a_{2j}]}, to suitable “vacuum” KP τ-functions, effecting thereby a continuous version of Darboux transformations, yielding new KP τ-functions, such that the Fredholm determinant equals the ratio of the two.

For the Airy kernel, the first equation in this hierarchy may be expressed as

D_{0}^{4}F^{A} − 4D_{1}D_{0}F^{A} + 2D_{0}F^{A} + 6(D_{0}^{2}F^{A})^{2} = 0,  (1.8)

where

F^{A} := ln τ^{A}  (1.9)

Equations for Fredholm Determinants Appearing in Random Matrices

and

D_{m} := ∑_{j=1}^{2n} a_{j}^{m} \frac{∂}{∂a_{j}} ,  m ∈ N,  (1.10)

while for the Bessel kernel, it is

D_{1}^{4}F_{α}^{B} − 2D_{1}^{4}F_{α}^{B} + (1 − α^{2})D_{1}^{2}F_{α}^{B} + D_{2}D_{1}F_{α}^{B} − \frac{1}{2}D_{2}F_{α}^{B} − 4(D_{1}F_{α}^{B})(D_{1}^{2}F_{α}^{B}) + 6(D_{1}^{2}F_{α}^{B})^{2} = 0,

(1.11)

with

F_{α}^{B} := ln τ_{α}^{B}.  (1.12)

No analogous equations were derived for the sine kernel, although in the special case where the intervals [a_{2j−1}, a_{2j}] are chosen symmetrically about the origin, the Fredholm determinant τ^{S} may be expressed [M, TW3] as a product τ B_{\frac{1}{2}}τ_{− \frac{1}{2}}^{B} of two Bessel kernel determinants.

In the case of a single interval, it is easy to see that equations (1.8) and (1.11) just give the τ-function form of the Painlevé equations P_{II} and P_{V}, respectively, to which the Tracy– Widom systems reduce in the case of the Airy and Bessel kernels. It seems reasonable to expect that analogous results hold for the general case, involving an arbitrary number of intervals. The purpose of this work is to show how the hierarchies of equations derived in [ASV1, ASV2] can in fact be deduced directly from the Tracy-Widom Hamiltonian systems for both the Airy and Bessel cases, and to also apply this approach to the sine kernel case. The main step is to recognize that the Hamiltonian systems imply isomonodromic deformation equations for associated families of rational covariant derivative operators on the Riemann sphere. It is known [JMU, JM] that such isomonodromic deformations give rise to bilinear equations for indexed sets of isomonodromic τ-functions related by Schlesinger transformations. The fact that for the systems associated with the Airy and Bessel kernels such equations may be written in terms of a single scalar τ-function is due to the presence of a pair of conserved quantities, allowing the elimination of the additional variables by fixing the level sets of these invariants. In the sine kernel case this is not possible, and the associated bilinear equations therefore involve coupled systems for τ^{S} together with a pair of additional variables (τ_{+}^{S}, τ_{−}^{S}).

In section 2, equations (1.8) and (1.11) are first derived directly from the Hamiltonian systems of [TW2-3]. In section 3, it is shown how the isomonodromic deformation equations following from the associated Hamiltonian systems may be used to derive the full hierarchy of τ-function equations for all these cases. In section 4, these results are related to the rational classical R-matrix approach to isomonodromic and isospectral systems developed in [AHP, H].

2. Deduction of τ-function equations from the Hamiltonian systems

To establish notation, following [TW1-3], we define the quantities:

x_{2j} := 2i\sqrt{λ_{j}}(I − K̂)^{−1}ϕ(a_{2j}),  x_{2j+1} := 2\sqrt{λ_{j}}(I − K̂)^{−1}ϕ(a_{2j+1})  (2.1a)

y_{2j} := i\sqrt{λ_{j}}(I − K̂)^{−1}ψ(a_{2j}),  y_{2j+1} := \sqrt{λ_{j}}(I − K̂)^{−1}ψ(a_{2j+1}),  (2.1b)
x_{0} := 2 ∑_{j=1}^{n} λ_{j} ∫_{a_{2j−1}}^{a_{2j}} ϕ(x)(I − K̂)^{−1}ψ(x)dx,  (2.1c)
y_{0} := ∑_{j=1}^{n} λ_{j} ∫_{a_{2j−1}}^{a_{2j}} ϕ(x)(I − K̂)^{−1}ϕ(x)dx,  (2.1d)

where, for the case of the sine kernel K̂ = K̂^{S},

ϕ(x) := \frac{sin(πx)}{π}, ψ(x) := cos(πx).  (2.2)

while for the Airy kernel K̂ = K̂^{A},

ϕ(x) := Ai(x), ψ(x) := \frac{dAi(x)}{dx},  (2.3)

and for the Bessel kernel K̂ = K̂_{α}^{B},

ϕ(x) := J_{α}(\sqrt{x}), ψ(x) := x\frac{dJ_{α}(\sqrt{x})}{dx}.  (2.4)

(An odd number of variables may also occur if we set one of the a_{j}’s equal to some fixed constant, say 0 or ∞, and eliminate the corresponding pair (q_{j}, p_{j}).) As shown in [TW1-3], the logarithmic derivatives of the associated Fredholm determinants are given by:

G_{j}^{S} := \frac{∂F^{S}}{∂a_{j}} = \frac{π^{2}}{4} x_{j}^{2} + y_{j}^{2} − \frac{1}{4} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})^{2}}{a_{j} − a_{k}}  (2.5)

for the sine kernel,

G_{j}^{A} := \frac{∂F^{A}}{∂a_{j}} = y_{j}^{2} + \frac{1}{4}(x_{0} − a_{j})x_{j}^{2} − y_{0}x_{j}y_{j} − \frac{1}{4} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})^{2}}{a_{j} − a_{k}}  (2.6)

for the Airy kernel, and

a_{j}G_{α,j}^{B} := a_{j} \frac{∂F_{α}^{B}}{∂a_{j}} = y_{j}^{2} − \frac{1}{16}(α^{2} − a_{j} + x_{0})x_{j}^{2} + \frac{1}{4}y_{0}x_{j}y_{j} − \frac{1}{4} ∑_{{k=1 \atop k≠j}}^{n} \frac{a_{k}(x_{j}y_{k} − y_{j}x_{k})^{2}}{a_{j} − a_{k}} (2.7)

for the Bessel kernel.

