The dependence on the monodromy data of the isomonodromic tau function

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 provided an algebraic construction of its derivatives with respect to isomonodromic times. However the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes’ parameters) was not derived. 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 formulæ 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 Ω (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 Ω 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 exemplifications in the setting of the Painlevé II equation and finite Töplitz/Hankel determinants are provided.