The dependence on the monodromy data of the isomonodromic tau function

Abstract

[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno did not derive the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters). We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing and monodromy-preserving) are dealt with at the same level. We thus provide variational formulae for the isomonodromic tau function with respect to the (generalized) monodromy data. The construction applies more generally: given any (sufficiently well-behaved) family of Riemann-Hilbert problems (RHP) where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form Omega (not necessarily closed) on the deformation space (Malgrange's differential), defined off Malgrange's divisor. We then introduce the notion of discrete Schlesinger transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Even if Omega is not closed, its difference evaluated along the original solution and the transformed one, is shown to be the logarithmic differential (on the deformation space) of a function. As a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker-Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector. Some exemples (Painleve' II and finite Toplitz/Hankel determinants) are provided.

34 pages, 7 figures. An important "Corrigendum" is now available as arXiv:1601.04790

References in corpus (2)

• Symplectic Manifolds and Isomonodromic Deformations
• Moment determinants as isomonodromic tau functions

Cited by in corpus (34)

• The transition between the gap probabilities from the Pearcey to the Airy process; a Riemann-Hilbert approach
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• Monodromy dependence and connection formulae for isomonodromic tau functions
• Connection problem for the sine-Gordon/Painlevé III tau function and irregular conformal blocks
• On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo-Miwa-Ueno differential
• Asymptotics for the partition function in two-cut random matrix models
• Riemann-Hilbert approach to multi-time processes; the Airy and the Pearcey case
• On some Hamiltonian properties of the isomonodromic tau functions
• Fredholm determinant solutions of the Painlevé II hierarchy and gap probabilities of determinantal point processes
• Tau functions as Widom constants
• $\mathcal{N}=2^{\ast }$ gauge theory, free fermions on the torus and Painlevé VI
• Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials
• On the connection problem for Painlevé I
• Fredholm determinants and pole-free solutions to the noncommutative Painleve' II equation
• Noncommutative Painlevé equations and systems of Calogero type
• Quantum spectral problems and isomonodromic deformations
• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Connection problem for the tau-function of the Sine-Gordon reduction of Painlevé-III equation via the Riemann-Hilbert approach
• Darboux transformations and random point processes
• Gap probabilities for the Generalized Bessel process: a Riemann-Hilbert approach
• Tau functions and the limit of block Toeplitz determinants
• The Kontsevich-Penner matrix integral, isomonodromic tau functions and open intersection numbers
• Asymptotics of the Tacnode process: a transition between the gap probabilities from the Tacnode to the Airy process
• Jánossy densities and Darboux transformations for the Stark and cylindrical KdV equations
• The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes' phenomenon
• Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus
• Matrix models for stationary Gromov-Witten invariants of the Riemann sphere
• The Malgrange Form and Fredholm Determinants
• On the tau function of the hypergeometric equation
• Circular quiver gauge theories, isomonodromic deformations and $W_N$ fermions on the torus
• A fully noncommutative Painlevé II hierarchy: Lax pair and solutions related to Fredholm determinants
• Global rigidity and exponential moments for soft and hard edge point processes
• Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure
• Asymptotics of the deformed higher order Airy-kernel determinants and applications