×
Nagoya, Hajime

On \(q\)-isomonodromic deformations and \(q\)-Nekrasov functions. (English) Zbl 1467.39006

SIGMA, Symmetry Integrability Geom. Methods Appl. 17, Paper 050, 21 p. (2021).
Summary: We construct a fundamental system of a \(q\)-difference Lax pair of rank \(N\) in terms of 5d Nekrasov functions with \(q=t\). Our fundamental system degenerates by the limit \(q\to 1\) to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at \(0\) and \(\infty \), we obtain several determinantal identities of the tau functions.

Cited in 2 Documents

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A12 Discrete version of topics in analysis
33E17 Painlevé-type functions
39A36 Integrable difference and lattice equations; integrability tests
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain

Keywords:

isomonodromic deformations; Nekrasov functions; Painlevé equations; determinantal identities
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Awata, H. and Feigin, B. and Shiraishi, J., Quantum algebraic approach to refined topological vertex, Journal of High Energy Physics, 2012, 3, no. 3, 041, 35 pages, (2012) · Zbl 1309.81112 · doi:10.1007/JHEP03(2012)041
[2] Alday, Luis F. and Gaiotto, Davide and Tachikawa, Yuji, Liouville correlation functions from four-dimensional gauge theories, Letters in Mathematical Physics, 91, 2, 167-197, (2010) · Zbl 1185.81111 · doi:10.1007/s11005-010-0369-5
[3] Bershtein, M. A. and Gavrilenko, P. G. and Marshakov, A. V., Cluster {T}oda lattices and {N}ekrasov functions, Theoretical and Mathematical Physics, 198, 2, 157-188, (2019) · Zbl 1421.81083 · doi:10.1134/S0040577919020016
[4] Bershtein, M. A. and Shchechkin, A. I., {\(q\)}-deformed {P}ainlev\'e {\( \tau \)} function and {\(q\)}-deformed conformal blocks, Journal of Physics. A. Mathematical and Theoretical, 50, 8, 085202, 22 pages, (2017) · Zbl 1360.81264 · doi:10.1088/1751-8121/aa5572
[5] Bershtein, M. and Shchechkin, A., Painlev\'e equations from {N}akajima–{Y}oshioka blowup relations, Letters in Mathematical Physics, 109, 11, 2359-2402, (2019) · Zbl 1428.81127 · doi:10.1007/s11005-019-01198-4
[6] Bonelli, Giulio and Lisovyy, Oleg and Maruyoshi, Kazunobu and Sciarappa, Antonio and Tanzini, Alessandro, On {P}ainlev\'e/gauge theory correspondence, Letters in Mathematical Physics, 107, 12, 2359-2413, (2017) · Zbl 1380.34130 · doi:10.1007/s11005-017-0983-6
[7] Bonelli, Giulio and Grassi, Alba and Tanzini, Alessandro, Quantum curves and {\(q\)}-deformed {P}ainlev\'e equations, Letters in Mathematical Physics, 109, 9, 1961-2001, (2019) · Zbl 1431.39003 · doi:10.1007/s11005-019-01174-y
[8] Bonelli, Giulio and Tanzini, Alessandro and Zhao, Jian, The {L}iouville side of the vortex, Journal of High Energy Physics, 2011, 9, no. 9, 096, 24 pages, (2011) · Zbl 1301.81189 · doi:10.1007/JHEP09(2011)096
[9] Bonelli, Giulio and Tanzini, Alessandro and Zhao, Jian, Vertices, vortices & interacting surface operators, Journal of High Energy Physics, 2012, 6, no. 6, 178, 22 pages, (2012) · Zbl 1397.81136 · doi:10.1007/JHEP06(2012)178
[10] Felder, Giovanni and M\"uller-Lennert, Martin, Analyticity of {N}ekrasov partition functions, Communications in Mathematical Physics, 364, 2, 683-718, (2018) · Zbl 1400.81138 · doi:10.1007/s00220-018-3270-1
