Abstract
In this paper we refine and extend the results of [1], where a connection between the AdS5 × S5 superstring lambda model on S1 = ∂D and a double Chern-Simons (CS) theory on D based on the Lie superalgebra \( \mathfrak{p}\mathfrak{s}\mathfrak{u} \)(2, 2|4) was suggested, after introduction of the spectral parameter z. The relation between both theories mimics the well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature. All the statements are now valid in the strong sense, i.e. valid on the whole phase space, making the connection between both theories precise. By constructing a z-dependent gauge field in the 2+1 Hamiltonian CS theory it is shown that: i) by performing a symplectic reduction of the CS theory the Maillet algebra satisfied by the extended Lax connection of the lambda model emerges as a boundary current algebra and ii) the Poisson algebra of the supertraces of z-dependent Wilson loops in the CS theory obey some sort of spectral parameter generalization of the Goldman bracket. The latter algebra is interpreted as the precursor of the (ambiguous) lambda model monodromy matrix Poisson algebra prior to the symplectic reduction. As a consequence, the problematic non-ultralocality of lambda models is avoided (for any value of the deformation parameter λ ⊂ [0, 1]), showing how the lambda model classical integrable structure can be understood as a byproduct of the symplectic reduction process of the z-dependent CS theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.M. Schmidtt, Integrable Lambda Models And Chern-Simons Theories, JHEP 05 (2017) 012 [arXiv:1701.04138] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, Phys. Rev. D 69 (2004) 046002 [hep-th/0305116] [INSPIRE].
J.M. Maldacena, The Large N Limit Of Superconformal Field Theories And Supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J.M. Maillet, New Integrable Canonical Structures in Two-dimensional Models, Nucl. Phys. B 269 (1986) 54 [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, JHEP 11 (2014) 009 [arXiv:1407.2840] [INSPIRE].
T.J. Hollowood, J.L. Miramontes and D.M. Schmidtt, An Integrable Deformation of the AdS 5 × S 5 Superstring, J. Phys. A 47 (2014) 495402 [arXiv:1409.1538] [INSPIRE].
D.M. Schmidtt, Exploring The Lambda Model Of The Hybrid Superstring, JHEP 10 (2016) 151 [arXiv:1609.05330] [INSPIRE].
R. Borsato, A.A. Tseytlin and L. Wulff, Supergravity Background Of λ-Deformed Model For AdS 2 × S 2 Supercoset, Nucl. Phys. B 905 (2016) 264 [arXiv:1601.08192].
Y. Chervonyi and O. Lunin, Supergravity background of the λ-deformed AdS 3 × S 3 supercoset, Nucl. Phys. B 910 (2016) 685 [arXiv:1606.00394] [INSPIRE].
R. Borsato and L. Wulff, Target space supergeometry of η and λ-deformed strings, JHEP 10 (2016) 045 [arXiv:1608.03570] [INSPIRE].
L.D. Faddeev and N. Yu. Reshetikhin, Integrability of the Principal Chiral Field Model in (1+1)-dimension, Annals Phys. 167 (1986) 227 [INSPIRE].
L. Freidel and J.M. Maillet, Quadratic algebras and integrable systems, Phys. Lett. B 262 (1991) 278 [INSPIRE].
L. Hlavaty and A. Kundu, Quantum integrability of nonultralocal models through Baxterization of quantized braided algebra, Int. J. Mod. Phys. A 11 (1996) 2143 [hep-th/9406215] [INSPIRE].
M. Semenov-Tian-Shansky and A. Sevostyanov, Classical and quantum nonultralocal systems on the lattice, hep-th/9509029 [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381 [hep-th/9412229] [INSPIRE].
D. Ridout and J. Teschner, Integrability of a family of quantum field theories related to sigma models, Nucl. Phys. B 853 (2011) 327 [arXiv:1102.5716] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Alleviating the non-ultralocality of coset sigma models through a generalized Faddeev-Reshetikhin procedure, JHEP 08 (2012) 019 [arXiv:1204.0766] [INSPIRE].
A. Melikyan and G. Weber, On the quantization of continuous non-ultralocal integrable systems, Nucl. Phys. B 913 (2016) 716 [arXiv:1611.02622] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, Alleviating the non-ultralocality of the AdS 5 × S 5 superstring, JHEP 10 (2012) 061 [arXiv:1206.6050] [INSPIRE].
F. Delduc, M. Magro and B. Vicedo, A lattice Poisson algebra for the Pohlmeyer reduction of the AdS 5 × S 5 superstring, Phys. Lett. B 713 (2012) 347 [arXiv:1204.2531] [INSPIRE].
