Damghan University Press Journal of Holography Applications in Physics 2783-4778 3 4 2023 11 01 q-discrete Painleve VI equations from M2-branes 57 80 369 10.22128/jhap.2023.747.1062 EN Tomoki Nosaka Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, China 0000-0002-1517-0526 Journal Article 2023 10 12 In this paper we review the novel connection between a theory of N M2-branes on (C^2/Z_2 x C/Z_2)/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_5) of D_5=SO(10). 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 M2-brane partition function satisfies a bilinear difference equation associated with W(D_5), 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. 112 no. 6, (2022) 109] and [Moriyama, JHEP 08 (2023) 191]. https://jhap.du.ac.ir/article_369_73e119a74f14445ab4c582d032e7f7da.pdf