[11] Fuji, Kenta and Suzuki, Takao, Drinfeld–{S}okolov hierarchies of type {\(A\)} and fourth order {P}ainlev\'e systems, Funkcialaj Ekvacioj. Serio Internacia, 53, 1, 143-167, (2010) · Zbl 1202.34156 · doi:10.1619/fesi.53.143
[12] Suzuki, Takao and Fuji, Kenta, Higher order {P}ainlev\'e systems of type {\(A\), Progress in mathematics of integrable systems, RIMS K\^oky\^uroku Bessatsu, B30, 181-208, (2012), Res. Inst. Math. Sci. (RIMS), Kyoto} · Zbl 1264.34174
[13] Gasper, George and Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96, xxvi+428, (2004), Cambridge University Press, Cambridge · Zbl 1129.33005 · doi:10.1017/CBO9780511526251
[14] Gamayun, O. and Iorgov, N. and Lisovyy, O., Conformal field theory of {P}ainlev\'e {VI}, Journal of High Energy Physics, 2012, 10, no. 10, 038, 25 pages, (2012) · Zbl 1397.81307 · doi:10.1007/JHEP10(2012)038
[15] Gamayun, O. and Iorgov, N. and Lisovyy, O., How instanton combinatorics solves {P}ainlev\'e {VI}, {V} and {III}s, Journal of Physics. A. Mathematical and Theoretical, 46, 33, 335203, 29 pages, (2013) · Zbl 1282.34096 · doi:10.1088/1751-8113/46/33/335203
[16] Gavrylenko, Pavlo and Iorgov, Nikolai and Lisovyy, Oleg, Higher-rank isomonodromic deformations and {\(W\)}-algebras, Letters in Mathematical Physics, 110, 2, 327-364, (2020) · Zbl 1433.81097 · doi:10.1007/s11005-019-01207-6
[17] Iorgov, N. and Lisovyy, O. and Teschner, J., Isomonodromic tau-functions from {L}iouville conformal blocks, Communications in Mathematical Physics, 336, 2, 671-694, (2015) · Zbl 1311.30029 · doi:10.1007/s00220-014-2245-0
[18] Ishikawa, Masao and Mano, Toshiyuki and Tsuda, Teruhisa, Determinant structure for {\( \tau \)}-function of holonomic deformation of linear differential equations, Communications in Mathematical Physics, 363, 3, 1081-1101, (2018) · Zbl 1406.34105 · doi:10.1007/s00220-018-3256-z
[19] Jimbo, Michio and Miwa, Tetsuji and Ueno, Kimio, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. {I}. {G}eneral theory and {\( \tau \)}-function, Physica D. Nonlinear Phenomena, 2, 2, 306-352, (1981) · Zbl 1194.34167 · doi:10.1016/0167-2789(81)90013-0
[20] Jimbo, Michio and Miwa, Tetsuji, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. {II}, Physica D. Nonlinear Phenomena, 2, 3, 407-448, (1981) · Zbl 1194.34166 · doi:10.1016/0167-2789(81)90021-X
[21] Jimbo, M. and Nagoya, H. and Sakai, H., C{FT} approach to the {\(q}-{P\)}ainlev\'e {VI} equation, Journal of Integrable Systems, 2, 1, xyx009, 27 pages, (2017) · Zbl 1400.39008 · doi:10.1093/integr/xyx009
[22] Jimbo, Michio and Sakai, Hidetaka, A {\(q\)}-analog of the sixth {P}ainlev\'e equation, Letters in Mathematical Physics, 38, 2, 145-154, (1996) · Zbl 0859.39006 · doi:10.1007/BF00398316
[23] Lisovyy, O. and Nagoya, H. and Roussillon, J., Irregular conformal blocks and connection formulae for {P}ainlev\'e {V} functions, Journal of Mathematical Physics, 59, 9, 091409, 27 pages, (2018) · Zbl 1404.33020 · doi:10.1063/1.5031841
