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Permutation centralizer algebras and multimatrix invariants

Paolo Mattioli* and Sanjaye Ramgoolam†

  • Centre for Research in String Theory, School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom

  • *p.mattioli@qmul.ac.uk
  • †s.ramgoolam@qmul.ac.uk
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Phys. Rev. D 93, 065040 – Published 21 March, 2016

DOI: https://doi.org/10.1103/PhysRevD.93.065040

Abstract

We introduce a class of permutation centralizer algebras which underly the combinatorics of multimatrix gauge-invariant observables. One family of such noncommutative algebras is parametrized by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of two-matrix models. The structure of the algebra, notably its dimension, its center and its maximally commuting subalgebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The center of the algebra allows efficient computation of a sector of multimatrix correlators. These generate the counting of a certain class of bicoloured ribbon graphs with arbitrary genus.

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References (53)

  1. J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, 1113 (1999); Adv. Theor. Math. Phys. 2, 231 (1998).
  2. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  3. S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  4. S. Corley, A. Jevicki, and S. Ramgoolam, Exact correlators of giant gravitons from dual N=4 SYM theory, Adv. Theor. Math. Phys. 5, 809 (2002).
  5. S. Corley and S. Ramgoolam, Finite factorization equations and sum rules for BPS correlators in N=4 SYM theory, Nucl. Phys. B641, 131 (2002).
  6. V. Balasubramanian, M.-x. Huang, T. S. Levi, and A. Naqvi, Open strings from N=4 super Yang-Mills, J. High Energy Phys. 08 (2002) 037.
  7. V. Balasubramanian, D. Berenstein, B. Feng, and M.-x. Huang, D-branes in Yang-Mills theory and emergent gauge symmetry, J. High Energy Phys. 03 (2005) 006.
  8. R. de Mello Koch, J. Smolic, and M. Smolic, Giant gravitons—with strings attached (I), J. High Energy Phys. 06 (2007) 074.
  9. R. de Mello Koch, J. Smolic, and M. Smolic, Giant gravitons—with strings attached (II), J. High Energy Phys. 09 (2007) 049.
  10. D. Bekker, R. de Mello Koch, and M. Stephanou, Giant gravitons—with strings attached. III., J. High Energy Phys. 02 (2008) 029.
  11. J. M. Maldacena and A. Strominger, AdS(3) black holes and a stringy exclusion principle, J. High Energy Phys. 12 (1998) 005.
  12. J. McGreevy, L. Susskind, and N. Toumbas, Invasion of the giant gravitons from anti-de Sitter space, J. High Energy Phys. 06 (2000) 008.
  13. V. Balasubramanian, M. Berkooz, A. Naqvi, and M. J. Strassler, Giant gravitons in conformal field theory, J. High Energy Phys. 04 (2002) 034.
  14. Y. Kimura and S. Ramgoolam, Branes, anti-branes and brauer algebras in gauge-gravity duality, J. High Energy Phys. 11 (2007) 078.
  15. T. W. Brown, P. Heslop, and S. Ramgoolam, Diagonal multi-matrix correlators and BPS operators in N=4 SYM, J. High Energy Phys. 02 (2008) 030.
  16. T. W. Brown, P. Heslop, and S. Ramgoolam, Diagonal free field matrix correlators, global symmetries and giant gravitons, J. High Energy Phys. 04 (2009) 089.
  17. R. Bhattacharyya, S. Collins, and R. d. M. Koch, Exact multi-matrix correlators, J. High Energy Phys. 03 (2008) 044.
  18. S. Collins, Restricted Schur Polynomials and Finite N Counting, Phys. Rev. D 79, 026002 (2009).
  19. P. Caputa, R. d. M. Koch, and P. Diaz, Operators, Correlators and free fermions for SO(N) and Sp(N), J. High Energy Phys. 06 (2013) 018.
  20. R. de Mello Koch, B. A. E. Mohammed, J. Murugan, and A. Prinsloo, Beyond the planar limit in ABJM, J. High Energy Phys. 05 (2012) 037.
  21. J. Pasukonis and S. Ramgoolam, From counting to construction of BPS states in N=4 SYM, J. High Energy Phys. 02 (2011) 078.
  22. J. Pasukonis and S. Ramgoolam, Quivers as calculators: Counting, correlators and Riemann surfaces, J. High Energy Phys. 04 (2013) 094
  23. P. Mattioli and S. Ramgoolam, Quivers, Words and fundamentals, J. High Energy Phys. 03 (2015) 105.
