q-deformed Painlevéτfunction andq-deformed conformal blocks

Abstract

We propose q-deformation of the Gamayun-Iorgov-Lisovyy formula for Painlevé τ function. Namely we propose the formula for τ function for q-difference Painlevé equation corresponding to A(1)′ 7 surface (and A (1) 1 symmetry) in Sakai classification. In this formula τ function equals the series of q-Virasoro Whittaker conformal blocks (equivalently Nekrasov partition functions for pure SU (2) 5d theory). [arXiv:q-alg/9605002v2].

Cluster integrable systems, q-Painlevé equations and their quantization

“…Note that this relation between q-Painlevé equations and cluster mutations (but without relation to cluster integrable systems) was already noticed in [49] for A equation in [7]. See also [33], where relations to cluster integrable systems were also mentioned.…”

“…Recall the meaning of other relations in this case: the spectral curve of relativistic Toda is the Seiberg-Witten curve of pure SU(2) 5d theory, the corresponding Nekrasov partition function equals to the Whittaker limit of conformal block for q-deformed Virasoro algebra. Finally, it has been proposed in [7], that certain sum of these Nekrasov 5d partition functions for pure theory in case q 1 q 2 = 1 (c = 1 in CFT terms) is equal to the tau-function of the q-Painlevé equation A (1) 7 , the same equation which we get by deautonomization. This is our main example, some of the statements in sections 3, 4 were checked explicitly only in this case.…”

“…The relation (4.25) was proposed in [7], the conformal q → 1 limit of the relations (4.22), (4.24), (4.25) should follow from the results of [8]. Note that appearance of operator a leads to the fact that bilinear relations contain conformal blocks with different parameters q 1 , q 2 (or different central charges).…”

“…In the paper [7] the formal solution of these equations was proposed, namely τ 1 = T (u, s; q|Z), τ 3 = is 1/2 T (uq, s; q|Z), where…”

“…The convergence of the series (B.3) was proven in several cases, in [7] for q 1 q 2 = 1 and infinite radius of convergence, in [17] for q 1 , q 2 > 1 or q 2 =q 1 , |q 1 | > 1 and nonzero radius of convergence.…”