q-discrete Painleve VI equations from M2-branes

Document Type : Regular article


Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, China


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].


Main Subjects

 [1] G. Bonelli, F. Globlek, N. Kubo, T. Nosaka, and A. Tanzini, “M2-branes and q-Painlevé equations”, Lett. Math. Phys. 112 no. 6, 109 (2022). DOI: 10.1007/s11005-022-01597-0
[2] S. Moriyama and T. Nosaka, “40 bilinear relations of q-Painlevé VI from N = 4 super Chern-Simons theory”, JHEP 08, 191 (2023). DOI: 10.1007/JHEP08(2023)191
[3] Y. Imamura and K. Kimura, “On the moduli space of elliptic Maxwell-Chern-Simons theories”, Prog. Theor. Phys. 120, 509–523 (2008). DOI: 10.1143/PTP.120.509
[4] Y. Imamura and K. Kimura, “N=4 Chern-Simons theories with auxiliary vector multiplets”, JHEP 10, 040 (2008). DOI: 10.1088/1126-6708/2008/10/040
[5] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee, and J. Park, “N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds”, JHEP 09, 002 (2008). DOI: 10.1088/1126-6708/2008/09/002
[6] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee, and J. Park, “N=4 Superconformal ChernSimons Theories with Hyper and Twisted Hyper Multiplets”, JHEP 07, 091 (2008).
[7] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena, “N=6 superconformal ChernSimons-matter theories, M2-branes and their gravity duals”, JHEP 0810, 091 (2008).
[8] O. Aharony, O. Bergman, and D. L. Jafferis, “Fractional M2-branes”, JHEP 11, 043 (2008).
[9] V. Pestun, “Localization of the four-dimensional N=4 SYM to a two-sphere and 1/8 BPS Wilson loops”, JHEP 12, 067 (2012). DOI: 10.1007/JHEP12(2012)067
[10] A. Kapustin, B. Willett, and I. Yaakov, “Nonperturbative Tests of Three-Dimensional Dualities”, JHEP 10, 013 (2010). DOI: 10.1007/JHEP10(2010)013
[11] C. P. Herzog, I. R. Klebanov, S. S. Pufu, and T. Tesileanu, “Multi-Matrix Models and Tri-Sasaki Einstein Spaces”, Phys. Rev. D 83, 046001 (2011). DOI: 10.1103/PhysRevD.83.046001
[12] I. R. Klebanov and A. A. Tseytlin, “Near extremal black hole entropy and ?uctuating three-branes”, Nucl. Phys. B 479, 319–335 (1996). DOI: 10.1016/0550-3213(96)00459-2
[13] M. Marino and P. Putrov, “ABJM theory as a Fermi gas”, J. Stat. Mech. 1203, P03001 (2012). DOI: 10.1088/1742-5468/2012/03/P03001
[14] S. Moriyama and T. Nosaka, “Superconformal Chern-Simons Partition Functions of Affine D-type Quiver from Fermi Gas”, JHEP 09, 054 (2015). DOI: 10.1007/JHEP09(2015)054
[15] B. Assel, N. Drukker, and J. Felix, “Partition functions of 3d Dˆ-quivers and their mirror duals from 1d free fermions”, JHEP 08, 071 (2015). DOI: 10.1007/JHEP08(2015)071
[16] H. Fuji, S. Hirano, and S. Moriyama, “Summing Up All Genus Free Energy of ABJM Matrix Model”, JHEP 08, 001 (2011). DOI: 10.1007/JHEP08(2011)001
[17] Y. Hatsuda, M. Marino, S. Moriyama, and K. Okuyama, “Non-perturbative e?ects and the refined topological string”, JHEP 09, 168 (2014). DOI: 10.1007/JHEP09(2014)168
[18] A. Grassi, Y. Hatsuda, and M. Marino, “Topological Strings from Quantum Mechanics”, Annales Henri Poincare 17 no. 11, 3177–3235 (2016). DOI: 10.1007/s00023-016-0479-4
[19] N. Kubo, “Fermi gas approach to general rank theories and quantum curves”, JHEP 10, 158 (2020). DOI: 10.1007/JHEP10(2020)158
[20] A. Grassi, Y. Hatsuda, and M. Marino, “Quantization conditions and functional equations in ABJ(M) theories”, J. Phys. A 49 no. 11, 115401 (2016). DOI: 10.1088/1751-8113/49/11/115401
[21] H. Sakai, “Rational Surfaces Associated with Affine Root Systems and Geometry of the Painlevé Equations”, Commun. Math. Phys. 220, 165–229 (2001). DOI: 10.1007/s002200100446
[22] G. Bonelli, A. Grassi, and A. Tanzini, “Quantum curves and q-deformed Painlevé equations”, Lett. Math. Phys. 109 no. 9, 1961–2001 (2019). DOI: 10.1007/s11005-019-01174-y
[23] K. Kajiwara, M. Noumi, and Y. Yamada, “Geometric Aspects of Painlevé Equations”, J. Phys. A 50 no. 7, 073001 (2017). DOI: 10.1088/1751-8121/50/7/073001
[24] T. Kitao, K. Ohta, and N. Ohta, “Three-dimensional gauge dynamics from brane configurations with (p,q)-Fivebrane”, Nucl. Phys. B 539, 79–106 (1999). DOI: 10.1016/S0550-3213(98)00726-3
[25] O. Bergman, A. Hanany, A. Karch, and B. Kol, “Branes and supersymmetry breaking in three-dimensional gauge theories”, JHEP 10, 036 (1999). DOI: 10.1088/1126-6708/1999/10/036
