×

Cluster integrable systems, \(q\)-Painlevé equations and their quantization. (English) Zbl 1387.83078

Summary: We discuss the relation between the cluster integrable systems and \(q\)-difference Painlevé equations. The Newton polygons corresponding to these integrable systems are all 16 convex polygons with a single interior point. The Painlevé dynamics is interpreted as deautonomization of the discrete flows, generated by a sequence of the cluster quiver mutations, supplemented by permutations of quiver vertices.
We also define quantum \(q\)-Painlevé systems by quantization of the corresponding cluster variety. We present formal solution of these equations for the case of pure gauge theory using \(q\)-deformed conformal blocks or 5-dimensional Nekrasov functions. We propose, that quantum cluster structure of the Painlevé system provides generalization of the isomonodromy/CFT correspondence for arbitrary central charge.

MSC:

83E30 String and superstring theories in gravitational theory
81T45 Topological field theories in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
81R12 Groups and algebras in quantum theory and relations with integrable systems

Software:

Quiver
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville correlation functions from four-dimensional gauge theories, Lett. Math. Phys.91 (2010) 167 [arXiv:0906.3219] [INSPIRE]. · Zbl 1185.81111 · doi:10.1007/s11005-010-0369-5
[2] M. Alim et al., \[N=2 \mathcal{N}=2\] quantum field theories and their BPS quivers, Adv. Theor. Math. Phys.18 (2014) 27 [arXiv:1112.3984] [INSPIRE]. · Zbl 1309.81142
[3] H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP01 (2010) 125 [arXiv:0910.4431] [INSPIRE]. · Zbl 1269.81157 · doi:10.1007/JHEP01(2010)125
[4] L. Bao, V. Mitev, E. Pomoni, M. Taki and F. Yagi, Non-lagrangian theories from brane junctions, JHEP01 (2014) 175 [arXiv:1310.3841] [INSPIRE]. · Zbl 1333.83014 · doi:10.1007/JHEP01(2014)175
[5] A. Berenstein and A. Zelevinsky, Quantum cluster algebras, Adv. in Math.195 (2005) 405 [math/0404446]. · Zbl 1124.20028
[6] M.A. Bershtein and A.I. Shchechkin, Bilinear equations on Painlevé τ functions from CFT, Commun. Math. Phys.339 (2015) 1021 [arXiv:1406.3008] [INSPIRE]. · Zbl 1332.34141 · doi:10.1007/s00220-015-2427-4
[7] M.A. Bershtein and A.I. Shchechkin, q-deformed Painlevé τ function and q-deformed conformal blocks, J. Phys.A 50 (2017) 085202 [arXiv:1608.02566] [INSPIRE]. · Zbl 1360.81264
[8] M.A. Bershtein and A.I. Shchechkin, Backlund transformation of Painlevé III(D8) τ function, J. Phys.A 50 (2017) 115205 [arXiv:1608.02568] [INSPIRE]. · Zbl 1396.33038
[9] G. Bonelli, A. Grassi and A. Tanzini, Quantum curves and q-deformed Painlevé equations, arXiv:1710.11603 [INSPIRE]. · Zbl 1431.39003
[10] M. Caorsi and S. Cecotti, Categorical webs and S-duality in \[4dN=2 \mathcal{N}=2\] QFT, arXiv:1707.08981 [INSPIRE]. · Zbl 1420.81025
[11] A.S. Carstea, A. Dzhamay and T. Takenawa, Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations, J. Phys.A 50 (2017) 405202 [arXiv:1702.04907]. · Zbl 1375.14144
[12] S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, Surveys in differential geometry18 (2013) [arXiv:1103.5832] [INSPIRE]. · Zbl 1320.81085
[13] T. Eguchi and H. Kanno, Five-dimensional gauge theories and local mirror symmetry, Nucl. Phys.B 586 (2000) 331 [hep-th/0005008] [INSPIRE]. · Zbl 1043.81714
[14] T. Eguchi and H. Kanno, Topological strings and Nekrasov’s formulas, JHEP12 (2003) 006 [hep-th/0310235] [INSPIRE]. · doi:10.1088/1126-6708/2003/12/006
