Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

Abstract

We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL (N , C). The corresponding operator acts in the direct sum of N (n − 3) copies of L 2 S 1 . Its kernel has a block integrable form and is expressed in terms of fundamental solutions of n−2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant n-point system via a decomposition of the punctured sphere into pairs of pants. For N = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to n = 4 gives a series representation of the general solution to Painlevé VI equation.

Argyres-Douglas theories and Liouville irregular states

“…The coefficients of this linear combination turn out to depend on c 1 and β 1 but are independent of β 0 . As pointed out 20 In other words, T (z)dz is invariant under this transformation. 21 As mentioned already, we here use a slightly different convention for |I…”

“…Since the stress tensor T is of dimension two, this also rescales T as T → −(c 2 ) −2 T . 20 As a result, we have…”

“…However, there has recently been remarkable progress in the study of the partition functions of AD theories. In particular, the authors of [15] have evaluated strong coupling expansions of the Nekrasov partition functions of the (A 1 , A 2 ), (A 1 , A 3 ) and (A 1 , D 4 ) AD theories via a remarkable relation to the tau-functions of Painlevé equations, inspired by [16] and subsequent works [17][18][19][20]. 2 After these works, the author of [24][25][26] has nicely shown that these strong coupling expansions can be reproduced from irregular conformal blocks of Virasoro algebra.…”

The τ-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

“…The minor expansion of the Fredholm determinant (1.2) in this particular basis gives rise to interesting combinatorics. In the case of Painlevé VI, V, III the combinatorics correspond to certain Nekrasov Partition functions of certain Gauge theories [13].…”

Induction of hierarchy and time through one-dimensional probability space with certain topologies

“…where the Jacobi theta function is ϑ 3 (z|τ ) = ∑ n∈Z e iπn 2 τ +2inz ; trM µ M ν = 2 cos 2πσ µν when the parameter space of (θ 0 , θ t , θ 1 , θ ∞ ) is M [24] [25] [22] [26]. For other algebraic solutions, see [27].…”