For use in what follows, we also define the quantities

R_{m}^{S} := D_{m}F^{S} = ∑_{j=1}^{2n} a_{j}^{m}G_{j}^{S}, m ∈ N  (2.8)

for the sine kernel case,

R_{m}^{A} := D_{m}F^{A} = ∑_{j=1}^{2n} a_{j}^{m}G_{j}^{A}, m ∈ N  (2.9)

for the Airy case and

R_{α,m}^{B} := D_{m}F_{α,j}^{B} = ∑_{j=1}^{2n} a_{j}^{m}G_{α,j}^{B}, m ∈ N  (2.10)

for the Bessel case. For all three cases, we define the following sequence of bilinear forms

P_{m} := ∑_{j=1}^{2n} a_{j}^{m}y_{j}^{2},  Q_{m} := ∑_{j=1}^{2n} a_{j}^{m}x_{j}^{2},  S_{m} := ∑_{j=1}^{2n} a_{j}^{m}x_{j}y_{j},  m ∈ N.  (2.11)

As explained below, the {G_{j}^{A}}’s and {G_{α,j}^{B}}’s may be viewed as sets of Poisson commuting, nonautonomous Hamiltonians on an auxiliary phase space with canonical coordinates {x_{0}, y_{0}, x_{j}, y_{j}}, such that the quantities defined in (2.1a)-(2.1d) satisfy the corresponding systems of Hamiltonian equations. These equations will then be shown to imply equations (1.8) and (1.11).

2a. The Airy kernel system.

The system of dynamical equations for this case is given [TW2] by

\frac{∂x_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} ,  j ≠ k,  (2.12a)
\frac{∂y_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} ,  j ≠ k,  (2.12b)
\frac{∂x_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} + 2y_{j} − y_{0}x_{j},  (2.12c)

\frac{∂y_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} + \frac{1}{2}(a_{j} − x_{0})x_{j} + y_{0}x_{j}y_{j},  (2.12d)

\frac{∂x_{0}}{∂a_{j}} = −x_{j}y_{j},  \frac{∂y_{0}}{∂a_{j}} = −\frac{1}{4}x_{j}^{2}.  (2.12e)

Viewing the a_{j}’s as multi-time parameters, this is a compatible system of nonautonomous Hamiltonian equations generated by the Poisson commuting Hamiltonians {G_{j}^{A}} defined in (2.6). There is an additional functionally independent Hamiltonian, defined by

G_{0}^{A} := y_{0}^{2} − x_{0} − \frac{1}{4}Q_{0},  (2.13)

which also Poisson commutes with all the G_{j}^{A}’s. Since G_{0}^{A} is not explicitly dependent on the parameters {a_{j}}, it follows that it is a conserved quantity. Since all the quantities {x_{0}, y_{0}, x_{j}, y_{j}} defined in (2.1a)-(2.1d) vanish in the limit {a_{j} → ∞, ∀j}, the invariant G_{0}^{A} must vanish on this particular solution. Therefore we may express x_{0} in terms of the other variables as

x_{0} = y_{0}^{2} − \frac{1}{4}Q_{0}.  (2.14)

The quantity R_{0}^{A} defined in (2.9) will just be denoted

R := R_{0}^{A} = ∑_{j=1}^{2n} G_{j}^{A} = P_{0} − \frac{1}{4}Q_{1} + \frac{1}{4}y_{0}^{2}Q_{0} − y_{0}S_{0} − \frac{1}{16}Q_{0}^{2},  (2.15)

where (2.14) has been used. In terms of R, equation (1.8) becomes

D_{0}^{3}R − 4D_{1}R + 2R + 6(D_{0}R)^{2} = 0.  (2.16)

It follows from the Poisson commutativity of the Hamiltonians {G_{j}^{A}}_{j=1...2n} that their Hamiltonian vector fields applied as derivations to R give zero, and hence along any integral surface of eqs. (2.12a)-(2.12e), the derivatives of R with respect to the a_{j}’s are just given by its explicit dependence on these parameters. This just comes from the Q_{1} term in expression (2.15), and therefore we have

\frac{∂R}{∂a_{j}} = −\frac{1}{4}x_{j}^{2}  (2.17)

Comparing with (2.12e), this implies that

G_{∞}^{A} := y_{0} − R  (2.18)

is a second conserved quantity. Since in the limit {a_{j} → ∞, ∀j}, both y_{0} and R vanish, G_{∞}^{A} must vanish for all values of the parameters, and therefore the invariant relation

y_{0} = R  (2.19)

is satisfied by this solution. Applying the operators D_{0}, D_{1} to R, it follows from (2.17) that

D_{0}R = −\frac{1}{4}Q_{0},  (2.20a)
D_{1}R = −\frac{1}{4}Q_{1}.  (2.20b)

Eqs. (2.12a)-(2.12e) also imply that application of D_{0} to {Q_{0}, S_{0}, x_{0}, y_{0}, Q_{1}} gives

D_{0}Q_{0} = 4S_{0} − 2y_{0}Q_{0},  D_{0}S_{0} = \frac{1}{2}Q_{1} − \frac{1}{2}x_{0}Q_{0} + 2P_{0},  (2.21a)
D_{0}x_{0} = −S_{0},  D_{0}y_{0} = −\frac{1}{4}Q_{0}.  (2.21b)
D_{0}Q_{1} = Q_{0} + 4S_{1} − 2y_{0}Q_{1}.  (2.21c)

Further application of D_{0} and D_{1}, using (2.20a), (2.21a), (2.21b) and (2.14), therefore gives

D_{0}^{2}R = \frac{1}{2}y_{0}Q_{0} − S_{0},  (2.22a)
D_{0}^{3}R = −\frac{1}{2}Q_{1} + 2y_{0}S_{0} − \frac{1}{2}y_{0}^{2}Q_{0} − \frac{1}{4}Q_{0}^{2} − 2P_{0}.  (2.22b)

Substituting (2.15), (2.20a), (2.20b) (2.22b), into (2.16) and using (2.14) shows that all terms cancel, verifying the equation.

2b. The Bessel kernel system.

In this case, the system of dynamical equations is given [TW3] by

\frac{∂x_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} ,  j ≠ k,  (2.23a)
\frac{∂y_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} ,  j ≠ k,  (2.23b)
a_{j} \frac{∂x_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{a_{k}(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} + 2y_{j} + \frac{1}{4}y_{0}x_{j},  (2.23c)

a_{j} \frac{∂y_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{a_{k}(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} + \frac{1}{8}(α^{2} − a_{j} + x_{0})x_{j} − \frac{1}{4}y_{0}y_{j},  (2.23d)

\frac{∂x_{0}}{∂a_{j}} = −x_{j}y_{j},  \frac{∂y_{0}}{∂a_{j}} = −\frac{1}{4}x_{j}^{2}.  (2.23e)

This is again a compatible system of nonautonomous Hamiltonian equations generated by the Poisson commuting Hamiltonians G_{α,j}^{B} defined in (2.7).