B. Vicedo, On integrable field theories as dihedral affine Gaudin models, arXiv:1701.04856 [INSPIRE].
C. Appadu, T.J. Hollowood and D. Price, Quantum Inverse Scattering and the Lambda Deformed Principal Chiral Model, J. Phys. A 50 (2017) 305401 [arXiv:1703.06699] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [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].
W. Goldman, Invariant Functions On Lie groups And Hamiltonian Flows Of Surface Group Representations, Invent. Math. 85 (1986) 263.
V.G. Turaev, Skein Quantization Of Poisson Algebras Of Loops On Surfaces, Annales Sci. Ecole Norm. Sup. 24 (1991) 635 ISSN: 0012-9593.
J.E. Nelson and R.F. Picken, Constant connections, quantum holonomies and the Goldman bracket, Adv. Theor. Math. Phys. 9 (2005) 407 [math-ph/0412007] [INSPIRE].
J.E. Nelson and R.F. Picken, A quantum Goldman bracket in 2+1 quantum gravity, J. Phys. A 41 (2008) 304011 [arXiv:0711.2271] [INSPIRE].
J.E. Nelson and R.F. Picken, A Quantum Goldman Bracket for Loops on Surfaces, Int. J. Mod. Phys. A 24 (2009) 2839 [arXiv:0903.4809] [INSPIRE].
C. Appadu, T.J. Hollowood, J.L. Miramontes, D. Price and D.M. Schmidtt, Giant magnons of string theory in the lambda background, JHEP 07 (2017) 098 [arXiv:1704.05437].
M. Grigoriev and A.A. Tseytlin, Pohlmeyer reduction of AdS 5 × S 5 superstring σ-model, Nucl. Phys. B 800 (2008) 450 [arXiv:0711.0155] [INSPIRE].
D.M. Schmidtt, Supersymmetry Flows, Semi-Symmetric Space sine-Gordon Models And The Pohlmeyer Reduction, JHEP 03 (2011) 021 [arXiv:1012.4713] [INSPIRE].
T.J. Hollowood and J.L. Miramontes, The AdS 5 xS 5 Semi-Symmetric Space sine-Gordon Theory, JHEP 05 (2011) 136 [arXiv:1104.2429] [INSPIRE].
T. Hori and K. Kamimura, Canonical Formulation of Superstring, Prog. Theor. Phys. 73 (1985) 476 [INSPIRE].
A.K. Das, J. Maharana, A. Melikyan and M. Sato, The algebra of transition matrices for the AdS 5 × S 5 superstring, JHEP 12 (2004) 055 [hep-th/0411200] [INSPIRE].
B. Vicedo, Hamiltonian dynamics and the hidden symmetries of the AdS 5 × S 5 superstring, JHEP 01 (2010) 102 [arXiv:0910.0221] [INSPIRE].
M. Magro, The Classical Exchange Algebra of AdS 5 × S 5, JHEP 01 (2009) 021 [arXiv:0810.4136] [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].
G. Itsios, K. Sfetsos, K. Siampos and A. Torrielli, The classical Yang-Baxter equation and the associated Yangian symmetry of gauged WZW-type theories, Nucl. Phys. B 889 (2014) 64 [arXiv:1409.0554] [INSPIRE].
O. Babelon and D. Bernard, Dressing symmetries, Commun. Math. Phys. 149 (1992) 279 [hep-th/9111036] [INSPIRE].
M.F. Atiyah and R. Bott. The Yang-Mills Equations Over Riemann Surfaces, Philos. Trans. Roy. Soc. Lond. A 308 (1982) 523.
M. Audin, Lectures on gauge theory and integrable systems, in J. Hurtubise, F. Lalonde and G. Sabidussi eds., Gauge Theory and Symplectic Geometry, NATO ASI Ser. C 488 (1997) 1.
M. Bañados and I.A. Reyes, A short review on Noether’s theorems, gauge symmetries and boundary terms, Int. J. Mod. Phys. D 25 (2016) 1630021 [arXiv:1601.03616] [INSPIRE].
T. Regge and C. Teitelboim, Role of Surface Integrals in the Hamiltonian Formulation of General Relativity, Annals Phys. 88 (1974) 286 [INSPIRE].
J.M. Evans and P.A. Tuckey, A geometrical approach to time dependent gauge fixing, Int. J. Mod. Phys. A 8 (1993) 4055 [hep-th/9208009] [INSPIRE].
S.H.H. Chowdhury, On Goldman bracket for G 2 gauge group, JHEP 02 (2016) 001 [arXiv:1310.4519] [INSPIRE].
D.M. Schmidtt, in progress.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1808.05994
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Schmidtt, D.M. Lambda models from Chern-Simons theories. J. High Energ. Phys. 2018, 111 (2018). https://doi.org/10.1007/JHEP11(2018)111
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2018)111