Dual isomonodromic tau functions and determinants of integrable Fredholm operators (Q2759647)

The author deals with the rational covariant derivative operators on the punctured Riemann sphere, having the form NEWLINE\[NEWLINED_\lambda= \frac{\partial}{\partial\lambda}- N(\lambda), \quad N(\lambda):= B+ \sum_{i=1}^n \frac{N_i}{\lambda-\alpha_i}, \tag{1}NEWLINE\]NEWLINE where \(B= \text{diag} (\beta_1,\dots, \beta_r)\), \(N_j\in \text{gl} (R,\mathbb{C})\). The author reviews the Hamiltonian approach to dual isomonodromic deformations in the setting of rational \(R\)-matrix structures on loop algebras. The construction of a particular class of solutions to the deformation equations follow from the matrix Riemann-Hilbert approach. The author interpretes the notion of duality in terms of the data defining the Riemann-Hilbert problem and Laplace-Fourier transforms of the corresponding Fredholm integral operators.NEWLINENEWLINEFor the entire collection see [Zbl 0967.00059].