There again exist two additional conserved quantities for this case. The first is defined by

G_{0}^{B} := x_{0} + \frac{1}{4}y_{0}^{2} + y_{0} + \frac{1}{4}Q_{1},  (2.24)

as may be seen directly by differentiating with respect to the a_{j}’s, using (2.23a)-(2.23e). Since all the quantities appearing in (2.24) vanish in the limit {a_{j} → 0, ∀j}, this difference must vanish, and therefore the invariant relation

x_{0} = −\frac{1}{4}y_{0}^{2} − y_{0} − \frac{1}{4}Q_{1}  (2.25)

is satisfied for this solution. The second conserved quantity is

G_{∞}^{B} := y_{0} + 4 ∑_{j=1}^{2n} a_{j}G_{α,j}^{B} = y_{0} + 4R_{α,1}^{B}
= y_{0} − \frac{1}{4}(α^{2} + x_{0})Q_{0} + \frac{1}{4}Q_{1} + y_{0}S_{0} + 4P_{0} + Q_{0}P_{0} − S_{0}^{2},  (2.26)

Again, due to the Poisson commutativity of the Hamiltonians defined in (2.7), the Hamiltonian vector fields generating the a_{j} deformations when applied to the term R_{α,1}^{B} give zero, and therefore only the explicit dependence of this term upon the parameters need be taken into account when verifying that differentiation of the sum gives zero. Since all the quantities appearing in (2.26) vanish in the limit {a_{j} → 0, ∀j}, the invariant G_{∞}^{B} must also vanish on this particular solution, and we therefore have the relation

y_{0} = −4R_{α,1}^{B} = −4D_{1}F_{α}^{B} = \frac{1}{4}(α^{2} + x_{0})Q_{0} − \frac{1}{4}Q_{1} − y_{0}S_{0} − 4P_{0} − Q_{0}P_{0} + S_{0}^{2}.  (2.27)
The quantities R_{α,1}^{B}, R_{α,2}^{B} are given by
R_{α,1}^{B} = D_{1}F_{α}^{B} = ∑_{j=1}^{2n} a_{j}G_{α,j}^{B}
= −\frac{1}{16}(α^{2} + x_{0})Q_{0} + \frac{1}{16}Q_{1} + \frac{1}{4}y_{0}S_{0} + P_{0} + \frac{1}{4}Q_{0}P_{0} − \frac{1}{4}S_{0}^{2},  (2.28a)
R_{α,2}^{B} = D_{2}F_{α}^{B} = ∑_{j=1}^{2n} a_{j}^{2}G_{α,j}^{B}
= −\frac{1}{16}(α^{2} + x_{0})Q_{1} + \frac{1}{16}Q_{2} + \frac{1}{4}y_{0}S_{1} + P_{1}.  (2.28b)

It again follows from the Poisson commutativity of the Hamiltonians {G_{α,j}^{B}} that the derivatives of R_{α,1}^{B} and R_{α,2}^{B} with respect to the parameters are given by their explicit dependence on these parameters, and hence

D_{1}^{2}F_{α}^{B} = D_{1}R_{α,1}^{B} = \frac{1}{16}Q_{1},  (2.29a)
D_{2}D_{1}F_{α}^{B} = D_{2}R_{α,1}^{B} = \frac{1}{16}Q_{2}.  (2.29b)

From (2.23a)-(2.23e), application of D_{1} to {Q_{1}, S_{1}, x_{0}, y_{0}} gives

D_{1}Q_{1} = Q_{1} + 4S_{1} + \frac{1}{2}y_{0}Q_{1},  D_{1}S_{1} = S_{1} + \frac{1}{8} (α^{2} + x_{0}) Q_{1} − \frac{1}{8}Q_{2} + 2P_{1}, (2.30a)
D_{1}x_{0} = −S_{1},  D_{1}y_{0} = −\frac{1}{4}Q_{1}.  (2.30b)

Further application of D_{1}, using (2.20a), (2.30a), (2.30b) and (2.25) therefore gives

D_{1}^{3}F_{α}^{B} = \frac{1}{16} (1 + \frac{y_{0}}{2} ) Q_{1} + \frac{1}{4}S_{1},  (2.31a)
D_{1}^{4}F_{α}^{B} = \frac{1}{16} (1 + \frac{α^{2}}{2} + \frac{y_{0}}{2} + \frac{y_{0}^{2}}{8} ) Q_{1} + \frac{1}{2}P_{1} + (\frac{1}{2} + \frac{y_{0}}{8} ) S_{1} − \frac{1}{64}Q_{1}^{2} − \frac{1}{32}Q_{2}. (2.31b)

Substitution of (2.28b), (2.29a), (2.29b), (2.31a), (2.31b) into (1.11), and use of (2.27) to replace the term −4D_{1}F_{α}^{B} by y_{0}, and (2.25) to eliminate x_{0}, shows that all the terms cancel, verifying the equation.

3. Deduction of the τ-function equations from isomonodromic deformations

In this section, we show how the full hierarchies of equations derived in [ASV1, ASV2] may be deduced from the Hamiltonian systems (2.12a)-(2.12e), (2.23a)-(2.23e) and also how the corresponding hierarchy is deduced for the case of the sine kernel. The key step is to recast these systems as isomonodromic deformation equations for an associated differential operator in an auxiliary spectral variable z ∈ P^{1}, having rational coefficients with poles at the points {z = a_{j}}, and to interpret the Fredholm determinants τ^{S}, τ^{A} and τ_{α}^{B} as isomonodromic τ-functions.

3a. The Airy kernel isomonodromic system.

