Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 058, 45 pages      arXiv:2011.13809

Integrable $\mathcal{E}$-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

Sylvain Lacroix ab and Benoît Vicedo c
a) II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
b) Zentrum für Mathematische Physik, Universität Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany
c) Department of Mathematics, University of York, York YO10 5DD, UK

Received December 07, 2020, in final form May 31, 2021; Published online June 10, 2021

We construct the actions of a very broad family of 2d integrable $\sigma$-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d fields $h$ and $\mathcal{L}$, with $\mathcal{L}$ depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for $\mathcal{L}$ in terms of $h$ this produces a 2d integrable field theory for the 2d field $h$ whose Lax connection is given by $\mathcal{L}(h)$. We construct a general class of solutions to this constraint and show that the resulting 2d integrable field theories can all naturally be described as $\mathcal{E}$-models.

Key words: 4d Chern-Simons theory; $\mathcal E$-models; affine Gaudin models; integrable $\sigma$-models.

pdf (772 kb)   tex (59 kb)  


  1. Ashwinkumar M., Png K.-S., Tan M.-C., 4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondence, arXiv:2008.06053.
  2. Bassi C., Lacroix S., Integrable deformations of coupled $\sigma$-models, J. High Energy Phys. 2020 (2020), no. 5, 059, 58 pages, arXiv:1912.06157.
  3. Benini M., Schenkel A., Vicedo B., Homotopical analysis of 4d Chern-Simons theory and integrable field theories, arXiv:2008.01829.
  4. Bittleston R., Skinner D., Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory, arXiv:2011.04638.
  5. Bykov D., Flag manifold sigma models and nilpotent orbits, Proc. Steklov Inst. Math. 309 (2020), 78-86, arXiv:1911.07768.
  6. Bykov D., Quantum flag manifold $\sigma$-models and Hermitian Ricci flow, arXiv:2006.14124.
  7. Bykov D., The $\mathbb{CP}^{n-1}$-model with fermions: a new look, arXiv:2009.04608.
  8. Costello K., Supersymmetric gauge theory and the Yangian, arXiv:1303.2632.
  9. Costello K., Integrable lattice models from four-dimensional field theories, in String-Math 2013, Proc. Sympos. Pure Math., Vol. 88, Amer. Math. Soc., Providence, RI, 2014, 3-23, arXiv:1308.0370.
  10. Costello K., Stefański Jr. B., Chern-Simons origin of superstring integrability, Phys. Rev. Lett. 125 (2020), 121602, 6 pages, arXiv:2005.03064.
  11. Costello K., Witten E., Yamazaki M., Gauge theory and integrability, I, ICCM Not. 6 (2018), 46-119, arXiv:1709.09993.
  12. Costello K., Witten E., Yamazaki M., Gauge theory and integrability, II, ICCM Not. 6 (2018), 120-146, arXiv:1802.01579.
  13. Costello K., Yamazaki M., Gauge theory and integrability, III, arXiv:1908.02289.
  14. Curtright T., Zachos C., Currents, charges, and canonical structure of pseudodual chiral models, Phys. Rev. D 49 (1994), 5408-5421, arXiv:hep-th/9401006.
  15. Delduc F., Hoare B., Kameyama T., Magro M., Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable $\sigma$-model, J. High Energy Phys. 2017 (2017), no. 10, 212, 20 pages, arXiv:1707.08371.
  16. Delduc F., Lacroix S., Magro M., Vicedo B., Assembling integrable $\sigma$-models as affine Gaudin models, J. High Energy Phys. 2019 (2019), no. 6, 017, 86 pages, arXiv:1903.00368.
  17. Delduc F., Lacroix S., Magro M., Vicedo B., A unifying 2D action for integrable $\sigma$-models from 4D Chern-Simons theory, Lett. Math. Phys. 110 (2020), 1645-1687, arXiv:1909.13824.
