Abstract
We establish a correspondence between a class of Wilson-’t Hooft lines in four-dimensional \( \mathcal{N} \) = 2 supersymmetric gauge theories described by circular quivers and transfer matrices constructed from dynamical L-operators for trigonometric quantum integrable systems. We compute the vacuum expectation values of the Wilson-’t Hooft lines in a twisted product space S1 × ϵ ℝ2 × ℝ by supersymmetric localization and show that they are equal to the Wigner transforms of the transfer matrices. A variant of the AGT correspondence implies an identification of the transfer matrices with Verlinde operators in Toda theory, which we also verify. We explain how these field theory setups are related to four-dimensional Chern-Simons theory via embedding into string theory and dualities.
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References
M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP 10 (2014) 062 [arXiv:1401.3379] [INSPIRE].
K. Maruyoshi and J. Yagi, Surface defects as transfer matrices, PTEP 2016 (2016) 113B01 [arXiv:1606.01041] [INSPIRE].
J. Yagi, Surface defects and elliptic quantum groups, JHEP 06 (2017) 013 [arXiv:1701.05562] [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
K. Hasegawa, Ruijsenaars’ commuting difference operators as commuting transfer matrices, Comm. Math. Phys. 187 (1997) 289.
R.J. Baxter, Eight vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. 2. Equivalence to a generalized ice-type lattice model, Annals Phys. 76 (1973) 25 [INSPIRE].
M. Jimbo, T. Miwa and M. Okado, Solvable lattice models whose states are dominant integral weights of \( {A}_{n-1}^{(1)} \), Lett. Math. Phys. 14 (1987) 123.
M. Jimbo, T. Miwa and M. Okado, Local state probabilities of solvable lattice models: an \( {A}_{n-1}^{(1)} \) family, Nucl. Phys. B 300 (1988) 74 [INSPIRE].
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
Y. Ito, T. Okuda and M. Taki, Line operators on S1 × R3 and quantization of the Hitchin moduli space, JHEP 04 (2012) 010 [Erratum ibid. 03 (2016) 085] [arXiv:1111.4221] [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].
N. Wyllard, AN − 1 conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [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].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge theory loop operators and Liouville theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
J. Gomis and B. Le Floch, ’t Hooft operators in gauge theory from Toda CFT, JHEP 11 (2011) 114 [arXiv:1008.4139] [INSPIRE].
E.P. Verlinde, Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
K. Costello, Supersymmetric gauge theory and the Yangian, arXiv:1303.2632 [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge theory and integrability, I, ICCM Not. 06 (2018) 46 [arXiv:1709.09993] [INSPIRE].
K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories, arXiv:1810.01970 [INSPIRE].
G. Felder, Elliptic quantum groups, in 11th international conference on mathematical physics (ICMP-11) (satellite colloquia: new problems in the general theory of fields and particles, Paris, France, 25–28 July 1994, pg. 211 [hep-th/9412207] [INSPIRE].
G. Felder, Conformal field theory and integrable systems associated to elliptic curves, in Proceedings of the international congress of mathematicians, Birkhäuser, Basel, Switzerland (1995), pg. 1247 [hep-th/9407154] [INSPIRE].
P. Etingof and A. Varchenko, Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Commun. Math. Phys. 196 (1998) 591 [q-alg/9708015].
J.-L. Gervais and A. Neveu, Novel triangle relation and absence of tachyons in Liouville string field theory, Nucl. Phys. B 238 (1984) 125 [INSPIRE].
R.J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26 (1971) 832 [INSPIRE].
R.J. Baxter, Partition function of the eight vertex lattice model, Annals Phys. 70 (1972) 193 [INSPIRE].
A.A. Belavin, Dynamical symmetry of integrable quantum systems, Nucl. Phys. B 180 (1981) 189 [INSPIRE].
V.V. Bazhanov and S.M. Sergeev, A master solution of the quantum Yang-Baxter equation and classical discrete integrable equations, Adv. Theor. Math. Phys. 16 (2012) 65 [arXiv:1006.0651] [INSPIRE].
V.V. Bazhanov and S.M. Sergeev, Elliptic gamma-function and multi-spin solutions of the Yang-Baxter equation, Nucl. Phys. B 856 (2012) 475 [arXiv:1106.5874] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
T.D. Brennan and G.W. Moore, Index-like theorems from line defect vevs, JHEP 09 (2019) 073 [arXiv:1903.08172] [INSPIRE].
T.D. Brennan, A. Dey and G.W. Moore, On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics, JHEP 09 (2018) 014 [arXiv:1801.01986] [INSPIRE].
T.D. Brennan, Monopole bubbling via string theory, JHEP 11 (2018) 126 [arXiv:1806.00024] [INSPIRE].
T.D. Brennan, A. Dey and G.W. Moore, ’t Hooft defects and wall crossing in SQM, JHEP 10 (2019) 173 [arXiv:1810.07191] [INSPIRE].
B. Assel and A. Sciarappa, On monopole bubbling contributions to ’t Hooft loops, JHEP 05 (2019) 180 [arXiv:1903.00376] [INSPIRE].
S.N.M. Ruijsenaars, Complete integrability of relativistic Calogero-Moser systems and elliptic function identities, Commun. Math. Phys. 110 (1987) 191 [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].
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
J. Yagi, Ω-deformation and quantization, JHEP 08 (2014) 112 [arXiv:1405.6714] [INSPIRE].
S. Hellerman, D. Orlando and S. Reffert, String theory of the Ω deformation, JHEP 01 (2012) 148 [arXiv:1106.0279] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].
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Maruyoshi, K., Ota, T. & Yagi, J. Wilson-’t Hooft lines as transfer matrices. J. High Energ. Phys. 2021, 72 (2021). https://doi.org/10.1007/JHEP01(2021)072
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DOI: https://doi.org/10.1007/JHEP01(2021)072