Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

Abstract

We derive series representations for the tau functions of the q-Painlevé V, III 1 , III 2 , and III 3 equations, as degenerations of the tau functions of the q-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of q-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the q-Painlevé V, III 1 , III 2 , and III 3 equations are written by our tau functions. We also prove that our tau functions for the q-Painlevé III 1 , III 2 , and III 3 equations satisfy the three-term bilinear equations for them.

BPS Quivers of Five-Dimensional SCFTs, Topological Strings and q-Painlevé Equations

“…In "Appendix A" we collect all the necessary formulas for five-dimensional Nekrasov functions, while in "Appendix B" we show how the bilinear equations of [30] can be also recovered from the cluster algebra approach. These are seemingly different from the ones of [26], but we show that they correspond to a different choice of initial coefficients for the cluster algebra. "Appendix C" collects few technical points related to improved saddle point expansion of NO partition functions used in the paper.…”

“…Notice that, given the geometrical datum of the toric Calabi-Yau, it is possible to obtain its associated quiver from the corresponding dimer model [22,23], and the A-cluster variables defined from this quiver lead to bilinear equations. In many cases these have been shown to be satisfied by dual partition functions of Topological String theory on this same Calabi-Yau [24], or by q-deformed Virasoro conformal blocks [21,[25][26][27]. These can also be rephrased in terms of K-theoretic blowup equations [24,28,29].…”

$${\mathcal {N}}$$ = $$2^*$$ Gauge Theory, Free Fermions on the Torus and Painlevé VI

“…Further developments along these lines are reported in [7][8][9][10][11][12][13]. This correspondence has been then broadened to the case of q-difference Painlevé equations, q-Virasoro algebra [14][15][16][17][18][19] and five dimensional N = 1 gauge theories and nonperturbative topological strings [20][21][22][23][24]. The four-dimensional theories in question can be seen as arising as the world-volume theories of stacks of M5 branes wrapped around a punctured Riemann surface C g,n described by the compactification of the relevant 6d N = (0, 2) superconformal field theory [25].…”

Painlevé equations from Nakajima–Yoshioka blowup relations

“…We will present the proof of this result in Subsection 4.1 (see Proposition 4.1). For another proof, see [MN18].…”

“…After the paper was written, we noticed preprint[MN18] with another (based on degeneration of the results of[JNS17]) proof of this conjecture .…”