Abstract
Bethe/Gauge correspondence as it is usually stated is ill-defined in five dimensions and needs a “non-perturbative” completion; a related problem also appears in three dimensions. It has been suggested that this problem, probably due to incompleteness of Omega background regularization in odd dimension, may be solved if we consider gauge theory on compact S 5 and S 3 geometries. We will develop this idea further by giving a full Bethe/Gauge correspondence dictionary on S 5 and S 3 focussing mainly on the eigenfunctions of (open and closed) relativistic 2-particle Toda chain and its quantized spectral curve: these are most properly written in terms of non-perturbatively completed NS open topological strings. A key ingredient is Faddeev’s modular double structure which is naturally implemented by the S 5 and S 3 geometries.
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References
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].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
S. Cecotti, Stringy cosmic strings and SUSY gauge theories, Phys. Lett. B 244 (1990) 23 [INSPIRE].
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].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin systems and the WKB approximation, arXiv:0907.3987 [INSPIRE].
N. Nekrasov, Five dimensional gauge theories and relativistic integrable systems, Nucl. Phys. B 531 (1998) 323 [hep-th/9609219] [INSPIRE].
S.N.M. Ruijsenaars, Relativistic Toda systems, Commun. Math. Phys. 133 (1990) 217.
S. Ruijsenaars, Finite-dimensional soliton systems, in Integrable and super-integrable systems, B.A. Kupershmidt ed., World Scientific, Singapore (1989).
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math. 244 (2006) 525 [hep-th/0306238] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in XVI th International congress on mathematical physics. Prague Czech Republic 3-8 August 3-8 2009, P. Exner ed., World Scientific Publishing Co. Pte. Ltd., Singapore (2010), pg. 265 [arXiv:0908.4052] [INSPIRE].
N. Nekrasov, Seminar series: mathematics and physics of Calogero-Moser-Sutherland systems, http://scgp.stonybrook.edu/video_portal/results.php?profile_id=356.
Y. Hatsuda and M. Mariño, Exact quantization conditions for the relativistic Toda lattice, JHEP 05 (2016) 133 [arXiv:1511.02860] [INSPIRE].
D. Krefl, Non-perturbative quantum geometry II, JHEP 12 (2014) 118 [arXiv:1410.7116] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Classical irregular block, N = 2 pure gauge theory and Mathieu equation, JHEP 12 (2014) 032 [arXiv:1407.0305] [INSPIRE].
M. Piatek and A.R. Pietrykowski, Classical limit of irregular blocks and Mathieu functions, JHEP 01 (2016) 115 [arXiv:1509.08164] [INSPIRE].
G. Basar and G.V. Dunne, Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP 02 (2015) 160 [arXiv:1501.05671] [INSPIRE].
G.V. Dunne and M. Ünsal, WKB and resurgence in the Mathieu equation, arXiv:1603.04924 [INSPIRE].
J. Kallen and M. Mariño, Instanton effects and quantum spectral curves, Annales Henri Poincaré 17 (2016) 1037 [arXiv:1308.6485] [INSPIRE].
A. Grassi, Y. Hatsuda and M. Mariño, Topological strings from quantum mechanics, arXiv:1410.3382 [INSPIRE].
G. Lockhart and C. Vafa, Superconformal partition functions and non-perturbative topological strings, arXiv:1210.5909 [INSPIRE].
H.-C. Kim and S. Kim, M 5-branes from gauge theories on the 5-sphere, JHEP 05 (2013 144 [arXiv:1206.6339] [INSPIRE].
H.-C. Kim, J. Kim and S. Kim, Instantons on the 5-sphere and M 5-branes, arXiv:1211.0144 [INSPIRE].
J. Källén and M. Zabzine, Twisted supersymmetric 5D Yang-Mills theory and contact geometry, JHEP 05 (2012) 125 [arXiv:1202.1956] [INSPIRE].
K. Hosomichi, R.-K. Seong and S. Terashima, Supersymmetric gauge theories on the five-sphere, Nucl. Phys. B 865 (2012) 376 [arXiv:1203.0371] [INSPIRE].
J. Källén, J. Qiu and M. Zabzine, The perturbative partition function of supersymmetric 5D Yang-Mills theory with matter on the five-sphere, JHEP 08 (2012) 157 [arXiv:1206.6008] [INSPIRE].
Y. Imamura, Supersymmetric theories on squashed five-sphere, Prog. Theor. Exp. Phys. 2013 (2013) 013B04 [arXiv:1209.0561] [INSPIRE].
