Mathematics

Abstract:In these two lectures I will try to give an elementary explanation of how one can get the representation of the general isomonodromic tau function on sphere with punctures as the Fredholm determinant of certain operator with matrix-valued integral kernel. I'm going to show the free-fermionic construction of the isomonodromic tau function, introduce generalized Wick theorem, and explain how to get the Fredholm determinant of our interest from the free-fermionic formulas. The second part will be devoted to the study of such obtained Fredholm determinant. It will be rewritten in terms of some projection operators, as matrix Toeplitz determinant, and after all differentiated explicitly in order to reproduce the definition of the Jimbo-Miwa-Ueno tau function.References:1) Alexander Alexandrov, Anton Zabrodin, "Free fermions and tau-functions", https://arxiv.org/abs/1212.60492) Nikita Nekrasov, Andrei Okounkov, "Seiberg-Witten Theory and Random Partitions", https://arxiv.org/abs/hep-th/03062383) Mikio Sato, Tetsuji Miwa, Michio Jimbo, "Holonomic quantum fields"4) O. Gamayun, N. Iorgov, O. Lisovyy, "Conformal field theory of Painlevé VI", https://arxiv.org/abs/1207.07875) P. Gavrylenko, A. Marshakov, "Free fermions, W-algebras and isomonodromic deformations ", https://arxiv.org/abs/1605.045546) P. Gavrylenko, O. Lisovyy, "Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions", https://arxiv.org/abs/1608.009587) M. Cafasso, P. Gavrylenko, O. Lisovyy, "Tau functions as Widom constants", https://arxiv.org/abs/1712.08546