Abstract
In this note we consider the symplectic reduction of a four-dimensional holomorphic Chern-Simons theory recently introduced in [1] for describing integrable field theories. We work out explicitly the case of the lambda deformed Principal Chiral Model (PCM) and show that the symplectic reduction works as a localization mechanism. The reduced Chern-Simons theory restricts to the set of poles of the twist function underlying the theory, where the known classical integrability of the lambda deformed PCM can be reconstructed from the phase space data associated to this set of points in the spectral space.
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References
K. Costello and M. Yamazaki, Gauge theory and integrability, III, arXiv:1908.02289 [INSPIRE].
B. Vicedo, On integrable field theories as dihedral affine Gaudin models, arXiv:1701.04856 [INSPIRE].
C. Klimčík, Yang-Baxter σ-models and dS/AdS T duality, JHEP12 (2002) 051 [hep-th/0210095] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, On classical q-deformations of integrable σ-models, JHEP11 (2013) 192 [arXiv:1308.3581] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, An integrable deformation of the AdS5 × S5superstring action, Phys. Rev. Lett.112 (2014) 051601 [arXiv:1309.5850] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Derivation of the action and symmetries of the q-deformed AdS5 × S5superstring, JHEP10 (2014) 132 [arXiv:1406.6286] [INSPIRE].
H.A. Benítez and V.O. Rivelles, Yang-Baxter deformations of the AdS5 × S5pure spinor superstring, JHEP02 (2019) 056 [arXiv:1807.10432] [INSPIRE].
R. Negrón and V.O. Rivelles, Yang-Baxter deformations of the AdS4 × CP3superstring σ-model, JHEP11 (2018) 043 [arXiv:1809.01174] [INSPIRE].
K. Sfetsos, Integrable interpolations: from exact CFTs to non-Abelian T-duals, Nucl. Phys.B 880 (2014) 225 [arXiv:1312.4560] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, Integrable deformations of strings on symmetric spaces, JHEP11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An integrable deformation of the AdS5 × S5superstring, J. Phys.A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
D.M. Schmidtt, Exploring the lambda model of the hybrid superstring, JHEP10 (2016) 151 [arXiv:1609.05330] [INSPIRE].
H.A. Benítez and D.M. Schmidtt, λ-deformation of the AdS5 × S5pure spinor superstring, JHEP10 (2019) 108 [arXiv:1907.13197] [INSPIRE].
A.A. Tseytlin, On a ‘universal’ class of WZW type conformal models, Nucl. Phys.B 418 (1994) 173 [hep-th/9311062] [INSPIRE].
K. Bardakci, L.M. Bernardo and N. Sochen, Integrable generalized Thirring model, Nucl. Phys.B 487 (1997) 513 [hep-th/9607018] [INSPIRE].
C. Klimčík, η and λ deformations as E-models, Nucl. Phys.B 900 (2015) 259 [arXiv:1508.05832] [INSPIRE].
B. Vicedo, Deformed integrable σ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles, J. Phys.A 48 (2015) 355203 [arXiv:1504.06303] [INSPIRE].
C. Appadu, T.J. Hollowood, J.L. Miramontes, D. Price and D.M. Schmidtt, Giant magnons of string theory in the lambda background, JHEP07 (2017) 098 [arXiv:1704.05437] [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys.B 326 (1989) 108 [INSPIRE].
G.W. Moore and N. Seiberg, Taming the conformal zoo, Phys. Lett.B 220 (1989) 422 [INSPIRE].
E. Witten, Non-Abelian bosonization in two-dimensions, Commun. Math. Phys.92 (1984) 455 [INSPIRE].
D.M. Schmidtt, Integrable lambda models and Chern-Simons theories, JHEP05 (2017) 012 [arXiv:1701.04138] [INSPIRE].
D.M. Schmidtt, Lambda models from Chern-Simons theories, JHEP10 (2019) 095 [Erratum ibid.10 (2019) 095] [arXiv:1808.05994] [INSPIRE].
K. Costello, Supersymmetric gauge theory and the Yangian, arXiv:1303.2632 [INSPIRE].
K. Costello, Integrable lattice models from four-dimensional field theories, Proc. Symp. Pure Math.88 (2014) 3 [arXiv:1308.0370] [INSPIRE].
E. Witten, Integrable lattice models from gauge theory, Adv. Theor. Math. Phys.21 (2017) 1819 [arXiv:1611.00592] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge theory and integrability, I, Not. Int. Congr. Chin. Math.6 (2018) 46 [arXiv:1709.09993] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge theory and integrability, II, Not. Int. Congr. Chin. Math.6 (2018) 120 [arXiv:1802.01579] [INSPIRE].
B. Vicedo, Holomorphic Chern-Simons theory and affine Gaudin models, arXiv:1908.07511 [INSPIRE].
F. Delduc, S. Lacroix, M. Magro and B. Vicedo, A unifying 2d action for integrable σ-models from 4d Chern-Simons theory, arXiv:1909.13824 [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, S-matrices and quantum group symmetry of k-deformed σ-models, J. Phys.A 49 (2016) 465201 [arXiv:1506.06601] [INSPIRE].
A. Alekseev, L.D. Faddeev and M. Semenov-Tian-Shansky, Hidden quantum groups inside Kac-Moody algebra, Commun. Math. Phys.149 (1992) 335 [INSPIRE].
J.M. Maillet, New integrable canonical structures in two-dimensional models, Nucl. Phys.B 269 (1986) 54 [INSPIRE].
T. Regge and C. Teitelboim, Role of surface integrals in the Hamiltonian formulation of general relativity, Annals Phys.88 (1974) 286 [INSPIRE].
M. Audin, Lectures on gauge theory and integrable systems, in Gauge theory and symplectic geometry, J. Hurtubise, F. Lalonde and G. Sabidussi eds., NATO Sci. Ser.C 488 (1997) 1 [INSPIRE].
G. Itsios, K. Sfetsos and K. Siampos, The all-loop non-Abelian Thirring model and its RG flow, Phys. Lett.B 733 (2014) 265 [arXiv:1404.3748] [INSPIRE].
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Schmidtt, D.M. Holomorphic Chern-Simons theory and lambda models: PCM case. J. High Energ. Phys. 2020, 60 (2020). https://doi.org/10.1007/JHEP04(2020)060
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DOI: https://doi.org/10.1007/JHEP04(2020)060