Y. Imamura, Perturbative partition function for squashed S 5, Prog. Theor. Exp. Phys. 2013 (2013) 073B01 [arXiv:1210.6308] [INSPIRE].
Y. Hatsuda, Comments on exact quantization conditions and non-perturbative topological strings, arXiv:1507.04799 [INSPIRE].
X. Wang, G. Zhang and M.-X. Huang, New exact quantization condition for toric Calabi-Yau geometries, Phys. Rev. Lett. 115 (2015) 121601 [arXiv:1505.05360] [INSPIRE].
S. Kharchev, D. Lebedev and M. Semenov-Tian-Shansky, Unitary representations of U q \( \left(\mathfrak{s}\mathfrak{l}\left(2,\ \mathrm{R}\right)\right) \) , the modular double and the multiparticle q deformed Toda chains, Commun. Math. Phys. 225 (2002) 573 [hep-th/0102180] [INSPIRE].
L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995) 249 [hep-th/9504111] [INSPIRE].
L.D. Faddeev, Modular double of quantum group, Math. Phys. Stud. 21 (2000) 149 [math/9912078] [INSPIRE].
A. Nedelin, F. Nieri and M. Zabzine, q-Virasoro modular double and 3d partition functions, arXiv:1605.07029 [INSPIRE].
M. Mariño and S. Zakany, Exact eigenfunctions and the open topological string, arXiv:1606.05297 [INSPIRE].
H.W. Braden and R. Sasaki, The Ruijsenaars-Schneider model, Prog. Theor. Phys. 97 (1997) 1003 [hep-th/9702182] [INSPIRE].
R.J. Baxter, Partition function of the eight vertex lattice model, Annals Phys. 70 (1972) 193 [INSPIRE].
R.J. Baxter, Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 1. Some fundamental eigenvectors, Annals Phys. 76 (1973) 1 [INSPIRE].
R.J. Baxter, Exactly solved models in statistical mechanics, (1982) [INSPIRE].
V.B. Kuznetsov and A.V. Tsyganov, Quantum relativistic toda chains, J. Math. Sci. 80 (1996) 1802.
B. Konstant, Quantization and representation theory, in Representation theory of Lie groups, Oxford U.K. 1977 34, U.K. (1979), pg. 287.
A.-K. Kashani-Poor, Quantization condition from exact WKB for difference equations, JHEP 06 (2016) 180 [arXiv:1604.01690] [INSPIRE].
S.-S. Kim and F. Yagi, 5d E n Seiberg-Witten curve via toric-like diagram, JHEP 06 (2015) 082 [arXiv:1411.7903] [INSPIRE].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, arXiv:1412.2781 [INSPIRE].
M. Bullimore, H.-C. Kim and P. Koroteev, Defects and quantum Seiberg-Witten geometry, JHEP 05 (2015) 095 [arXiv:1412.6081] [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid surface operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].
D. Gaiotto and E. Witten, S-duality of boundary conditions in N = 4 super Yang-Mills theory, Adv. Theor. Math. Phys. 13 (2009) 721 [arXiv:0807.3720] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 theories on the squashed 3-sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic blocks in three dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
F. Nieri, S. Pasquetti and F. Passerini, 3d and 5d gauge theory partition functions as q-deformed CFT correlators, Lett. Math. Phys. 105 (2015) 109 [arXiv:1303.2626] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY gauge theories on squashed three-spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
Y. Imamura and S. Yokoyama, Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 04 (2011) 007 [arXiv:1101.0557] [INSPIRE].
S. Kim, The complete superconformal index for N = 6 Chern-Simons theory, Nucl. Phys. B 821 (2009) 241 [Erratum ibid. B 864 (2012) 884] [arXiv:0903.4172] [INSPIRE].
A. Kapustin and B. Willett, Generalized superconformal index for three dimensional field theories, arXiv:1106.2484 [INSPIRE].
S. Garoufalidis and R. Kashaev, Evaluation of state integrals at rational points, Commun. Num. Theor. Phys. 09 (2015) 549 [arXiv:1411.6062] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
M. Bullimore and H.-C. Kim, The superconformal index of the (2, 0) theory with defects, JHEP 05 (2015) 048 [arXiv:1412.3872] [INSPIRE].
A. Givental and Y.-P. Lee, Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups, math/0108105.
D. Gaiotto and P. Koroteev, On three dimensional quiver gauge theories and integrability, JHEP 05 (2013) 126 [arXiv:1304.0779] [INSPIRE].