The Hamiltonian system (2.12a)-(2.12e) implies that the compatibility conditions

\frac{∂A_{j}}{∂a_{k}} = \frac{[A_{j}, A_{k}]}{a_{j} − a_{k}}, j ≠ k,  (3.1a)
\frac{∂A_{j}}{∂a_{j}} = [a_{j}B + C, A_{j}] − ∑_{{k=1 \atop k≠j}}^{2n} \frac{[A_{j}, A_{k}]}{a_{j} − a_{k}},  (3.1b)

\frac{∂C}{∂a_{k}} = [B, A_{j}]  (3.1c)

are satisfied for the following overdetermined system [HTW]

\frac{∂Ψ^{A}}{∂z} = X^{A}(z)Ψ,  (3.2a)
\frac{∂Ψ^{A}}{∂a_{j}} = −\frac{A_{j}}{z − a_{j}} Ψ^{A}, j = 1, … 2n,  (3.2b)
X^{A}(z) := zB + C + ∑_{j=1}^{2n} \frac{A_{j}}{z − a_{j}} ,  (3.2c)

where Ψ^{A}(z, a_{1}, … a_{2n}) is a 2 × 2 matrix, invertible where defined, and

A_{j} := −\frac{1}{2} ( \begin{matrix}x_{j}y_{j} & y_{j}^{2} \\ −x_{j}^{2} & x_{j}y_{j} \end{matrix} ) ,  (3.3a)
B := ( \begin{matrix}0 & −\frac{1}{2} \\ 0 & 0 \end{matrix} ) ,  C := ( \begin{matrix}y_{0} & \frac{x_{0}}{2} \\ −2 & −y_{0} \end{matrix} ) .  (3.3b)

This implies the invariance of the monodromy of the operator \frac{∂}{∂z} − X^{A}(z) under changes in the parameters {a_{j}}. In view of eq. (2.6), according to the constructions of [JMU, JM], the Fredholm determinant τ^{A} is just the isomonodromic τ-function of the system (3.1a)-(3.2c).

Now define the sequence of 2 × 2 matrices

B_{m} := ∑_{j=1}^{2n} a_{j}^{m}A_{j} = −\frac{1}{2} ( \begin{matrix}S_{m} & P_{m} \\ −Q_{m} & −S_{m} \end{matrix} ) ,  m ∈ N,  (3.4)

where the quantities P_{m}, Q_{m}, S_{m} were defined in (2.11). Expanding X^{A}(z) for large z gives

X^{A}(z) = zB + C + ∑_{m=0}^{∞} \frac{B_{m}}{z^{m+1}} .  (3.5)

Since

G_{j}^{A} = \frac{1}{2}res_{z=∞}tr((X^{A})^{2}(z)),  (3.6)

we have

\frac{1}{2}tr((X^{A})^{2}(z)) = z + G_{0}^{A} + ∑_{m=0}^{∞} \frac{R_{m}^{A}}{z^{m+1}} ,  (3.7)

where

R_{m}^{A} := ∑_{j=1}^{2n} a_{j}^{m}G_{j}^{A} = tr(BB_{m+1} + CB_{m}) + \frac{1}{2}tr ∑_{k=0}^{m−1} B_{k}B_{m−k−1}  (3.8)

(with the last term absent if m = 0) are the quantities defined in (2.9).

Using the fact that the Hamiltonian vector fields generating the a_{j} deformations give zero when applied to the G_{j}^{A}’s, and hence also the R_{m}^{A}’s, it follows that the effect of applying the operators D_{k} to R_{m}^{A} gives just the explicit derivatives,

D_{k}R_{m}^{A} = (m + 1)tr(BB_{m+k}) + m tr(CB_{m+k−1}) + ∑_{l=1}^{m−1} l tr(B_{l+k−1}B_{m−l−1})  (3.9)

(with the sum in the last term absent if m = 0 and the second term absent if m + k = 0).

Applying the operator D_{m} to Ψ^{A}, using (3.2b) and (3.4) gives the sequence of equations

D_{m}Ψ^{A} = − ∑_{k=0}^{∞} \frac{B_{m+k}}{z^{k+1}} Ψ^{A},  m ∈ N.  (3.10)

The compatibility of these equations with (3.2a) implies the following equations for the matrices {B_{m}, C}.

D_{k}B_{m} = mB_{m+k−1} + [C, B_{m+k}] + [B, B_{m+k+1}] + ∑_{l=0}^{m−1}[B_{l}, B_{m+k−l−1}], (3.11a)
D_{k}C = [B, B_{k}],  k, m ∈ N  (3.11b)

(where the first term of (3.11a) is absent if m + k = 0 and the last term is absent if m = 0).

The strategy for deriving the hierarchy of equations for τ^{A} is to now choose a k-value (k_{1}) in (3.9), (3.11a), (3.11b) and use these equations, together with (3.8) to express all the relevant matrix elements of the B_{m}’s for m ≤ k in terms of the R_{k}’s for k < k_{1} and the corresponding D_{k}’s applied repeatedly to them. Equations (3.11a), (3.11b) for k = k_{1} may then be expressed entirely in terms of these quantities, and hence in terms of repeated applications of the operators D_{k} to F^{A} = ln τ^{A}. An essential step in this procedure is to also eliminate the additional variables x_{0}, y_{0} from the equations through use of the invariant conditions (2.14), (2.19).

For example, choosing k_{1} = 1, we note that for m = 0, eq. (3.8) reduces to (2.15) while for k = 0, 1 and m = 0, (3.9) reduces to (2.20a), (2.20b) and for k = 0, m = 0, eqs. (3.11a), (3.11b) give (2.21a), (2.21b). Combining these with the invariant relations (2.14), (2.19) allows us to express the relevant matrix elements of C, B_{0} and B_{1} as

x_{0} = D_{0}R + R^{2},  y_{0} = R,  (3.12a)
Q_{0} = −4D_{0}R,  S_{0} = −2RD_{0}R − D_{0}^{2}R,  (3.12b)
P_{0} = \frac{1}{2}R − \frac{1}{4}D_{0}^{3}R − RD_{0}^{2}R − \frac{1}{2}(D_{0}R)^{2} − R^{2}D_{0}R  (3.12c)
Q_{1} = −2R − 6(D_{0}R)^{2} − D_{0}^{3}R.  (3.12d)

Substituting these in eq. (3.11b) for k = 1 gives (2.16). Similarly, eq. (3.11a) for k = 1, m = 0 and eq (3.8) for m = 1 produce the following expressions for the relevant matrix elements of B_{1} and B_{2}.