  18. Demulder S., Driezen S., Sevrin A., Thompson D.C., Classical and quantum aspects of Yang-Baxter Wess-Zumino models, J. High Energy Phys. 2018 (2018), no. 3, 041, 38 pages, arXiv:1711.00084.
  19. Demulder S., Hassler F., Thompson D.C., Doubled aspects of generalised dualities and integrable deformations, J. High Energy Phys. 2019 (2019), no. 2, 189, 54 pages, arXiv:1810.11446.
  20. Demulder S., Hassler F., Thompson D.C., An invitation to Poisson-Lie $T$-duality in double field theory and its applications, PoS Proc. Sci. (2019), PoS(CORFU2018), 113, 30 pages, arXiv:1904.09992.
  21. Evans J.M., Integrable sigma-models and Drinfeld-Sokolov hierarchies, Nuclear Phys. B 608 (2001), 591-609, arXiv:hep-th/0101231.
  22. Evans J.M., Hassan M., MacKay N.J., Mountain A.J., Local conserved charges in principal chiral models, Nuclear Phys. B 561 (1999), 385-412, arXiv:hep-th/9902008.
  23. Evans J.M., Mountain A.J., Commuting charges and symmetric spaces, Phys. Lett. B 483 (2000), 290-298, arXiv:hep-th/0003264.
  24. Fradkin E.S., Tseytlin A.A., Quantum equivalence of dual field theories, Ann. Physics 162 (1985), 31-48.
  25. Fridling B.E., Jevicki A., Dual representations and ultraviolet divergences in nonlinear $\sigma$-models, Phys. Lett. B 134 (1984), 70-74.
  26. Fukushima O., Sakamoto J.-i., Yoshida K., Comments on $\eta$-deformed principal chiral model from 4D Chern-Simons theory, Nuclear Phys. B 957 (2020), 115080, 37 pages, arXiv:2003.07309.
  27. Fukushima O., Sakamoto J.-i., Yoshida K., Yang-Baxter deformations of the $\rm AdS_5\times S^5$ supercoset sigma model from 4D Chern-Simons theory, J. High Energy Phys. 2020 (2020), no. 9, 100, 22 pages, arXiv:2005.04950.
  28. Hoare B., Lacroix S., Yang-Baxter deformations of the principal chiral model plus Wess-Zumino term, J. Phys. A: Math. Theor. 53 (2020), 505401, 53 pages, arXiv:2009.00341.
  29. Hoare B., Tseytlin A.A., On integrable deformations of superstring sigma models related to ${\rm AdS}_n\times S^n$ supercosets, Nuclear Phys. B 897 (2015), 448-478, arXiv:1504.07213.
  30. Klimčík C., Yang-Baxter $\sigma$-models and dS/AdS $T$-duality, J. High Energy Phys. 2002 (2002), no. 12, 051, 23 pages, arXiv:hep-th/0210095.
  31. Klimčík C., On integrability of the Yang-Baxter $\sigma$-model, J. Math. Phys. 50 (2009), 043508, 11 pages, arXiv:0802.3518.
  32. Klimčík C., $\eta$ and $\lambda$ deformations as $\mathcal{E}$-models, Nuclear Phys. B 900 (2015), 259-272, arXiv:1508.05832.
  33. Klimčík C., Yang-Baxter $\sigma$-model with WZNW term as ${\mathcal E}$-model, Phys. Lett. B 772 (2017), 725-730, arXiv:1706.08912.
  34. Klimčík C., Affine Poisson and affine quasi-Poisson T-duality, Nuclear Phys. B 939 (2019), 191-232, arXiv:1809.01614.
  35. Klimčík C., Dressing cosets and multi-parametric integrable deformations, J. High Energy Phys. 2019 (2019), no. 7, 176, 44 pages, arXiv:1903.00439.
  36. Klimčík C., Strong integrability of the bi-YB-WZ model, Lett. Math. Phys. 110 (2020), 2397-2416, arXiv:2001.05466.
  37. Klimčík C., Ševera P., Dual non-abelian duality and the Drinfel'd double, Phys. Lett. B 351 (1995), 455-462, arXiv:hep-th/9502122.
  38. Klimčík C., Ševera P., Dressing cosets, Phys. Lett. B 381 (1996), 56-61, arXiv:hep-th/9602162.
  39. Klimčík C., Ševera P., Non-abelian momentum-winding exchange, Phys. Lett. B 383 (1996), 281-286, arXiv:hep-th/9605212.
  40. Klimčík C., Ševera P., Poisson-Lie $T$-duality and loop groups of Drinfeld doubles, Phys. Lett. B 372 (1996), 65-71, arXiv:hep-th/9512040.
  41. Lacroix S., Constrained affine Gaudin models and diagonal Yang-Baxter deformations, J. Phys. A: Math. Theor. 53 (2020), 255203, 91 pages, arXiv:1907.04836.