[15] P. Etingof, Whittaker functions on quantum groups and q-deformed Toda operators, Amer. Math. Soc. Transl.194 (1999) 9 [math/9901053]. · Zbl 1157.33327
[16] B. Feigin et al., Kernel function and quantum algebras, RIMS Kōkyūroku1689 (2010) 133 [arXiv:1002.2485]. · Zbl 1375.81197
[17] G. Felder and M. Müller-Lennert, Analyticity of Nekrasov partition functions, arXiv:1709.05232 [INSPIRE]. · Zbl 1400.81138
[18] V.V. Fock and A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ec. Norm. Super.42 (2009) 865 [math/0311245] [INSPIRE]. · Zbl 1225.53070
[19] V. Fock and A. Goncharov, Cluster X-varieties, amalgamation and Poisson-Lie groups, in Algebraic Geometry Theory and Number Theory, Progress in Mathematics volume 253, Birkhäuser, Boston U.S.A. (2006), math/0508408. · Zbl 1162.22014
[20] V. Fock and A. Marshakov, A note on quantum groups and relativistic Toda theory, Nucl. Phys. Proc. Suppl.B 56 (1997) 208. · Zbl 0925.58029
[21] V.V. Fock and A. Marshakov, Loop groups, clusters, dimers and integrable systems, in Geometry and quantization of moduli spaces, V. Fock et al. eds., Birkhäuser, Boston U.S.A. (2016), arXiv:1401.1606 [INSPIRE]. · Zbl 1417.37248
[22] S. Fomin and A. Zelevinsky, Cluster algebras IV: coefficients, Compos. Math.143 (2007) 112 [math/0602259]. · Zbl 1127.16023
[23] S. Franco, Y.-H. He, C. Sun and Y. Xiao, A comprehensive survey of brane tilings, Int. J. Mod. Phys.A 32 (2017) 1750142 [arXiv:1702.03958] [INSPIRE]. · Zbl 1375.81197 · doi:10.1142/S0217751X17501421
[24] D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, Adv. Math.234 (2013) 239 [arXiv:0907.3987] [INSPIRE]. · Zbl 1358.81150 · doi:10.1016/j.aim.2012.09.027
[25] O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP10 (2012) 038 [Erratum ibid.1210 (2012) 183] [arXiv:1207.0787] [INSPIRE]. · Zbl 1397.81307
[26] O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys.A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE]. · Zbl 1282.34096
[27] P. Gavrylenko, Isomonodromic τ -functions and WNconformal blocks, JHEP09 (2015) 167 [arXiv:1505.00259] [INSPIRE]. · Zbl 1388.81664
[28] A.B. Goncharov and R. Kenyon, Dimers and cluster integrable systems, Ann. Sci. Ec. Norm. Sup46 (2013) 747 [arXiv:1107.5588] [INSPIRE]. · Zbl 1288.37025 · doi:10.24033/asens.2201
[29] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov and A. Morozov, Integrability and Seiberg-Witten exact solution, Phys. Lett.B 355 (1995) 466 [hep-th/9505035] [INSPIRE]. · Zbl 0997.81567
[30] B. Grammaticos and A. Ramani, Discrete Painlevé equations: a review, Lect. Notes Phys.644 (2004) 245. · Zbl 1064.39019
[31] K. Hasegawa, Quantizing the Bäcklund transformations of Painlevé equations and the quantum discrete Painlevé VI equation, Adv. Studies Pure Math.61 (2011) 275 [math/0703036]. . · Zbl 1241.81114
[32] A. Hanany and R.-K. Seong, Brane tilings and reflexive polygons, Fortsch. Phys.60 (2012) 695 [arXiv:1201.2614] [INSPIRE]. · Zbl 1258.81068 · doi:10.1002/prop.201200008
[33] A. Hone, R. Inoue Discrete Painlevé equations from Y -systems, J. Phys.47 (2014) 474007 [arXiv:1405.5379]. · Zbl 1326.13015
[34] A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys.10 (2006) 1 [hep-th/0306032] [INSPIRE]. · Zbl 1101.81088
[35] N. Iorgov, O. Lisovyy and J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Commun. Math. Phys.336 (2015) 671 [arXiv:1401.6104] [INSPIRE]. · Zbl 1311.30029 · doi:10.1007/s00220-014-2245-0
[36] M. Jimbo, H. Nagoya and H. Sakai, CFT approach to the q-Painlevé VI equation, J. Int. Syst.2 (2017) 1 [arXiv:1706.01940]. · Zbl 1400.39008