S_{1} =D_{0}R − D_{0}D_{1}R + 2(D_{0}R)(D_{0}^{2}R) − R(D_{0}^{3}R) − 2R^{2} + 2RD_{1}R
+ R^{2}D_{0}^{2}R − 6R(D_{o}R)^{2} − 2R^{3}D_{0}R  (3.13a)
Q_{2} = − 2R_{1}^{A} − D_{1}D_{0}^{2}R − 6RD_{0}R + 2RD_{0}D_{1}R − 3(D_{0}R)(D_{0}^{3}R) + \frac{1}{4}(D_{0}^{2}R)^{2}
− 2R^{3} − 16(D_{0}R)^{3} − R^{2}(D_{0}^{3}R) − 4R(D_{0}R)(D_{0}^{2}R)
− 2R^{3}D_{0}R − 4R^{2}(D_{0}R)^{2} − 2R^{2}(D_{0}R)(D_{0}^{2}R) + 4R^{4}(D_{0}R).  (3.13b)

Substitution of (3.13b) in eq. (3.11a) (or (3.9)) for k = 2, m = 0, thus gives

\begin{matrix}4D_{2}R − 2R_{1}^{A} − D_{1}D_{0}^{2}R − 6RD_{0}R + 2RD_{0}D_{1}R − 3(D_{0}R)(D_{0}^{3}R) + \frac{1}{4}(D_{0}^{2}R)^{2} \\ − 2R^{3} − 16(D_{0}R)^{3} − R^{2}(D_{0}^{3}R) − 4R(D_{0}R)(D_{0}^{2}R) \\ − 2R^{3}D_{0}R − 4R^{2}(D_{0}R)^{2} − 2R^{2}(D_{0}R)(D_{0}^{2}R) + 4R^{4}(D_{0}R) = 0. \end{matrix}  (3.14)

as the next equation of the hierarchy. The remaining equations may similarly be expressed in terms of the derivations D_{k} acting upon F^{A}.

3b. The Bessel kernel isomonodromic system.

The Bessel kernel case is so similar to the above that only the pertinent equations will be given, without repeating any details of the procedure. Define for this case, the matrices

X^{B}(z) := B̃ + \frac{(C_{α} − ∑_{j=1}^{2n} A_{j})}{z} + ∑_{j=1}^{2n} \frac{A_{j}}{z − a_{j}} ,  (3.15a)
B̃ := ( \begin{matrix}0 & \frac{1}{8} \\ 0 & 0 \end{matrix} ) ,  C_{α} := −\frac{1}{4} ( \begin{matrix}y_{0} & \frac{1}{2}(x_{0} + α^{2}) \\ 8 & −y_{0} \end{matrix} ) .  (3.15b)

where the A_{j}’s are again defined as in (3.3a).

The Hamiltonian system (2.23a)-(2.23e) implies that the compatibility conditions

\frac{∂A_{j}}{∂a_{k}} = \frac{[A_{j}, A_{k}]}{a_{j} − a_{k}}, j ≠ k,  (3.16a)
a_{j} \frac{∂A_{j}}{∂a_{j}} = [C_{α}, A_{j}] − ∑_{{k=1 \atop k≠j}}^{2n} \frac{a_{k}[A_{j}, A_{k}]}{a_{j} − a_{k}},  (3.16b)

\frac{∂C_{α}}{∂a_{k}} = [B̃, A_{j}]  (3.16c)

are satisfied for the system

\frac{∂Ψ^{B}}{∂z} = X^{B}(z)Ψ^{B}  (3.17a)
\frac{∂Ψ^{B}}{∂a_{j}} = −\frac{A_{j}}{z − a_{j}} Ψ^{B}, j = 1, … 2n,  (3.17b)

where Ψ^{B}(z, a_{1}, … a_{2n}) is again a 2 × 2 matrix, invertible where defined. This again implies the invariance of the monodromy of the operator \frac{∂}{∂z} −X^{B}(z) under changes in the parameters {a_{j}}. In view of eq. (2.7), the Fredholm determinant τ_{α}^{B} is again an isomonodromic τ-function for the system (3.16a)-(3.17b).

Defining the sequence of 2 × 2 matrices {B_{m}, m ∈ N} as in (3.4), and expanding X^{B}(z) for large z gives

X^{B}(z) = B̃ + \frac{C_{α}}{z} + ∑_{m=1}^{∞} \frac{B_{m}}{z^{m+1}} ,  (3.19)

and

\frac{1}{2}tr((X^{B})^{2}(z)) = −\frac{1}{4}z + \frac{(G_{0}^{B} − G_{∞}^{B} + α^{2})}{4z^{2}} + ∑_{m=1}^{∞} \frac{R_{α,m}^{B}}{z^{m+1}} ,  (3.20)

where

R_{α,1}^{B} = \frac{1}{4}(G_{∞}^{B} − G_{0}^{B} − α^{2}) + \frac{1}{2}tr(C_{α}^{2} + 2B̃B_{1}),  (3.21a)
R_{α,m}^{B} = tr(B̃B_{m} + C_{α}B_{m−1}) + \frac{1}{2}tr ∑_{k=1}^{m−2} B_{k}B_{m−k−1},  m ≥ 2  (3.21b)

are the quantities defined in (2.10) and G_{0}^{B}, G_{∞}^{B} are the conserved quatities defined in (2.24), (2.26), which vanish on the particular solutions defined by (2.1a)-(2.1d).

The fact that the Hamiltonian vector fields generating the a_{j} deformations give zero when applied to the G_{α,j}^{B}’s, and R_{α,m}^{B}’s again implies that the effect of applying the operators D_{k} to the R_{α,m}^{B}’s is to evaluate only explicit derivatives with respect to the parameters, giving

D_{k}R_{α,1}^{B} = \frac{1}{2}tr(B̃B_{k})
D_{k}R_{α,m}^{B} = mtr(B̃B_{m+k−1}) + (m − 1)tr(C_{α}B_{m+k−2}) + ∑_{l=1}^{m−2} l tr(B_{l+k−1}B_{m−l−1}), m ≥ 2
(3.22)

(with the sum in the last term absent if m = 2).

Applying the operator D_{m} to Ψ^{B}, using (3.4) and (3.17b), again gives the sequence of equations

D_{m}Ψ^{B} = − ∑_{k=0}^{∞} \frac{B_{m+k}}{z^{k+1}} Ψ^{B},  m ∈ N.  (3.23)

whose compatibility with (3.17a) implies the following equations for the matrices {B_{m}, C_{α}},

D_{k}B_{m} = mB_{m+k−1} + [C_{α}, B_{m+k−1}] + [B̃, B_{m+k}] + ∑_{l=1}^{m−1}[B_{l}, B_{m+k−l−1}], (3.24a)
D_{k}C_{α} = [B̃, B_{k}],  k, m ∈ N, m ≥ 1.  (3.24b)