[24] Matsuhira, Yuya and Nagoya, Hajime, Combinatorial expressions for the tau functions of {\(q}-{P\)}ainlev\'e {V} and {III} equations, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 15, 074, 17 pages, (2019) · Zbl 1423.39009 · doi:10.3842/SIGMA.2019.074
[25] Nagao, Hidehito, A variation of the {\(q}-{P\)}ainlev\'e system with affine {W}eyl group symmetry of type {\(E_7^{(1)}\)}, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 13, 092, 18 pages, (2017) · Zbl 1387.14096 · doi:10.3842/SIGMA.2017.092
[26] Nagao, Hidehito and Yamada, Yasuhiko, Variations of the {\(q\)}-{G}arnier system, Journal of Physics. A. Mathematical and Theoretical, 51, 13, 135204, 19 pages, (2018) · Zbl 1392.35266 · doi:10.1088/1751-8121/aaae31
[27] Nagao, Hidehito and Yamada, Yasuhiko, Study of {\(q\)}-{G}arnier system by {P}ad\'e method, Funkcialaj Ekvacioj. Serio Internacia, 61, 1, 109-133, (2018) · Zbl 1423.14218 · doi:10.1619/fesi.61.109
[28] Nagoya, Hajime, Irregular conformal blocks, with an application to the fifth and fourth {P}ainlev\'e equations, Journal of Mathematical Physics, 56, 12, 123505, 24 pages, (2015) · Zbl 1328.81195 · doi:10.1063/1.4937760
[29] Nagoya, Hajime, Remarks on irregular conformal blocks and {P}ainlev\'e {III} and {II} tau functions, Proceedings of the {M}eeting for {S}tudy of {N}umber {T}heory, {H}opf {A}lgebras and {R}elated {T}opics, 105-124, (2019), Yokohama Publ., Yokohama
[30] Nakajima, Hiraku and Yoshioka, Kota, Instanton counting on blowup. {I}. 4-dimensional pure gauge theory, Inventiones Mathematicae, 162, 2, 313-355, (2005) · Zbl 1100.14009 · doi:10.1007/s00222-005-0444-1
[31] Nieri, Fabrizio and Pasquetti, Sara and Passerini, Filippo, 3d and 5d gauge theory partition functions as {\(q\)}-deformed {CFT} correlators, Letters in Mathematical Physics, 105, 1, 109-148, (2015) · Zbl 1305.81112 · doi:10.1007/s11005-014-0727-9
[32] Park, K., A certain generalization of {\(q\)}-hypergeometric functions and their related monodromy preserving deformation, Journal of Integrable Systems, 3, 1, xyy019, 14 pages, (2018) · Zbl 1422.39030 · doi:10.1093/integr/xyy019
[33] Sakai, Hidetaka, A {\(q\)}-analog of the {G}arnier system, Funkcialaj Ekvacioj. Serio Internacia, 48, 2, 273-297, (2005) · Zbl 1117.33014 · doi:10.1619/fesi.48.273
[34] Schlesinger, Ludwig, \"Uber eine {K}lasse von {D}ifferentialsystemen beliebiger {O}rdnung mit festen kritischen {P}unkten, Journal f\"ur die Reine und Angewandte Mathematik. [Crelle’s Journal], 141, 96-145, (1912) · JFM 43.0385.01 · doi:10.1515/crll.1912.141.96
[35] Suzuki, Takao, Affine {W}eyl group symmetry of the {G}arnier system, Funkcialaj Ekvacioj. Serio Internacia, 48, 2, 203-230, (2005) · Zbl 1118.33012 · doi:10.1619/fesi.48.203
[36] Suzuki, Takao, A class of higher order {P}ainlev\'e systems arising from integrable hierarchies of type {\(A\)}, Algebraic and Geometric Aspects of Integrable Systems and Random Matrices, Contemp. Math., 593, 125-141, (2013), Amer. Math. Soc., Providence, RI · Zbl 1287.34082 · doi:10.1090/conm/593/11875
[37] Tsuda, Teruhisa, U{C} hierarchy and monodromy preserving deformation, Journal f\"ur die Reine und Angewandte Mathematik. [Crelle’s Journal], 690, 1-34, (2014) · Zbl 1339.37060 · doi:10.1515/crelle-2012-0022
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.