Abstract
Representation theory provides an efficient framework to count and classify invariants in tensor models of (gauge) symmetry G d = U(N1) ⊗ · · · ⊗ U(N d ) . We show that there are two natural ways of counting invariants, one for arbitrary G d and another valid for large rank of G d . We construct basis of invariant operators based on the counting, and compute correlators of their elements. The basis associated with finite rank of G d diagonalizes two-point function. It is analogous to the restricted Schur basis used in matrix models. We comment on future directions for investigation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Einstein, B. Podolsky and N. Rosen, Can quantum mechanical description of physical reality be considered complete?, Phys. Rev. 47 (1935) 777 [INSPIRE].
M. Christandl and G. Mitchison, The spectra of quantum states and the Kronecker coefficients of the symmetric group, Commun. Math. Phys. 261 (2006) 789 [quant-ph/0409016].
P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2D gravity and random matrices, Phys. Rept. 254 (1995) 1 [hep-th/9306153] [INSPIRE].
J. Ambjørn, B. Durhuus and T. Jonsson, Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A 6 (1991) 1133 [INSPIRE].
M. Gross, Tensor models and simplicial quantum gravity in > 2D, Nucl. Phys. Proc. Suppl. A 25 (1992) 144 [INSPIRE].
N. Sasakura, Tensor model for gravity and orientability of manifold, Mod. Phys. Lett. A 6 (1991) 2613 [INSPIRE].
R. Gurau, Colored group field theory, Commun. Math. Phys. 304 (2011) 69 [arXiv:0907.2582] [INSPIRE].
R. Gurau and J.P. Ryan, Colored tensor models — a review, SIGMA 8 (2012) 020 [arXiv:1109.4812] [INSPIRE].
R. Gurau, The 1/N expansion of colored tensor models, Annales Henri Poincaré 12 (2011) 829 [arXiv:1011.2726] [INSPIRE].
R. Gurau and V. Rivasseau, The 1/N expansion of colored tensor models in arbitrary dimension, EPL 95 (2011) 50004 [arXiv:1101.4182] [INSPIRE].
R. Gurau, The complete 1/N expansion of colored tensor models in arbitrary dimension, Annales Henri Poincaré 13 (2012) 399 [arXiv:1102.5759] [INSPIRE].
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett. 70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk at KITP Strings Seminar and Entanglement 2015 program, KITP, Santa Barbara U.S.A., 12 February 2015.
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP Strings Seminar and Entanglement 2015 program, KITP, Santa Barbara U.S.A., 7 April 2015.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP Strings Seminar and Entanglement 2015 program, KITP, Santa Barbara U.S.A., 27 May 2015.
J. Polchinski and V. Rosenhaus, The spectrum in the Sachdev-Ye-Kitaev model, JHEP 04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-local holography in the SYK model, JHEP 07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
S.R. Das, A. Jevicki and K. Suzuki, Three dimensional view of the SYK/AdS duality, JHEP 09 (2017) 017 [arXiv:1704.07208] [INSPIRE].
E. Witten, An SYK-like model without disorder, arXiv:1610.09758 [INSPIRE].
R. Gurau, The complete 1/N expansion of a SYK-like tensor model, Nucl. Phys. B 916 (2017) 386 [arXiv:1611.04032] [INSPIRE].
S. Corley, A. Jevicki and S. Ramgoolam, Exact correlators of giant gravitons from dual N = 4 SYM theory, Adv. Theor. Math. Phys. 5 (2002) 809[hep-th/0111222] [INSPIRE].
V. Balasubramanian, D. Berenstein, B. Feng and M.-X. Huang, D-branes in Yang-Mills theory and emergent gauge symmetry, JHEP 03 (2005) 006 [hep-th/0411205] [INSPIRE].
R. Bhattacharyya, S. Collins and R. de Mello Koch, Exact multi-matrix correlators, JHEP 03 (2008) 044 [arXiv:0801.2061] [INSPIRE].
R. Bhattacharyya, R. de Mello Koch and M. Stephanou, Exact multi-restricted Schur polynomial correlators, JHEP 06 (2008) 101 [arXiv:0805.3025] [INSPIRE].
Y. Kimura and S. Ramgoolam, Branes, anti-branes and Brauer algebras in gauge-gravity duality, JHEP 11 (2007) 078 [arXiv:0709.2158] [INSPIRE].
Y. Kimura, Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra, JHEP 12 (2009) 044 [arXiv:0910.2170] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal multi-matrix correlators and BPS operators in N = 4 SYM, JHEP 02 (2008) 030 [arXiv:0711.0176] [INSPIRE].
T.W. Brown, P.J. Heslop and S. Ramgoolam, Diagonal free field matrix correlators, global symmetries and giant gravitons, JHEP 04 (2009) 089 [arXiv:0806.1911] [INSPIRE].
R. de Mello Koch, P. Diaz and H. Soltanpanahi, Non-planar anomalous dimensions in the sl(2) sector, Phys. Lett. B 713 (2012) 509 [arXiv:1111.6385] [INSPIRE].
R. de Mello Koch, P. Diaz and N. Nokwara, Restricted Schur polynomials for fermions and integrability in the SU(2|3) sector, JHEP 03 (2013) 173 [arXiv:1212.5935] [INSPIRE].
R. de Mello Koch, J. Smolic and M. Smolic, Giant gravitons — with strings attached (I), JHEP 06 (2007) 074 [hep-th/0701066] [INSPIRE].
R. de Mello Koch, J. Smolic and M. Smolic, Giant gravitons — with strings attached (II), JHEP 09 (2007) 049 [hep-th/0701067] [INSPIRE].
D. Bekker, R. de Mello Koch and M. Stephanou, Giant gravitons — with strings attached (III), JHEP 02 (2008) 029 [arXiv:0710.5372] [INSPIRE].
R. de Mello Koch, G. Mashile and N. Park, Emergent threebrane lattices, Phys. Rev. D 81 (2010) 106009 [arXiv:1004.1108] [INSPIRE].
V. De Comarmond, R. de Mello Koch and K. Jefferies, Surprisingly simple spectra, JHEP 02 (2011) 006 [arXiv:1012.3884] [INSPIRE].
M.W. Hero and J.F. Willenbring, Stable Hilbert series as related to the measurement of quantum entanglement, Discrete Math. 309 (2009) 6508.
J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, arXiv:1307.6490 [INSPIRE].
P. Diaz and S.-J. Rey, work to appear.
I.G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, Oxford U.K., (1995).
A. Mironov and A. Morozov, Correlators in tensor models from character calculus, Phys. Lett. B 774 (2017) 210 [arXiv:1706.03667] [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: 1706.02667
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
Diaz, P., Rey, SJ. Orthogonal bases of invariants in tensor models. J. High Energ. Phys. 2018, 89 (2018). https://doi.org/10.1007/JHEP02(2018)089
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2018)089