The hierarchy of equations for τ_{α}^{B} is derived in the same way as for the Airy case. For example, eqs. (3.21a) and (3.21b) for k = 2 reduce to (2.28a), (2.28b), while (3.22) for k = 1, 2, m = 1 reduces to (2.29a), (2.29b), and eqs. (3.24a), (3.24b) for k = 1, 2, m = 1 give (2.30a), (2.30b). Combining these with the invariant relations (2.25), (2.27) allows us to express the relevant matrix elements of C_{α}, B_{1} and B_{2} as

x_{0} = −4(D_{1}R_{α,1}^{B} + (R_{α,1}^{B})^{2} − 4R_{α,1}^{B}),  y_{0} = −4R_{α,1}^{B},  (3.25a)
Q_{1} = 16D_{1}R_{α,1}^{B},  (3.25b)
S_{1} = 8R_{α,1}^{B}D_{1}R_{α,1}^{B} − 4D_{1}R_{α,1}^{B} + 4D_{1}^{2}R_{α,1}^{B},  (3.25c)
P_{1} = R_{α,2}^{B} + α^{2}R_{α,1}^{B} + 4(R_{α,1}^{B})^{2}D_{1}R_{α,1}^{B} − 4(D_{1}R_{α,1}^{B})^{2}
+ 4R_{α,1}^{B}D_{1}^{2}R_{α,1}^{B} − D_{2}R_{α,1}^{B},  (3.25d)
Q_{2} = 16D_{2}R_{α,1}^{B}  (3.25e)

Substituting these in eqs. (3.24a), (3.24b) for k = 2 gives (1.11). Similar calculations for higher values of k yield the further equations of the Bessel hierachy.

3c. The sine kernel system.

For this case, the quantities defined in (2.1a)-(2.1b) satisfy the system of dynamical equations defined in [JMMS, TW1]

\frac{∂x_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} ,  j ≠ k,  (3.25a)
\frac{∂y_{j}}{∂a_{k}} = −\frac{1}{2} \frac{(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} ,  (3.25b)
\frac{∂x_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})x_{k}}{a_{j} − a_{k}} + 2y_{j},  (3.25c)

\frac{∂y_{j}}{∂a_{j}} = \frac{1}{2} ∑_{{k=1 \atop k≠j}}^{n} \frac{(x_{j}y_{k} − y_{j}x_{k})y_{k}}{a_{j} − a_{k}} − \frac{π^{2}}{2} x_{j}.  (3.25d)

This is again a compatible system of nonautonomous Hamiltonian equations, generated by the Poisson commuting Hamiltonians {G_{j}^{S}} defined in (2.5). They imply that the compatibility conditions

\frac{∂A_{j}}{∂a_{k}} = \frac{[A_{j}, A_{k}]}{a_{j} − a_{k}}, j ≠ k  (3.25a)
\frac{∂A_{j}}{∂a_{j}} = [B_{S}, A_{j}] − ∑_{{k=1 \atop k≠j}}^{2n} \frac{[A_{j}, A_{k}]}{a_{j} − a_{k}}, j ≠ k  (3.25b)

are satisfied for the system

\frac{∂Ψ^{S}}{∂z} = X^{S}(z)Ψ^{S},  (3.26a)
\frac{∂Ψ^{S}}{∂a_{j}} = −\frac{A_{j}}{z − a_{j}} Ψ^{S}, j = 1, … 2n,  (3.26b)

where

X^{S}(z) := B_{S} + ∑_{j=1}^{2n} \frac{A_{j}}{z − a_{j}} ,  (3.27a)
B_{S} := ( \begin{matrix}0 & \frac{π^{2}}{2} \\ −2 & 0 \end{matrix} ) ,  (3.27b)

with the A_{j}’s again defined as in (3.4). As in the previous cases, this implies the invariance of the monodromy of the operator \frac{∂}{∂z} − X^{S}(z). In view of eq. (2.5), the Fredholm determinant τ^{S} is an isomonodromic τ-function for the system (3.25a)-(3.26b).

Expanding X^{S}(z) for large z gives

X^{S}(z) = B_{S} + ∑_{m=0}^{∞} \frac{B_{m}}{z^{m+1}} ,  (3.28)

with the matrices {B_{m}, m ∈ N} again defined as in (3.4), and

\frac{1}{2}tr((X^{S})^{2}(z)) = −π^{2} + ∑_{m=0}^{∞} \frac{R_{m}^{S}}{z^{m+1}} ,  (3.29)

where

R_{m}^{S} := ∑_{j=1}^{2n} a_{j}^{m}G_{j}^{S} = tr(B_{S}B_{m}) + \frac{1}{2}tr ∑_{k=0}^{m−1} B_{k}B_{m−k−1}, m ∈ N.  (3.30)

Applying the operators D_{k} to R_{m}^{S} again just differentiates explicitly with respect to the parameters, giving

D_{k}R_{m}^{S} = m tr(B_{S}B_{m+k−1}) + ∑_{l=1}^{m−1} l tr(B_{l+k−1}B_{m−l−1})  (3.31)

(with the first term absent if k + m = 0 and the sum in the last term absent if m = 0). Applying D_{m} to Ψ^{S}, using (3.28) and (3.26b), gives the sequence of equations

D_{m}Ψ^{S} = − ∑_{k=0}^{∞} \frac{B_{m+k}}{z^{k+1}} Ψ^{S},  m ∈ N,  (3.32)

whose compatibility with (3.26a) implies the following equations for the matrices {B_{m}},

D_{k}B_{m} = mB_{m+k−1} + [B_{S}, B_{m+k}] + ∑_{l=0}^{m−1}[B_{l}, B_{m+k−l−1}].  (3.33)

The hierarchy of equations for τ^{S} is derived in the same way as for the Airy and Bessel cases, except that we no longer have two conserved quantities like G_{0}^{A,B}, G_{∞}^{A,B}. To derive a closed system of equations, we are obliged to include two further dependent variables τ_{±}^{S}, which we choose as the nonvanishing entries of the matrix [B_{S}, B_{0}]τ^{S},

τ_{+}^{S} := (2P_{0} − \frac{π^{2}}{2} Q_{0})τ^{S},  τ_{−}^{S} := S_{0}τ^{S}.  (3.34)

The remaining component of B_{0}, which cancels in the commutator [B_{S}, B_{0}], is

R_{0}^{S} = tr(B_{S}B_{0}) = P_{0} + \frac{π^{2}}{4} Q_{0} = 0,  (3.35)

where the first equality follows from choosing m = 0 in (3.30). This provides a single conserved quantity that vanishes for the particular solution defined by (2.1a)-(2.1b).