  24. R. de Mello Koch, R. Kreyfelt, and N. Nokwara, Finite N quiver gauge theory, Phys. Rev. D 89, 126004 (2014).
  25. D. Berenstein, Extremal chiral ring states in the AdS/CFT correspondence are described by free fermions for a generalized oscillator algebra, Phys. Rev. D 92, 046006 (2015).
  26. Y. Kimura, Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras, Commun. Math. Phys. 337, 1 (2015).
  27. R. de Mello Koch and S. Ramgoolam, A double coset ansatz for integrability in AdS/CFT, J. High Energy Phys. 06 (2012) 083.
  28. W. Carlson, R. d. M. Koch, and H. Lin, Nonplanar Integrability, J. High Energy Phys. 03 (2011) 105.
  29. R. d. M. Koch, B. A. E. Mohammed, and S. Smith, Nonplanar integrability: Beyond the SU(2) sector, Int. J. Geom. Methods Mod. Phys. 26, 4553 (2011).
  30. R. d. M. Koch, M. Dessein, D. Giataganas, and C. Mathwin, Giant graviton oscillators, J. High Energy Phys. 10 (2011) 009.
  31. Y. Kimura and S. Ramgoolam, Enhanced symmetries of gauge theory and resolving the spectrum of local operators, Phys. Rev. D 78, 126003 (2008).
  32. P. Diaz, Novel charges in CFT‘s, J. High Energy Phys. 09 (2014).
  33. R. Goodman and N. Wallach, Representations and Invariants of the Classical Groups, Encyclopedia of Mathematics and Its Applications (Cambridge University Press, Cambridge, England, 2000).
  34. A. Ram, Ph.D. in Mathematics, University of California, San Diego, Representation theory; dissertation, chapter 1. http://www.ms.unimelb.edu.au/ram/Preprints/dissertationChapt1.pdf, 1990.
  35. Y. Kimura and S. Ramgoolam, Enhanced symmetries of gauge theory and resolving the spectrum of local operators, Phys. Rev. D 78, 126003 (2008).
  36. R. Bhattacharyya, R. de Mello Koch, and M. Stephanou, Exact multi-restricted Schur polynomial correlators, J. High Energy Phys. 06 (2008) 101.
  37. M. Hamermesh, Group Theory and Its Application to Physical Problems (Dover Publications, New York, 1989).
  38. F. Larsen and E. J. Martinec, U(1) charges and moduli in the D1—D5 system, J. High Energy Phys. 06 (1999) 019.
  39. S. Danz, J. Ellers, and J. Murray, The centralizer of a subgroup in a group algebra, Proc. Edinburgh Math. Soc. 56, 49 (2013).
  40. W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics/Readings in Mathematics (Springer New York, 1991).
  41. M. Bianchi, F. A. Dolan, P. J. Heslop, and H. Osborn, N=4 superconformal characters and partition functions, Nucl. Phys. B767, 163 (2007).
  42. D. Vaman and H. L. Verlinde, Bit strings from N=4 gauge theory, J. High Energy Phys. 11 (2003) 041.
  43. T. Brown, Cut-and-join operators and N=4 super Yang-Mills, J. High Energy Phys. 05 (2010) 058.
  44. J. Pasukonis and S. Ramgoolam, From counting to construction of BPS states in N=4 SYM, J. High Energy Phys. 02 (2011) 078.
  45. V. Balasubramanian, J. de Boer, V. Jejjala, and J. Simon, The Library of Babel: On the origin of gravitational thermodynamics, J. High Energy Phys. 12 (2005) 006.
  46. V. Balasubramanian, B. Czech, K. Larjo, and J. Simon, Integrability versus information loss: A simple example, J. High Energy Phys. 11 (2006) 001.
  47. R. d. M. Koch and S. Ramgoolam, From matrix models and quantum fields to Hurwitz space and the absolute Galois Group, arXiv:1002.1634.
  48. G. W. Moore and G. Segal, D-branes and K-theory in 2D topological field theory, arXiv:hep-th/0609042.
  49. D. Berenstein and R. G. Leigh, Resolution of stringy singularities by noncommutative algebras, J. High Energy Phys. 06 (2001) 030.
  50. Y. Kimura, Noncommutative frobenius algebras and open strings, Talk at Queen Mary University of London, 2014 (unpublished).
  51. J. Ben Geloun and S. Ramgoolam, Counting tensor model observables and branched covers of the 2-sphere, arXiv:1307.6490.
  52. A. Mironov, A. Morozov, and S. Natanzon, A Hurwitz theory avatar of open-closed strings, Eur. Phys. J. C 73, 2324 (2013).
  53. I. MacDonald, The Theory of Groups (Clarendon Press, Oxford, England, 1968).

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Phys. Rev. D 93, 065040– Published 21 March, 2016
  • Received 7 February 2016

© 2016 American Physical Society

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