Abstract
We consider the compactifcation of 5d non-simply laced fractional quiver gauge theory constructed in [1]. In contrast to the simply laced quivers, here two Ω-background parameters play different roles, so that we can take two possible Nekrasov-Shatashvili limits. We demonstrate how different quantum integrable systems can emerge from these two limits, using BC2-quiver as the simplest illustrative example for our general results. We also comment possible connections with compactified 3d non-simply laced quiver gauge theory.
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References
T. Kimura and V. Pestun, Fractional quiver W-algebras, arXiv:1705.04410 [INSPIRE].
A. Gorsky et al., Integrability and Seiberg-Witten exact solution, Phys. Lett. B 355 (1995) 466 [hep-th/9505035] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson, C. Itzykson et al. eds., World Scientific, Singapore (1997), hep-th/9607163 [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of integrable systems and four dimensional gauge theories, in the proceedings of the 16th International Congress on Mathematical Physics (ICMP09), August 3-8, Prague, Czech Republic (2009), arXiv:0908.4052 [INSPIRE].
R. Poghossian, Deforming SW curve, JHEP 04 (2011) 033 [arXiv:1006.4822] [INSPIRE].
F. Fucito et al., Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves, JHEP 05 (2011) 098 [arXiv:1103.4495] [INSPIRE].
F. Fucito, J.F. Morales and D. Ricci Pacifici, Deformed Seiberg-Witten Curves for ADE quivers, JHEP 01 (2013) 091 [arXiv:1210.3580] [INSPIRE].
A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, N = 2 supersymmetric QCD and integrable spin chains: rational case N f < 2N c, Phys. Lett. B 380 (1996) 75 [hep-th/9603140] [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, Commun. Math. Phys. 357 (2018) 519 [arXiv:1312.6689] [INSPIRE].
T. Kimura and V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (2018) 1351 [arXiv:1512.08533] [INSPIRE].
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].
H. Awata and Y. Yamada, Five-dimensional AGT conjecture and the deformed Virasoro algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE].
E. Frenkel and N. Reshetikhin, Deformations of \( \mathcal{W} \) -algebras associated to simple Lie algebras, Comm. Math. Phys. 197 (1998) 1 [q-alg/9708006].
T. Kimura and V. Pestun, Quiver elliptic W-algebras, Lett. Math. Phys. 108 (2018) 1383 [arXiv:1608.04651] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters, JHEP 03 (2016) 181 [arXiv:1512.05388] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence II: instantons at crossroads, moduli and compactness theorem, Adv. Theor. Math. Phys. 21 (2017) 503 [arXiv:1608.07272] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence III: gauge origami partition function and qq-characters, Commun. Math. Phys. 358 (2018) 863 [arXiv:1701.00189] [INSPIRE].
N. Nekrasov, BPS/CFT correspondence IV: σ-models and defects in gauge theory, arXiv:1711.11011 [INSPIRE].
N. Nekrasov, BPS/CFT correspondence V: BPZ and KZ equations from qq-characters, arXiv:1711.11582 [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].
J.-E. Bourgine et al., Coherent states in quantum \( {\mathcal{W}}_{1+\infty } \) algebra and qq-character for 5d Super Yang-Mills, PTEP 2016 (2016) 123B05 [arXiv:1606.08020] [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of \( \mathcal{W} \) -algebras, Contemp. Math. 248 (1999) 163 [math/9810055].
A. Marshakov and N. Nekrasov, Extended Seiberg-Witten theory and integrable hierarchy, JHEP 01 (2007) 104 [hep-th/0612019] [INSPIRE].
H. Nakajima and K. Yoshioka, Lectures on instanton counting, CRM Proc. Lec. Notes 38 (2003) 31 [math/0311058] [INSPIRE].
J. Shiraishi, H. Kubo, H. Awata and S. Odake, A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 (1996) 33 [q-alg/9507034] [INSPIRE].
E. Frenkel and N. Reshetikhin, Quantum affine algebras and deformations of the Virasoro and W-algebras, Comm. Math. Phys. 178 (1996) 237 [q-alg/9505025].
P. Bouwknegt and K. Pilch, On deformed W algebras and quantum affine algebras, Adv. Theor. Math. Phys. 2 (1998) 357 [math/9801112] [INSPIRE].
A. Kuniba, T. Nakanishi and J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A 44 (2011) 103001 [arXiv:1010.1344] [INSPIRE].
N. Yu. Reshetikhin and P.B. Wiegmann, Towards the classification of completely integrabl quantum field theories, Phys. Lett. B 189 (1987) 125 [INSPIRE].
A. Dey, A. Hanany, P. Koroteev and N. Mekareeya, On three-dimensional quiver gauge theories of Type B, JHEP 09 (2017) 067 [arXiv:1612.00810] [INSPIRE].
K. Ito and H. Shu, ODE/IM correspondence for modified \( {B}_2^{{}^{(1)}} \) affine Toda field equation, Nucl. Phys. B 916 (2017) 414 [arXiv:1605.04668] [INSPIRE].
N. Dorey, S. Lee and T.J. Hollowood, Quantization of integrable systems and a 2d/4d duality, JHEP 10 (2011) 077 [arXiv:1103.5726] [INSPIRE].
H.-Y. Chen, N. Dorey, T.J. Hollowood and S. Lee, A new 2d/4d duality via integrability, JHEP 09 (2011) 040 [arXiv:1104.3021] [INSPIRE].
H.-Y. Chen, T.J. Hollowood and P. Zhao, A 5d/3d duality from relativistic integrable system, JHEP 07 (2012) 139 [arXiv:1205.4230] [INSPIRE].
H.-Y. Chen and A. Sinkovics, On integrable structure and geometric transition in supersymmetric gauge theories, JHEP 05 (2013) 158 [arXiv:1303.4237] [INSPIRE].
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Chen, HY., Kimura, T. Quantum integrability from non-simply laced quiver gauge theory. J. High Energ. Phys. 2018, 165 (2018). https://doi.org/10.1007/JHEP06(2018)165
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DOI: https://doi.org/10.1007/JHEP06(2018)165