To derive the hierarchy of τ-function equations , we first combine eqs. (3.34)-(3.35), which allows us to express the matrix elements of B_{0} as

Q_{0} = −\frac{τ_{+}^{S}}{π^{2}τ^{S}} ,  P_{0} = \frac{τ_{+}^{S}}{4τ^{S}} ,  S_{0} = \frac{τ_{−}^{S}}{τ^{S}} .  (3.36)

Eq. (3.33) for k = 0, m = 0 gives

D_{0}P_{0} = −π^{2}S_{0}, D_{0}Q_{0} = 4S_{0}, D_{0}S_{0} = 2P_{0} − \frac{π^{2}}{2} Q_{0},  (3.37)

and substituting (3.35), (3.36) in (3.37) gives

D_{0}τ^{S} = 0,  (3.38a)
D_{0}τ_{−}^{S} = τ_{+}^{S}, D_{0}τ_{+}^{S} = −4π^{2}τ_{−}^{S}.  (3.38b)

These equations are the lowest ones in the sine kernel hierarchy; note that they are linear because of the vanishing of the invariant R_{0}^{S}. To obtain higher, nonlinear equations, we first note that eq. (3.30) for m = 1 gives

R_{1}^{S} = P_{1} + \frac{π^{2}}{4} Q_{1} + \frac{1}{4}(S_{0}^{2} − Q_{0}P_{0}),  (3.39)

while (3.31) for k = 0, 1, m = 1 reduces to

D_{0}R_{1}^{S} = R_{0} = 0,  (3.40a)
D_{1}R_{1}^{S} = P_{1} + \frac{π^{2}}{4} Q_{1}.  (3.40b)

The first of these just gives the equation

D_{0}D_{1}τ^{S} = 0,  (3.41)

which already follows from (3.38a). The second, combined with eq. (3.39) and eq. (3.33) for k = 1, m = 0 gives the further equation

τ^{S}D_{1}^{2}τ^{S} − (D_{1}τ^{S})^{2} = τ^{S}D_{1}τ^{S} − \frac{1}{4}(τ_{−}^{S})^{2} − \frac{1}{16π^{2}} (τ_{+}^{S})^{2}.  (3.42)

Equation (3.33) for k = 1, m = 0 gives

D_{1}S_{0} = 2P_{1} − \frac{π^{2}}{2} Q_{1},  D_{1}P_{0} = −π^{2}S_{1},  D_{1}Q_{0} = 4S_{1}.  (3.43)

Solving these, together with (3.40b), gives the following expressions for the matrix entries of B_{1}:

Q_{1} = \frac{2}{π^{2}} \frac{D_{1}τ^{S}}{τ^{S}} − \frac{1}{2π^{2}} (\frac{τ_{−}^{S}}{τ^{S}} )^{2} − \frac{1}{8π^{4}} (\frac{τ_{−}^{S}}{τ^{S}} )^{2} − \frac{1}{π^{2}} D_{1} (\frac{τ_{−}^{S}}{τ^{S}} ) ,  (3.44a)
P_{1} = \frac{1}{2} \frac{D_{1}τ^{S}}{τ^{S}} − \frac{1}{8} (\frac{τ_{−}^{S}}{τ^{S}} )^{2} − \frac{1}{32π^{2}} (\frac{τ_{−}^{S}}{τ^{S}} )^{2} + \frac{1}{4}D_{1} (\frac{τ_{−}^{S}}{τ^{S}} ) ,  (3.44b)
S_{1} = −\frac{1}{4π^{2}} D_{1} (\frac{τ_{+}^{S}}{τ^{S}} ) .  (3.44c)

Combining eq. (3.33) for (k = 1, m = 1) and for (k = 2, m = 0) gives

D_{2}Q_{0} = D_{1}Q_{1} − Q_{1},  D_{2}P_{0} = D_{1}P_{1} − P_{1},  D_{2}S_{0} = D_{1}S_{1} − S_{1},  (3.45)

Substitution of (3.36), (3.44a)-(3.44c) into (3.45) gives the next equations of the hierarchy. Repeating this procedure for higher (k, m) values similarly generates the higher equations.

4. Classical R-Matrix approach and relation to isospectral flows

In [ASV1, ASV2], a key step in deriving the hierarchies of equations for the Fredholm determinants τ^{A} and τ_{α}^{B} was to begin with certain bilinear equations satisfied by KP τfunctions with respect to the flow parameters {t_{1}, t_{2}, …} and to then use Virasoro constraints to replace the t_{m}-derivations at vanishing t-values by the operators D_{m}. In this section, we show how the classical R-matrix approach to the underlying isomonodromic deformation equations developed in [H] provides a direct link with commuting isospectral flows in the loop algebra 𝔰̃𝔩(2), without the requirement that these arise as reduced KP flows. This fits into the broader framework of commutative isospectral flows in loop algebras with respect to the rational R-matrix Poisson (or Adler-Kostant-Symes) structure [RS, AV, AHP] (and allows us to include the sine kernel case, which does not appear as a reduced KP flow).

First we recall [H, HTW] that the isomonodromic deformation equations (3.1a)-(3.1c), (3.16a)-(3.16c), (3.25a)-(3.25b) may be viewed as Hamiltonian equations on the space of sets {A_{j}}_{j=1...2n} of 𝔰𝔩(2) elements, with respect to the Lie Poisson bracket, extended in the Airy and Bessel cases by the canonical variables (x_{0}, y_{0}). (The particular form (3.3a) for the A_{j}’s just represents a canonical parametrization on the symplectic leaves for which the Casimir invariants {trA_{j}^{2}} all vanish.) The formulae (3.2c), (3.15a), (3.27a) define a Poisson embedding of this space into the space 𝔰̃𝔩(2)_{R}^{∗} of rational, traceless 2 × 2 matrices depending rationally on the auxiliary loop variable z, with respect to the Lie Poisson bracket on 𝔰̃𝔩(2) corresponding to the Lie bracket:

[X, Y]_{R} := \frac{1}{2}[RX, Y] + \frac{1}{2}[X, RY],  (4.1)

where

R := P_{+} − P_{−}  (4.2)

is the classical R−matrix, given by the difference of the projection operators

\begin{aligned}P_{+} : 𝔰̃𝔩(2) & → 𝔰̃𝔩_{+}(2)  P_{+} : 𝔰̃𝔩(2) → 𝔰̃𝔩_{+}(2) \\ P_{−} : X & → X_{+}  P_{−} : X → X_{−} \end{aligned}  (4.3)

to the subalgebras 𝔰̃𝔩_{+}(2), 𝔰̃𝔩_{−}(2) consisting respectively of the nonnegative and negative terms in the Laurent expanson of X(z) for large z. The space 𝔰̃𝔩(2)_{R}^{∗} is identified as a subspace of 𝔰̃𝔩(2) through the trace-residue pairing

