Integrable Fredholm Operators and Dual Isomonodromic Deformations

The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed as nonautonomous Hamiltonian systems on an auxiliary symplectic vector space M, whose Poisson quotient, under a parametric family of Hamiltonian group actions, is identified with a Poisson submanifold of the loop algebra Lgl_R(r) with respect to the rational R-matrix structure. The matrix Riemann-Hilbert problem method is used to identify the auxiliary space M with the data defining the integral kernel of the resolvent operator at the endpoints of the curve segments. A second associated isomonodromic family of covariant derivative operators D_z is derived, having rank n=2m, and r finite regular singular points at the values of the exponents defining the kernel of K. This family is similarly embedded into the algebra Lgl_R(n) through a dual parametric family of Poisson quotients of M. The operators D_z are shown to be analogously associated to the integral operator obtained from K through a Fourier-Laplace transform.Comment: PlainTeX 32g

Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solutions associated to nonlinear sigma models (quantum cohomology) or Landau-Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa.Comment: 35 pages, 3 figures. Minor revisions for compatibility with the recently posted Part II (arXiv:1312.4825

Integro-Difference Equation for a correlation function of the spin-12{1\over2} Heisenberg XXZ chain

We consider the Ferromagnetic-String-Formation-Probability correlation function (FSFP) for the spin-$1\over 2$ Heisenberg XXZ chain. We construct a completely integrable system of integro-difference equations (IDE), which has the FSFP as a $\tau$-function. We derive the associated Riemann-Hilbert problem and obtain the large distance asymptotics of the FSFP correlator in some limiting cases.Comment: 14 pages, latex+epsf, 1 figur

Determinant representation for a quantum correlation function of the lattice sine-Gordon model

We consider a completely integrable lattice regularization of the sine-Gordon model with discrete space and continuous time. We derive a determinant representation for a correlation function which in the continuum limit turns into the correlation function of local fields. The determinant is then embedded into a system of integrable integro-differential equations. The leading asymptotic behaviour of the correlation function is described in terms of the solution of a Riemann Hilbert Problem (RHP) related to the system of integro-differential equations. The leading term in the asymptotical decomposition of the solution of the RHP is obtained.Comment: 30 pages Latex2e, 2 Figures, epsf. Significantly extended and revised versio

Random Words, Toeplitz Determinants and Integrable Systems. I

It is proved that the limiting distribution of the length of the longest weakly increasing subsequence in an inhomogeneous random word is related to the distribution function for the eigenvalues of a certain direct sum of Gaussian unitary ensembles subject to an overall constraint that the eigenvalues lie in a hyperplane.Comment: 15 pages, no figure

The tt*-Toda equations of A_n type

In previous articles we have studied the A_n tt*-Toda equations (topological-antitopological fusion equations of Toda type) of Cecotti and Vafa, giving details mainly for n=3. Here we give a proof of the existence and uniqueness of global solutions for any n, and a new treatment of their asymptotic data, monodromy data, and Stokes data.Comment: 68 pages, 7 figures. Typographical errors have been corrected in this versio

Connection formulae for the radial Toda equations I

This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of type $A_n$. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the general solution at zero and infinity. To reach this goal we are using a fusion of the PDE analysis and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou which is applicable to 2D Toda in view of its Lax integrability. A principal technical challenge is the extension of the nonlinear steepest descent analysis to Riemann-Hilbert problems of matrix rank greater than $2$. In this paper, we meet this challenge for the case $n=2$ (the rank $3$ case) and it already captures the principal features of the general $n$ case.Comment: 68 pages, 16 figure