Abstract
We study the mixed topological/holomorphic Chern-Simons theory of Costello, Witten and Yamazaki on an orbifold (Σ×ℂ)/ℤ2, obtaining a description of lattice integrable systems in the presence of a boundary. By performing an order ℏ calculation we derive a formula for the the asymptotic behaviour of K-matrices associated to rational, quasi-classical R-matrices. The ℤ2-action on Σ × ℂ fixes a line L, and line operators on L are shown to be labelled by representations of the twisted Yangian. The OPE of such a line operator with a Wilson line in the bulk is shown to give the coproduct of the twisted Yangian. We give the gauge theory realisation of the Sklyanin determinant and related conditions in the RTT presentation of the boundary Yang-Baxter equation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Costello, Supersymmetric gauge theory and the Yangian, arXiv:1303.2632 [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, I, arXiv:1709.09993 [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, II, arXiv:1802.01579 [INSPIRE].
A. Belavin and V. Drinfel’d, Solutions of the Classical Yang-Baxter Equation for Simple Lie Algebras, Func. Anal. Appl. 16 (1982) 159.
V. Drinfel’d, Hopf Algebras and the Quantum Yang-Baxter Equation, in Yang-Baxter Equation in Integrable Systems, World Scientific, Singapore (1990), pg. 264.
P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method. Recent developments, Lect. Notes Phys. 151 (1982) 61 [INSPIRE].
V. Drinfeld, Quantum Groups, Zap. Nauch. Sem. POMI 155 (1986) 18.
E.K. Sklyanin, Boundary Conditions for Integrable Quantum Systems, J. Phys. A 21 (1988) 2375 [INSPIRE].
G. Olshanskii, Twisted Yangians and Infinite-Dimensional Classical Lie Algebras, in Quantum Groups, Springer, Heidelberg Germany (1992), pp. 104.
A. Molev, M. Nazarov and G. Olshansky, Yangians and classical Lie algebras, Russ. Math. Surveys 51 (1996) 205 [hep-th/9409025] [INSPIRE].
P. Kulish, Yang-Baxter Equation and Reflection Equations in Integrable Models, in Low-Dimensional Models in Statistical Physics and Quantum Field Theory, Springer, Heidelberg Germany (1996), pp. 125.
H.-Q. Zhou, Quantum Integrability for the One-Dimensional Hubbard Open Chain, Phys. Rev. B 54 (1996) 41.
X.-W. Guan, M.-S. Wang and S.-D. Yang, Lax pair and boundary K-matrices for the one-dimensional Hubbard model, Nucl. Phys. B 485 (1997) 685 [INSPIRE].
M. Shiroishi and M. Wadati, Integrable Boundary Conditions for the One-Dimensional Hubbard Model, J. Phys. Soc. Jap. 66 (1997) 2288.
A. de la Rosa Gomez and N. MacKay, Twisted Yangian Symmetry of the Open Hubbard Model, J. Phys. A 47 (2014) 305203 [arXiv:1404.2095].
D. Berenstein and S. Vazquez, Integrable Open Spin Chains from Giant Gravitons, JHEP 06 (2005) 059 [hep-th/0501078] [INSPIRE].
D. Hofman and J. Maldacena, Reflecting Magnons, JHEP 11 (2007) 063 [arXiv:0708.2272] [INSPIRE].
V. Regelskis, Quantum Algebras and Integrable Boundaries in AdS/CFT, Ph.D. Thesis, University of York, York U.K. (2012).
I.V. Cherednik, Factorizing Particles on a Half Line and Root Systems, Theor. Math. Phys. 61 (1984) 977 [INSPIRE].
P. Fendley, Kinks in the Kondo problem, Phys. Rev. Lett. 71 (1993) 2485 [cond-mat/9304031] [INSPIRE].
H. Saleur, Lectures on Nonperturbative Field Theory and Quantum Impurity Problems, cond-mat/9812110.
G.W. Delius, N.J. MacKay and B.J. Short, Boundary remnant of Yangian symmetry and the structure of rational reflection matrices, Phys. Lett. B 522 (2001) 335 [Erratum ibid. B 524 (2002) 401] [hep-th/0109115] [INSPIRE].
N.J. MacKay, Rational K matrices and representations of twisted Yangians, J. Phys. A 35 (2002) 7865 [math/0205155] [INSPIRE].
N.J. MacKay and B.J. Short, Boundary scattering, symmetric spaces and the principal chiral model on the half line, Commun. Math. Phys. 233 (2003) 313 [Erratum ibid. 245 (2004) 425] [hep-th/0104212] [INSPIRE].
N.J. MacKay, Introduction to Yangian symmetry in integrable field theory, Int. J. Mod. Phys. A 20 (2005) 7189 [hep-th/0409183] [INSPIRE].
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge U.K. (1995).
A. Molev and E. Ragoucy, Representations of Reflection Algebras, Rev. Math. Phys. 14 (2002) 317 [math/0107213].
R. Thomas, Gauge Theory on Calabi-Yau Manifolds, Ph.D. Thesis, University of Oxford, Oxford U.K. (1997).
M. Ashwinkumar, M.-C. Tan and Q. Zhao, Branes and Categorifying Integrable Lattice Models, arXiv:1806.02821 [INSPIRE].
K. Costello and J. Yagi, Unification of integrability in supersymmetric gauge theories, arXiv:1810.01970 [INSPIRE].
P. Hořava, Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions, J. Geom. Phys. 21 (1996) 1 [hep-th/9404101].
G.W. Delius and N.J. MacKay, Quantum group symmetry in sine-Gordon and affine Toda field theories on the half line, Commun. Math. Phys. 233 (2003) 173 [hep-th/0112023] [INSPIRE].
D. Arnaudon, J. Avan, N. Crampé, A. Doikou, L. Frappat and É. Ragoucy, General boundary conditions for the sl(N) and sl(M|N) open spin chains, J. Stat. Mech. 0408 (2004) P08005 [math-ph/0406021] [INSPIRE].
D. Arnaudon, J. Avan, N. Crampé, A. Doikou, L. Frappat and É. Ragoucy, Classification of reflection matrices related to (super) Yangians and application to open spin chain models, Nucl. Phys. B 668 (2003) 469 [math/0304150] [INSPIRE].
S. Belliard and V. Regelskis, Drinfeld J-Presentation of Twisted Yangians, SIGMA 13 (2017) 011 [arXiv:1401.2143] [INSPIRE].
N. Guay and V. Regelskis, Twisted Yangians for Symmetric Pairs of Types b, c, d, Math. Zeit. 284 (2016) 131 [arXiv:1407.5247].
S. Belliard and N. Crampe, Coideal Algebras from Twisted Manin Triples, J. Geom. Phys. 62 (2012) 2009 [arXiv:1202.2312].
L.A. Takhtajan and L.D. Faddeev, The Quantum method of the inverse problem and the Heisenberg XYZ model, Russ. Math. Surveys 34 (1979) 11 [INSPIRE].
J. Ellis, TikZ-Feynman: Feynman Diagrams with TikZ, Comput. Phys. Commun. 210 (2017) 103 [arXiv:1601.05437] [INSPIRE].
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: 1903.03601
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
Bittleston, R., Skinner, D. Gauge theory and boundary integrability. J. High Energ. Phys. 2019, 195 (2019). https://doi.org/10.1007/JHEP05(2019)195
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)195