[26] T. Furukawa, K. Matsumura, S. Moriyama, and T. Nakanishi, “Duality cascades and a?ne Weyl groups”, JHEP 05, 132 (2022). DOI: 10.1007/JHEP05(2022)132
[27] S. Moriyama and T. Nosaka, “Exact Instanton Expansion of Superconformal Chern-Simons Theories from Topological Strings”, JHEP 05, 022 (2015). DOI: 10.1007/JHEP05(2015)022
[28] S. Moriyama, S. Nakayama, and T. Nosaka, “Instanton Effects in Rank Deformed Superconformal Chern-Simons Theories from Topological Strings”, JHEP 08, 003 (2017). DOI: 10.1007/JHEP08(2017)003
[29] S. Moriyama, T. Nosaka, and K. Yano, “Superconformal Chern-Simons Theories from del Pezzo Geometries” JHEP 11, 089 (2017). DOI: 10.1007/JHEP11(2017)089
[30] N. Kubo, S. Moriyama, and T. Nosaka, “Symmetry Breaking in Quantum Curves and Super Chern-Simons Matrix Models”, JHEP 01, 210 (2019). DOI: 10.1007/JHEP01(2019)210
[31] N. Kubo and S. Moriyama, “Hanany-Witten Transition in Quantum Curves”, JHEP 12, 101 (2019). DOI: 10.1007/JHEP12(2019)101
[32] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and threedimensional gauge dynamics”, Nucl.Phys. B 492, 152–190 (1997). DOI: 10.1016/S0550-3213(97)00157-0
[33] B. Assel, “Hanany-Witten effect and SL(2, Z) dualities in matrix models”, JHEP 10, 117 (2014). DOI: 10.1007/JHEP10(2014)117
[34] M. Honda and N. Kubo, “Non-perturbative tests of duality cascades in three dimensional supersymmetric gauge theories”, JHEP 07, 012 (2021). DOI: 10.1007/JHEP07(2021)012
[35] S. Moriyama and T. Nosaka, in preparation.
[36] R. Kashaev, M. Marino, and S. Zakany, “Matrix Models from Operators and Topological Strings, 2”, Annales Henri Poincare 17 no. 10, 2741–2781 (2016). DOI: 10.1007/s00023-016-0471-z
[37] M. Bershtein and A. Shchechkin, “Painlevé equations from Nakajima–Yoshioka blowup relations”, Lett. Math. Phys. 109 no. 11, 2359–2402 (2019). DOI: 10.1007/s11005-019-01198-4
[38] M. Bershtein, P. Gavrylenko, and A. Marshakov, “Cluster Toda chains and Nekrasov functions”, Theor. Math. Phys. 198 no. 2, 157–188 (2019). DOI: 10.1134/S0040577919020016
[39] T. Nosaka, “SU(N) q-Toda equations from mass deformed ABJM theory”, JHEP 06, 060 (2021). DOI: 10.1007/JHEP06(2021)060
[40] M. Marino and P. Putrov, “Exact Results in ABJM Theory from Topological Strings”, JHEP 06, 011 (2010). DOI: 10.1007/JHEP06(2010)011
[41] M. Honda and K. Okuyama, “Exact results on ABJ theory and the re?ned topological string”, JHEP 08, 148 (2014). DOI: 10.1007/JHEP08(2014)148
[42] A. Grassi and M. Marino, “M-theoretic matrix models”, JHEP 02, 115 (2015). DOI: 10.1007/JHEP02(2015)115
[43] S. H. Katz, A. Klemm, and C. Vafa, “Geometric engineering of quantum field theories”, Nucl. Phys. B 497, 173–195 (1997). DOI: 10.1016/S0550-3213(97)00282-4
[44] G. Bonelli, O. Lisovyy, K. Maruyoshi, A. Sciarappa, and A. Tanzini, “On Painlevé/-gauge theory correspondence”, Lett. Matth. Phys. 107 no. 12, 2359–2413 (2017). DOI: 10.1007/s11005-017-0983-6
[45] M. Jimbo, H. Nagoya, and H. Sakai, “CFT approach to the q-Painlevé VI equation“, J. Integrab. Syst. 2 no. 1, 1 (2017). DOI: 10.48550/arXiv.1706.01940
[46] N. Seiberg, “Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics”, Phys. Lett. B 388, 753–760 (1996). DOI: 10.1016/S0370-2693(96)01215-4
[47] V. Mitev, E. Pomoni, M. Taki, and F. Yagi, “Fiber-Base Duality and Global Symmetry Enhancement”, JHEP 04, 052 (2015). DOI: 10.1007/JHEP04(2015)052
[48] T. Tsuda and T. Masuda, “q-Painlevé VI Equation Arising from q-UC Hierarchy”, Communications in Mathematical Physics 262 no. 3, 595–609 (2006) 595–609. DOI: 10.1007/s00220-005-1461-z
[49] G. Bonelli, A. Grassi, and A. Tanzini, “Seiberg–Witten theory as a Fermi gas”, Lett. Math. Phys. 107 no. 1, 1–30 (2017). DOI: 10.1007/s11005-016-0893-z
[50] A. B. Zamolodchikov, “Painlevé III and 2-d polymers”, Nucl. Phys. B 432, 427–456 (1994). DOI: 10.1016/0550-3213(94)90029-9
[51] C. A. Tracy and H. Widom, “Fredholm determinants and the mkdv/sinh-gordon hierarchies”, Commun. Math. Phys. 179, 1–9 (1996). DOI: 10.1007/BF02103713
[52] B. M. McCoy, C. A. Tracy, and T. T. Wu, “Painlevé Functions of the Third Kind”, J. Math. Phys. 18, 1058 (1977). DOI: 10.1063/1.523367 
Volume 3, Issue 4
November 2023
Pages 57-80
  • Receive Date: 12 October 2023
  • Revise Date: 12 November 2023
  • Accept Date: 12 November 2023