Abstract
In the context of the AdS3/CFT2 correspondence, we investigate the Higgs branch CFT2. Witten showed that states localised near the small instanton singularity can be described in terms of vector multiplet variables. This theory has a planar, weak-coupling limit, in which anomalous dimensions of single-trace composite operators can be calculated. At one loop, the calculation reduces to finding the spectrum of a spin-chain with nearest-neighbour interactions. This CFT2 spin-chain matches precisely the one that was previously found as the weak-coupling limit of the integrable system describing the AdS3 side of the duality. We compute the one-loop dilatation operator in a non-trivial compact subsector and show that it corresponds to an integrable spin-chain Hamiltonian. This provides the first direct evidence of integrability on the CFT2 side of the correspondence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
G. Arutyunov and S. Frolov, Foundations of the AdS 5 × S 5 Superstring. Part I, J. Phys. A 42 (2009) 254003 [arXiv:0901.4937] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
A. Babichenko, B. Stefanski Jr. and K. Zarembo, Integrability and the AdS 3 /CF T 2 correspondence, JHEP 03 (2010) 058 [arXiv:0912.1723] [INSPIRE].
P. Sundin and L. Wulff, Classical integrability and quantum aspects of the AdS 3 × S 3 × S 3 × S 1 superstring, JHEP 10 (2012) 109 [arXiv:1207.5531] [INSPIRE].
A. Cagnazzo and K. Zarembo, B-field in AdS 3 /CF T 2 Correspondence and Integrability, JHEP 11 (2012) 133 [arXiv:1209.4049] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
A. Sfondrini, Towards integrability for AdS 3 /CF T 2, J. Phys. A 48 (2015) 023001 [arXiv:1406.2971] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., Towards the All-Loop Worldsheet S Matrix for AdS 3 × S 3 × T 4, Phys. Rev. Lett. 113 (2014) 131601 [arXiv:1403.4543] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete AdS 3 × S 3 × T 4 worldsheet S matrix, JHEP 10 (2014) 066 [arXiv:1406.0453] [INSPIRE].
T. Lloyd, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The complete worldsheet S matrix of superstrings on AdS 3 × S 3 × T 4 with mixed three-form flux, Nucl. Phys. B 891 (2015) 570 [arXiv:1410.0866] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini and B. Stefanski Jr., The AdS 3 × S 3 × S 3 × S 1 worldsheet S matrix, arXiv:1506.00218 [INSPIRE].
J.R. David and B. Sahoo, Giant magnons in the D1-D5 system, JHEP 07 (2008) 033 [arXiv:0804.3267] [INSPIRE].
J.R. David and B. Sahoo, S-matrix for magnons in the D1-D5 system, JHEP 10 (2010) 112 [arXiv:1005.0501] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
O. Ohlsson Sax and B. Stefanski Jr., Integrability, spin-chains and the AdS 3 /CF T 2 correspondence, JHEP 08 (2011) 029 [arXiv:1106.2558] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, A dynamic SU(1|1)2 S-matrix for AdS 3 /CFT 2, JHEP 04 (2013) 113 [arXiv:1211.5119] [INSPIRE].
R. Borsato, O. Ohlsson Sax and A. Sfondrini, All-loop Bethe ansatz equations for AdS 3 /CFT 2, JHEP 04 (2013) 116 [arXiv:1212.0505] [INSPIRE].
R. Borsato, O. Ohlsson Sax, A. Sfondrini, B. Stefanski Jr. and A. Torrielli, The all-loop integrable spin-chain for strings on AdS 3 × S 3 × T 4 : the massive sector, JHEP 08 (2013) 043 [arXiv:1303.5995] [INSPIRE].
O. Ohlsson Sax, B. Stefanski Jr. and A. Torrielli, On the massless modes of the AdS 3 /CFT 2 integrable systems, JHEP 03 (2013) 109 [arXiv:1211.1952] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for \( \mathcal{N}=4 \) super Yang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
L. Brink, J.H. Schwarz and J. Scherk, Supersymmetric Yang-Mills Theories, Nucl. Phys. B 121 (1977) 77 [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J.M. Maldacena, \( \mathcal{N}=6 \) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].