T. Dimofte, Complex Chern-Simons theory at level k via the 3d-3d correspondence, Commun. Math. Phys. 339 (2015) 619 [arXiv:1409.0857] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge theories labelled by three-manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-manifolds and 3d indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS states, Adv. Theor. Math. Phys. 17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
L.A. Takhtajan and L.D. Faddeev, The spectral theory of a functional-difference operator in conformal field theory, Izv. Math. 79 (2015) 388.
J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].
D. Tong and K. Wong, Instantons, Wilson lines and D-branes, Phys. Rev. D 91 (2015) 026007 [arXiv:1410.8523] [INSPIRE].
N. Nekrasov and V. Pestun, Seiberg-Witten geometry of four dimensional N = 2 quiver gauge theories, arXiv:1211.2240 [INSPIRE].
N. Nekrasov, V. Pestun and S. Shatashvili, Quantum geometry and quiver gauge theories, arXiv:1312.6689 [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic field realization of SH c and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE].
T. Kimura and V. Pestun, Quiver W -algebras, arXiv:1512.08533 [INSPIRE].
H.-C. Kim, Line defects and 5d instanton partition functions, JHEP 03 (2016) 199 [arXiv:1601.06841] [INSPIRE].
L.F. Alday and Y. Tachikawa, Affine SL(2) conformal blocks from 4d gauge theories, Lett. Math. Phys. 94 (2010) 87 [arXiv:1005.4469] [INSPIRE].
H. Kanno and Y. Tachikawa, Instanton counting with a surface operator and the chain-saw quiver, JHEP 06 (2011) 119 [arXiv:1105.0357] [INSPIRE].
R. Kashaev and M. Mariño, Operators from mirror curves and the quantum dilogarithm, Commun. Math. Phys. 346 (2016) 967 [arXiv:1501.01014] [INSPIRE].
A. Laptev, L. Schimmer and L.A. Takhtajan, Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves, arXiv:1510.00045 [INSPIRE].
A. Grassi, J. Kallen and M. Mariño, The topological open string wavefunction, Commun. Math. Phys. 338 (2015) 533 [arXiv:1304.6097] [INSPIRE].
M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Krefl and C. Vafa, Quantum geometry of refined topological strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE].
M. Aganagic and S. Shakirov, Knot homology and refined Chern-Simons index, Commun. Math. Phys. 333 (2015) 187 [arXiv:1105.5117] [INSPIRE].
M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [INSPIRE].
D. Gang, E. Koh and K. Lee, Superconformal index with duality domain wall, JHEP 10 (2012) 187 [arXiv:1205.0069] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
B. Fang and C.-C.M. Liu, Open Gromov-Witten invariants of toric Calabi-Yau 3-folds, Commun. Math. Phys. 323 (2013) 285 [arXiv:1103.0693] [INSPIRE].
Y. Hatsuda and K. Okuyama, Exact results for ABJ Wilson loops and open-closed duality, arXiv:1603.06579 [INSPIRE].
Y. Hatsuda, M. Honda, S. Moriyama and K. Okuyama, ABJM Wilson loops in arbitrary representations, JHEP 10 (2013) 168 [arXiv:1306.4297] [INSPIRE].
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [INSPIRE].
R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric gauge theories, intersecting branes and free fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [INSPIRE].
D. Maulik and A. Okounkov, Quantum groups and quantum cohomology, arXiv:1211.1287 [INSPIRE].
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].
M. Sato, T. Miwa and M. Jimbo, Studies on holonomic quantum fields, II, Proc. Japan Acad. A 53 (1977) 147.
T. Miwa, Painlevé property of monodromy preserving deformation equations and the analyticity of τ functions I, (1980) [INSPIRE].
G.W. Moore, Geometry of the string equations, Commun. Math. Phys. 133 (1990) 261 [INSPIRE].
S. Cecotti and C. Vafa, Ising model and N = 2 supersymmetric theories, Commun. Math. Phys. 157 (1993) 139 [hep-th/9209085] [INSPIRE].
A.B. Zamolodchikov, Painleve III and 2D polymers, Nucl. Phys. B 432 (1994) 427 [hep-th/9409108] [INSPIRE].
G. Bonelli, A. Grassi and A. Tanzini, Seiberg-Witten theory as a Fermi gas, arXiv:1603.01174 [INSPIRE].
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Sciarappa, A. Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings. J. High Energ. Phys. 2016, 14 (2016). https://doi.org/10.1007/JHEP10(2016)014
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DOI: https://doi.org/10.1007/JHEP10(2016)014