q-discrete Painleve VI equations from M۲-branes

In this paper we review the novel connection between a theory of N M۲-branes on (C^۲/Z_۲ x C/Z_۲)/Z_k and a discrete integrable system. Besides the IR duality induced by the Hanany-Witten transitions in the type IIB brane construction, the Fermi gas formalism tells us that the partition function of this theory enjoys a larger discrete symmetry which is the Weyl group W(D_۵) of D_۵=SO(۱۰). The Fermi gas formalism, together with the topological string/spectral theory correspondence and the connection between the integrable systems and the Nekrasov partition functions recently found, further suggests that the grand partition function of this M۲-brane partition function satisfies a bilinear difference equation associated with W(D_۵), called q-deformed Painleve VI. By using the exact values of the partition functions we identify the explicit expression of the bilinear equations and confirm that these equations are indeed satisfied for higher order in the chemical potential dual to the rank N. This article is based on [Bonelli, Globlek, Kubo, Nosaka, Tanzini, Lett. Math. Phys. ۱۱۲ no. ۶, (۲۰۲۲) ۱۰۹] and [Moriyama, JHEP ۰۸ (۲۰۲۳) ۱۹۱].