  42. Lacroix S., Magro M., Vicedo B., Local charges in involution and hierarchies in integrable sigma-models, J. High Energy Phys. 2017 (2017), no. 9, 117, 62 pages, arXiv:1703.01951.
  43. Lacroix S., Vicedo B., Integrable ${\mathcal E}$-models, 4d Chern-Simons theory and affine Gaudin models, II: Hamiltonian aspects, in preparation.
  44. Maillet J.-M., Kac-Moody algebra and extended Yang-Baxter relations in the ${\rm O}(N)$ nonlinear $\sigma$-model, Phys. Lett. B 162 (1985), 137-142.
  45. Maillet J.-M., New integrable canonical structures in two-dimensional models, Nuclear Phys. B 269 (1986), 54-76.
  46. Nappi C.R., Some properties of an analog of the chiral model, Phys. Rev. D 21 (1980), 418-420.
  47. Penna R.F., A twistor action for integrable systems, arXiv:2011.05831.
  48. Polyakov A., Wiegmann P.B., Theory of nonabelian Goldstone bosons in two dimensions, Phys. Lett. B 131 (1983), 121-126.
  49. Schmidtt D.M., Integrable lambda models and Chern-Simons theories, J. High Energy Phys. 2017 (2017), no. 5, 012, 23 pages, arXiv:1701.04138.
  50. Schmidtt D.M., Lambda models from Chern-Simons theories, J. High Energy Phys. 2018 (2018), no. 11, 111, 49 pages, arXiv:1808.05994.
  51. Schmidtt D.M., Holomorphic Chern-Simons theory and lambda models: PCM case, J. High Energy Phys. 2020 (2020), no. 4, 060, 25 pages, arXiv:1912.07569.
  52. Ševera P., Poisson-Lie $T$-duality as a boundary phenomenon of Chern-Simmons theory, J. High Energy Phys. 2016 (2016), no. 5, 044, 18 pages, arXiv:1602.05126.
  53. Ševera P., On integrability of 2-dimensional $\sigma$-models of Poisson-Lie type, J. High Energy Phys. 2017 (2017), no. 11, 015, 9 pages, arXiv:1709.02213.
  54. Sfetsos K., Poisson-Lie $T$-duality and supersymmetry, Nuclear Phys. B Proc. Suppl. 56B (1997), 302-309, arXiv:hep-th/9611199.
  55. Sfetsos K., Canonical equivalence of non-isometric $\sigma$-models and Poisson-Lie $T$-duality, Nuclear Phys. B 517 (1998), 549-566, arXiv:hep-th/9710163.
  56. Sfetsos K., Duality-invariant class of two-dimensional field theories, Nuclear Phys. B 561 (1999), 316-340, arXiv:hep-th/9904188.
  57. Sfetsos K., Integrable interpolations: from exact CFTs to non-Abelian $T$-duals, Nuclear Phys. B 880 (2014), 225-246, arXiv:1312.4560.
  58. Sfetsos K., Siampos K., Thompson D.C., Generalised integrable $\lambda$- and $\eta$-deformations and their relation, Nuclear Phys. B 899 (2015), 489-512, arXiv:1506.05784.
  59. Squellari R., Dressing cosets revisited, Nuclear Phys. B 853 (2011), 379-403, arXiv:1105.0162.
  60. Tian J., Comments on $\lambda$-deformed models from 4D Chern-Simons theory, arXiv:2005.14554.
  61. Tian J., He Y.-J., Chen B., $\lambda$-deformed ${\rm AdS}_5\times S^5$ superstring from 4D Chern-Simons theory, arXiv:2007.00422.
  62. Vavřín Z., Confluent Cauchy and Cauchy-Vandermonde matrices, Linear Algebra Appl. 258 (1997), 271-293.
  63. Vicedo B., Deformed integrable $\sigma$-models, classical $R$-matrices and classical exchange algebra on Drinfel'd doubles, J. Phys. A: Math. Theor. 48 (2015), 355203, 33 pages, arXiv:1504.06303.
  64. Vicedo B., On integrable field theories as dihedral affine Gaudin models, Int. Math. Res. Not. 2020 (2020), 4513-4601, arXiv:1701.04856.
  65. Vicedo B., 4D Chern-Simons theory and affine Gaudin models, Lett. Math. Phys. 111 (2021), 24, 21 pages, arXiv:1908.07511.
  66. Vizman C., The group structure for jet bundles over Lie groups, J. Lie Theory 23 (2013), 885-897, arXiv:1304.5024.
  67. Witten E., Integrable lattice models from gauge theory, Adv. Theor. Math. Phys. 21 (2017), 1819-1843, arXiv:1611.00592.
  68. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (1978), 1017-1027.

Previous article  Next article  Contents of Volume 17 (2021)