< X, Y >:= res_{z=∞}tr(X(z)Y (z)).  (4.4)

In this setting, the isomonodromic deformation equations (3.1a)-(3.1c), (3.16a)-(3.16c), (3.25a)-(3.25b) may all be expressed in the form

\frac{∂X}{∂a_{j}} = −[(dG_{j})_{−}, X] + \frac{∂(dG_{j})_{−}}{∂z} ,  (4.5a)
(dG_{j})_{−} = −\frac{A_{j}}{z − a_{j}} ,  (4.5b)

where X denotes X^{S}, X^{A} or X^{B}, and G_{j} denotes G_{j}^{S}, G_{j}^{A} or G_{α,J}^{B} respectively. Viewing the Hamiltonians {G_{j}} as spectral invariants defined on the space 𝔰̃𝔩(2), eq. (4.5a) follows from the Adler-Kostant-Symes theorem, in view of the relations

\frac{∂_{0}X}{∂a_{j}} = −\frac{∂(dG_{j})_{−}}{∂z} ,  (4.6)

where \frac{∂_{0}X}{∂a_{j}} denotes the derivative with respect to the explicit dependence on the parameters {a_{j}} only.

Rather than using the spectral invariants {G_{j}} as Hamiltonians, we consider the Hamiltonian equations generated by the linear combinations R_{m}^{S}, R_{m}^{A} or R_{α,m}^{B} defined in (2.8), (2.9) (2.10), which are all of the form

D_{m}X = −[(dR_{m})_{−}, X] + \frac{∂(dR_{m})_{−}}{∂z} ,  (4.7)

with the respective identifications for X and {R_{m}}. These are just equations (3.11a), (3.11b), (3.24a), (3.24b) or (3.33), depending on the identification, since

R_{m} = \frac{1}{2}res_{z=∞}z^{m}trX^{2}(z),  (4.8)

and therefore dR_{m}, viewed as an element of 𝔰̃𝔩(2), is just

dR_{m} = z^{m}X(z) = ∑_{k=0}^{∞} \frac{B_{k}}{z^{k−m+1}} .  (4.9)

implying

(dR_{m})_{−} = ∑_{k=0}^{∞} \frac{B_{m+k}}{z^{k+1}} .  (4.10)

If, instead of the nonautonomous systems occurring here because of the identifications of the a_{j}’s as multi-time parameters, we consider the autonomous systems generated by the same set of Hamiltonians {R_{0}, R_{1}, …}, denoting the corresponding flow parameters {t_{0}, t_{1}, …}, the resulting equations have the isospectral form

\frac{∂X}{∂t_{m}} = ±[(dR_{m})_{±}, X],  (4.11)

where either of the projections (dR_{m})_{±} may be used, since the differential dR_{m}, given by (4.10), commutes with X. Although these systems are generated by the same Hamiltonians as the nonautonomous systems (4.7), they of course do not generate isomonodromic deformations of the operator \frac{∂}{∂z} − X(z), and in fact are not even compatible with the systems (4.7); however, they are compatible amongst themselves, generating commuting isospectral Hamiltonian flows. The close relationship between the autonomous and associated nonautonomous systems implies a correspondence between the structure of the resulting hierarchies.

To see this, we substitute the expressions (3.2c), (3.15a) and (3.27a) for X(z) and (4.10) for dR_{m} into (4.11) to obtain the systems

\frac{∂B_{m}}{∂t_{m}} = [C, B_{m+k}] + [B, B_{m+k+1}] + ∑_{l=0}^{m−1}[B_{l}, B_{m+k−l−1}],  (4.12a)
\frac{∂C}{∂t_{m}} = [B, B_{k}],  k, m ∈ N  (4.12b)

for X = X^{A},

\frac{∂B_{m}}{∂t_{m}} = [C, B_{m+k−1}] + [B̃, B_{m+k}] + + ∑_{l=1}^{m−1}[B_{l}, B_{m+k−l−1}],  (4.13a)
\frac{∂C_{α}}{∂t_{m}} = [B̃, B_{k}],  k, m ∈ N, m ≥ 1.  (4.13b)

for X = X_{α}^{B} and

\frac{∂B_{m}}{∂t_{m}} = [B_{S}, B_{m+k}] + ∑_{l=0}^{m−1}[B_{l}, B_{m+k−l−1}]  (4.15)

for X = X^{S}. These only differ from the equations (3.24a), (3.11b), (3.24a), (3.24b) and (3.33) by the absence of the term mB_{m+k−1} in the right hand side of (4.12b), (4.13b), (4.15) and the replacement

D_{m} → \frac{∂}{∂t_{m}}  (4.16)

for the derivation on the left hand side. The procedure for deriving hierarchies for such systems is well known in the isospectral context (see, e.g. [FNR] for details); the recursive procedure used in section 3 above is just the analog of this approach applied to the isomonodromic systems (3.24a), (3.11b), (3.24a), (3.24b) and (3.33).

As a final point, it should be noted that almost nothing in the derivation of the τ-function equations of sections 2 and 3 depended on the fact that the specific τ-functions involved were equal to the Fredholm determinants (1.2), (1.4), (1.6). Everything just followed from the general form of the isomonodromic deformation equations (3.1a)-(3.1c), (3.16a)-(3.16c) and (3.25a)-(3.25b), the only features specific to the identifications of τ^{A}, τ_{α}^{B}, τ^{S} as Fredholm determinants being the fact that the matrix residues A_{j} were of rank 1 (as seen from the parametrization (3.3a)) and the invariants G_{0}^{A}, G_{∞}^{A}, G_{0}^{B}, G_{∞}^{B} vanished. By allowing these invariants, as well as the constants {detA_{j}}, to take arbitrary values, an identical procedure leads to equations for the τ-functions of the general isomonodromic systems, which only differ from the ones derived in sections 2 and 3 by the nonzero constant values of the two additional invariants G_{0}^{A}, G_{∞}^{A} or G_{0}^{B} and G_{∞}^{B}. For example, eq. (2.16) is replaced in the general case by

D_{0}^{3}R − 4D_{1}R + 2R + +4(g_{∞}^{2} − g_{0})D_{0}R − 2g_{∞}(D_{0}^{2}R + 2RD_{0}R) + 6(D_{0}R)^{2} = 0, (4.17)

where g_{0}, g_{∞} are the values taken by the invariants G_{0}^{A}, G_{∞}^{A}, respectively. The other equations of these hierarchies may similarly expressed in a way that allows arbitrary values for these constants.

Acknowledgements. The author would like to thank P. van Moerbeke for helpful discussions relating to this work.

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