Quantization of Calogero-Painlevé System and Multi-Particle Quantum Painlevé Equations II-VI

We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.

From Heun Class Equations to Painlevé Equations

In the first part of our paper we discuss linear 2nd order differential equations in the complex domain, especially Heun class equations, that is, the Heun equation and its confluent cases. The second part of our paper is devoted to Painlevé I-VI equations. Our philosophy is to treat these families of equations in a unified way. This philosophy works especially well for Heun class equations. We discuss its classification into 5 supertypes, subdivided into 10 types (not counting trivial cases). We also introduce in a unified way deformed Heun class equations, which contain an additional nonlogarithmic singularity. We show that there is a direct relationship between deformed Heun class equations and all Painlevé equations. In particular, Painlevé equations can be also divided into 5 supertypes, and subdivided into 10 types. This relationship is not so easy to describe in a completely unified way, because the choice of the ''time variable'' may depend on the type. We describe unified treatments for several possible ''time variables''.

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

We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with $$\tau $$ τ -functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and $$N_f=2$$ N f = 2 on a circle.