P.C. Argyres, M.R. Plesser and A.D. Shapere, N = 2 moduli spaces and N = 1 dualities for SO(n c ) and USp(2n c ) super-QCD, Nucl. Phys. B 483 (1997) 172 [hep-th/9608129] [INSPIRE].
D.-E. Diaconescu and N. Seiberg, The Coulomb branch of (4, 4) supersymmetric field theories in two-dimensions, JHEP 07 (1997) 001 [hep-th/9707158] [INSPIRE].
P. Fayet and J. Iliopoulos, Spontaneously Broken Supergauge Symmetries and Goldstone Spinors, Phys. Lett. B 51 (1974) 461 [INSPIRE].
M.R. Douglas, Branes within branes, hep-th/9512077 [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
E. Witten, On the conformal field theory of the Higgs branch, JHEP 07 (1997) 003 [hep-th/9707093] [INSPIRE].
O. Aharony and M. Berkooz, IR dynamics of d = 2, \( \mathcal{N}=\left(4,\;4\right) \) gauge theories and DLCQ of “little string theories”, JHEP 10 (1999) 030 [hep-th/9909101] [INSPIRE].
M.F. Sohnius, Introducing Supersymmetry, Phys. Rept. 128 (1985) 39 [INSPIRE].
J. Wess and J. Bagger, Supersymmetry and supergravity, Princeton University Press (1992).
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
S. Gates Jr., C.M. Hull and M. Roček, Twisted Multiplets and New Supersymmetric Nonlinear σ-models, Nucl. Phys. B 248 (1984) 157 [INSPIRE].
N. Beisert, The Dilatation operator of N = 4 super Yang-Mills theory and integrability, Phys. Rept. 405 (2004) 1 [hep-th/0407277] [INSPIRE].
J. de Boer, Six-dimensional supergravity on S 3 × AdS 3 and 2-D conformal field theory, Nucl. Phys. B 548 (1999) 139 [hep-th/9806104] [INSPIRE].
S.R. Coleman, 1/N , in the proceedings of the 1979 Erice School of Subnuclear Physics, Erice, Italy, July 31 - August 11 1979 and in Aspects of Symmetry. Selected Erice Lectures, Cambridge University Press (1985).
A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear σ-models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].
N.Y. Reshetikhin, A Method Of Functional Equations In The Theory Of Exactly Solvable Quantum Systems, Lett. Math. Phys. 7 (1983) 205 [INSPIRE].
N.Y. Reshetikhin, Integrable Models of Quantum One-dimensional Magnets With O(N ) and Sp(2k) Symmetry, Theor. Math. Phys. 63 (1985) 555 [INSPIRE].
N. Beisert, The SU(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487 [hep-th/0310252] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, A Spin Chain for the Symmetric Product CFT 2, JHEP 05 (2010) 099 [arXiv:0912.0959] [INSPIRE].
B. Hoare and A.A. Tseytlin, Massive S-matrix of AdS 3 × S 3 × T 4 superstring theory with mixed 3-form flux, Nucl. Phys. B 873 (2013) 395 [arXiv:1304.4099] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
D. Tong, The holographic dual of AdS 3 × S 3 × S 3 × S 1, JHEP 04 (2014) 193 [arXiv:1402.5135] [INSPIRE].
K. Lang and W. Rühl, The Critical O(N ) σ-model at dimension 2 < d < 4 and order 1/N 2 : Operator product expansions and renormalization, Nucl. Phys. B 377 (1992) 371 [INSPIRE].
A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N ) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [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: 1411.3676
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Sax, O.O., Sfondrini, A. & Stefanski, B. Integrability and the conformal field theory of the Higgs branch. J. High Energ. Phys. 2015, 103 (2015). https://doi.org/10.1007/JHEP06(2015)103
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2015)103