×

Generalized dynamical spin chain and 4-loop integrability in \({\mathcal N}=6\) superconformal Chern-Simons theory. (English) Zbl 1203.82060

Summary: We revisit unitary representation of centrally extended \({\mathfrak psu}(2|2)\) excitation superalgebra. We find most generally that ‘pseudo-momentum’, not lattice momentum, diagonalizes spin chain Hamiltonian and leads to generalized dynamic spin chain. All known results point to lattice momentum diagonalization for \({\mathcal N}=4\) super-Yang-Mills theory. Having different interacting structure, we ask if \({\mathcal N}=6\) superconformal Chern-Simons theory provides an example of pseudo-momentum diagonalization. For \(SO(6)\) sector, we study maximal shuffling and next-to-maximal shuffling terms in the dilatation operator and compare them with results expected from \({\mathfrak psu}(2|2)\) superalgebra and integrability. At two loops, we rederive maximal shuffling term (3-site) and find perfect agreement with known results. At four loops, we first find absence of next-to-maximal shuffling term (4-site), in agreement with prediction based on integrability. We next extract maximal shuffling term (5-site), the most relevant term for checking the possibility of pseudo-momentum diagonalization. Curiously, we find that result agrees with integrability prediction based on lattice momentum, as in \({\mathcal N}=4\) super-Yang-Mills theory. Consistency of our results is fully ensured by checks of renormalizability up to six loops.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics
81T13 Yang-Mills and other gauge theories in quantum field theory
17A70 Superalgebras
16S20 Centralizing and normalizing extensions

Software:

AMBRE
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Aharony, O.; Bergman, O.; Jafferis, D. L.; Maldacena, J., \(N = 6\) superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP, 0810, 091 (2008) · Zbl 1245.81130
[2] Maldacena, J. M., The large \(N\) limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys.. Adv. Theor. Math. Phys., Int. J. Theor. Phys., 38, 1113 (1999) · Zbl 0969.81047
[3] Minahan, J. A.; Zarembo, K., The Bethe-ansatz for \(N = 4\) super-Yang-Mills, JHEP, 0303, 013 (2003)
[4] Beisert, N.; Kristjansen, C.; Staudacher, M., The dilatation operator of \(N = 4\) super-Yang-Mills theory, Nucl. Phys. B, 664, 131 (2003) · Zbl 1051.81044
[5] Bena, I.; Polchinski, J.; Roiban, R., Hidden symmetries of the \(AdS(5) \times S^5\) superstring, Phys. Rev. D, 69, 046002 (2004)
[6] Beisert, N.; Staudacher, M., The \(N = 4\) SYM integrable super spin chain, Nucl. Phys. B, 670, 439 (2003) · Zbl 1058.81581
[7] Beisert, N., The \(su(2 | 3)\) dynamic spin chain, Nucl. Phys. B, 682, 487 (2004) · Zbl 1036.82513
[8] Serban, D.; Staudacher, M., Planar \(N = 4\) gauge theory and the Inozemtsev long range spin chain, JHEP, 0406, 001 (2004)
[9] Eden, B.; Jarczak, C.; Sokatchev, E., A three-loop test of the dilatation operator in \(N = 4\) SYM, Nucl. Phys. B, 712, 157 (2005) · Zbl 1109.81354
[10] Staudacher, M., The factorized S-matrix of CFT/AdS, JHEP, 0505, 054 (2005)
[11] Beisert, N.; Staudacher, M., Long-range \(PSU(2, 2 | 4)\) Bethe ansaetze for gauge theory and strings, Nucl. Phys. B, 727, 1 (2005)
[12] Rej, A.; Serban, D.; Staudacher, M., Planar \(N = 4\) gauge theory and the Hubbard model, JHEP, 0603, 018 (2006) · Zbl 1226.81272
[13] Beisert, N.; Eden, B.; Staudacher, M., Transcendentality and crossing, J. Stat. Mech., 0701, P021 (2007)
[14] Arutyunov, G.; Frolov, S.; Zamaklar, M., The Zamolodchikov-Faddeev algebra for \(AdS(5) \times S^5\) superstring, JHEP, 0704, 002 (2007)
[15] Beisert, N., The \(su(2 | 2)\) dynamic S-matrix, Adv. Theor. Math. Phys., 12, 945 (2008)
[16] Beisert, N., The analytic Bethe ansatz for a chain with centrally extended \(su(2 | 2)\) symmetry, J. Stat. Mech., 0701, P017 (2007)
[17] Gomis, J.; Sorokin, D.; Wulff, L., The complete \(AdS_4 \times CP^3\) superspace for the type IIA superstring and D-branes, JHEP, 0903, 015 (2009)
[18] Bak, D.; Rey, S. J., Integrable spin chain in superconformal Chern-Simons theory, JHEP, 0810, 053 (2008) · Zbl 1245.81259
[19] Minahan, J. A.; Zarembo, K., The Bethe ansatz for superconformal Chern-Simons, JHEP, 0809, 040 (2008) · Zbl 1245.81102
[20] Sundin, P., The \(AdS_4 \times CP^3\) string and its Bethe equations in the near plane wave limit, JHEP, 0902, 046 (2009) · Zbl 1245.81226
[21] Grignani, G.; Harmark, T.; Orselli, M., The \(SU(2) \times SU(2)\) sector in the string dual of \(N = 6\) superconformal Chern-Simons theory · Zbl 1192.81269
[22] Papathanasiou, G.; Spradlin, M., The morphology of \(N = 6\) Chern-Simons theory
[23] Ahn, C.; Nepomechie, R. I., Two-loop test of the \(N = 6\) Chern-Simons theory S-matrix
[24] Astolfi, D.; Puletti, V. G.M.; Grignani, G.; Harmark, T.; Orselli, M., Finite-size corrections in the \(SU(2) \times SU(2)\) sector of type IIA string theory on \(AdS_4 \times CP^3\) · Zbl 1323.81073
[25] McLoughlin, T.; Roiban, R.; Tseytlin, A. A., Quantum spinning strings in \(AdS_4 \times CP^3\): Testing the Bethe Ansatz proposal, JHEP, 0811, 069 (2008)
[26] Zarembo, K., Worldsheet spectrum in \(AdS_4/CFT_3\) correspondence
[27] Abbott, M. C.; Aniceto, I.; Sax, O. O., Dyonic Giant magnons in \(CP^3\): Strings and curves at finite \(J\)
[28] Janik, R. A., The \(AdS(5) \times S^5\) superstring worldsheet S-matrix and crossing symmetry, Phys. Rev. D, 73, 086006 (2006)
[29] Bargheer, T.; Beisert, N.; Loebbert, F.
[30] Gross, D. J.; Mikhailov, A.; Roiban, R., Operators with large \(R\) charge in \(N = 4\) Yang-Mills theory, Annals Phys., 301, 31 (2002) · Zbl 1014.81041
[31] J. Minahan, talk at integrability in Gauge and string theory, August 11-15, 2008, Utrecht, Netherlands (http://www.science.uu.nl/IGST08/video/Minahan.html; J. Minahan, talk at integrability in Gauge and string theory, August 11-15, 2008, Utrecht, Netherlands (http://www.science.uu.nl/IGST08/video/Minahan.html
[32] Arutyunov, G.; Frolov, S.; Zamaklar, M., Finite-size effects from giant magnons, Nucl. Phys. B, 778, 1 (2007) · Zbl 1200.83114
[33] Bak, D.; Gang, D.; Rey, S. J., Integrable spin chain of superconformal \(U(M) \times U(N)\) Chern-Simons theory, JHEP, 0810, 038 (2008) · Zbl 1245.81258
[34] Bak, D., Zero modes for the boundary Giant magnons, Phys. Lett. B, 672, 284 (2009)
[35] Hohenegger, S.; Kirsch, I., A note on the holography of Chern-Simons matter theories with flavour
[36] Bierenbaum, I.; Weinzierl, S., The massless two-loop two-point function, Eur. Phys. J. C, 32, 67 (2003) · Zbl 1099.81534
[37] Czakon, M., Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun., 175, 559 (2006) · Zbl 1196.81054
[38] Smirnov, A. V.; Smirnov, V. A., On the resolution of singularities of multiple Mellin-Barnes integrals · Zbl 1188.81090
[39] Gluza, J.; Kajda, K.; Riemann, T., AMBRE — a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun., 177, 879 (2007) · Zbl 1196.81131
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.