[37] N. Joishi, N. Nakazono and Y. Shi, Lattice equations arising from discrete Painlevé systems (I): (A2 + A1)(1)and (A1 + A1′)(1)cases, J. Math. Phys.56 (2015) 092705 [arXiv:1401.7044].
[38] K. Kajiwara et al., Cubic pencils and Painlevé hamiltonians, Funkcial. Ekvac.48 (2005) 147 [nlin/0403009]. · Zbl 1161.34064
[39] K. Kajiwara, M. Noumi, and Y. Yamada, Geometric aspects of Painlevé equations, J. Phys.A 50 (2017) 073001 [arXiv:1509.08186]. · Zbl 1441.34095
[40] O. Kruglinskaya and A. Marshakov, On Lie groups and Toda lattices, J. Phys.A 48 (2015) 125201 [arXiv:1404.6507] [INSPIRE]. · Zbl 1312.37041
[41] I.M. Krichever and D.H. Phong, On the integrable geometry of soliton equations and N = 2 supersymmetric gauge theories, J. Diff. Geom.45 (1997) 349 [hep-th/9604199] [INSPIRE]. · Zbl 0889.58044
[42] G. Kuroki, Quantum groups and quantization of Weyl group symmetries of Painlevé systems, Adv. Studies Pure Math.61 (2011) 289 [arXiv:0808.2604]. · Zbl 1247.81213
[43] A. Marshakov, Lie groups, cluster variables and integrable systems, J. Geom. Phys.67 (2013) 16 [arXiv:1207.1869] [INSPIRE]. · Zbl 1266.53083 · doi:10.1016/j.geomphys.2012.12.003
[44] A. Mironov and A. Morozov, q-Painleve equation from Virasoro constraints, arXiv:1708.07479 [INSPIRE]. · Zbl 1398.81220
[45] S. Mizoguchi and Y. Yamada, W (E10) symmetry, M-theory and Painlevé equations, Phys. Lett.B 537 (2002) 130 [hep-th/0202152] [INSPIRE]. · Zbl 0995.81112
[46] G. Musiker and C. Stump, A compendium on the cluster algebra and quiver package in Sage, Sem. Lothar. Comb.65 (2011) [arXiv:1102.4844]. · Zbl 1293.13008
[47] H. Nagoya and Y. Yamada, Symmetries of quantum Lax equations for the Painlevé equations, Ann. Henri Poincaré15 (2014) 313 [arXiv:1206.5963] [INSPIRE]. · Zbl 1288.81062 · doi:10.1007/s00023-013-0237-9
[48] N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys.B 531 (1998) 323 [hep-th/9609219] [INSPIRE]. · Zbl 0961.81116
[49] N. Okubo, Bilinear equations and q-discrete Painlevé equations satisfied by variables and coefficients in cluster algebras, J. Phys.A 48 (2015) 355201 [arXiv:1505.03067]. · Zbl 1338.37091
[50] N. Okubo, Co-primeness preserving higher dimensional extension of q-discrete Painlevé I, II equations, arXiv:1704.05403.
[51] H. Sakai, Rational surfaces associated with affine root systems and geometry of the Painlevé equations, Comm. Math.Phys.220 (2001) 165. · Zbl 1010.34083
[52] N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics, Phys. Lett.B 388 (1996) 753 [hep-th/9608111] [INSPIRE].
[53] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE]. · Zbl 0996.81510
[54] T. Tsuda, τ functions of q-Painlevé III and IV equations, Lett. Math. Phys.75 (2006) 39. · Zbl 1119.39018
[55] T. Tsuda, Tropical Weyl group action via point configurations and τ -functions of the q-Painlevé equations, Lett. Math. Phys.77 (2006) 21. · Zbl 1118.39007 · doi:10.1007/s11005-006-0052-z
[56] T. Tsuda, Universal character and q-difference Painlevé equations with affine Weyl groups, Math. Ann.345 (2009) 395 [arXiv:0811:3112]. · Zbl 1193.34177 · doi:10.1007/s00208-009-0359-z
[57] T. Tsuda and T. Masuda, q-Painlevé VI equation arising from q-UC hierarchy, Comm. Math. Phys.262 (2006) 595. · Zbl 1104.32007
[58] Y. Yamada, Amoebae of type E, talk at the VII Workshop on Geometric Correspondences of Gauge Theories, July 10-14, SISSA, Trieste, Italy (2017). · Zbl 1288.81062
[59] A. Yanagida, Norm of the Whittaker vector of the deformed Virasoro algebra, Eur. J. Phys.36 (2015) 035019 [arXiv:1411.0642].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.