Journal of High Energy Physics

, 2018:15 | Cite as

Chaos from equivariant fields on fuzzy S4

  • Ü. H. Coşkun
  • S. Kürkçüoğlu
  • G. C. Toga
  • G. Ünal
Open Access
Regular Article - Theoretical Physics


We examine the 5d Yang-Mills matrix model in 0 + 1-dimensions with U(4N) gauge symmetry and a mass deformation term. We determine the explicit SU(4) ≈ SO(6) equivariant parametrizations of the gauge field and the fluctuations about the classical four concentric fuzzy four sphere configuration and obtain the low energy reduced actions(LEAs) by tracing over the S F 4 s for the first five lowest matrix levels. The LEAs so obtained have potentials bounded from below indicating that the equivariant fluctuations about the S F 4 do not lead to any instabilities. These reduced systems exhibit chaos, which we reveal by computing their Lyapunov exponents. Using our numerical results, we explore various aspects of chaotic dynamics emerging from the LEAs. In particular, we model how the largest Lyapunov exponents change as a function of the energy. We also show that, in the Euclidean signature, the LEAs support the usual kink type soliton solutions, i.e. instantons in 1+ 0-dimensions, which may be seen as the imprints of the topological fluxes penetrating the concentric S F 4 s due to the equivariance conditions, and preventing them to shrink to zero radius. Relaxing the Gauss law constraint in the LEAs in the manner recently discussed by Maldacena and Milekhin leads to Goldstone bosons.


Gauge Symmetry M(atrix) Theories Spontaneous Symmetry Breaking 


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.


  1. [1]
    T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
  2. [2]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 super-Yang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    K. Dasgupta, M.M. Sheikh-Jabbari and M. Van Raamsdonk, Matrix perturbation theory for M-theory on a pp wave, JHEP 05 (2002) 056 [hep-th/0205185] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    B. Ydri, Review of M(atrix)-theory, type IIB matrix model and matrix string theory, arXiv:1708.00734 [INSPIRE].
  5. [5]
    E. Kiritsis, String theory in a nutshell, Princeton University Press, Princeton U.S.A. (2011).Google Scholar
  6. [6]
    B. Ydri, Lectures on matrix field theory, Lect. Notes Phys. 929 (2017) 1 [arXiv:1603.00924].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    N. Iizuka, D. Kabat, S. Roy and D. Sarkar, Black hole formation in fuzzy sphere collapse, Phys. Rev. D 88 (2013) 044019 [arXiv:1306.3256] [INSPIRE].
  8. [8]
    K.N. Anagnostopoulos, M. Hanada, J. Nishimura and S. Takeuchi, Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature, Phys. Rev. Lett. 100 (2008) 021601 [arXiv:0707.4454] [INSPIRE].
  9. [9]
    S. Catterall and T. Wiseman, Black hole thermodynamics from simulations of lattice Yang-Mills theory, Phys. Rev. D 78 (2008) 041502 [arXiv:0803.4273] [INSPIRE].
  10. [10]
    M. Hanada, Y. Hyakutake, J. Nishimura and S. Takeuchi, Higher derivative corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics, Phys. Rev. Lett. 102 (2009) 191602 [arXiv:0811.3102] [INSPIRE].ADSCrossRefGoogle Scholar
  11. [11]
    C. Asplund, D. Berenstein and D. Trancanelli, Evidence for fast thermalization in the plane-wave matrix model, Phys. Rev. Lett. 107 (2011) 171602 [arXiv:1104.5469] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    C.T. Asplund, D. Berenstein and E. Dzienkowski, Large N classical dynamics of holographic matrix models, Phys. Rev. D 87 (2013) 084044 [arXiv:1211.3425] [INSPIRE].
  13. [13]
    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    G. Gur-Ari, M. Hanada and S.H. Shenker, Chaos in classical D0-brane mechanics, JHEP 02 (2016) 091 [arXiv:1512.00019] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    D. Berenstein and D. Kawai, Smallest matrix black hole model in the classical limit, Phys. Rev. D 95 (2017) 106004 [arXiv:1608.08972] [INSPIRE].
  16. [16]
    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Berkowitz et al., Precision lattice test of the gauge/gravity duality at large-N , Phys. Rev. D 94 (2016) 094501 [arXiv:1606.04951] [INSPIRE].
  18. [18]
    Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
  19. [19]
    V.G. Filev and D. O’Connor, The BFSS model on the lattice, JHEP 05 (2016) 167 [arXiv:1506.01366] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    V.G. Filev and D. O’Connor, A computer test of holographic flavour dynamics, JHEP 05 (2016) 122 [arXiv:1512.02536] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    E. Rinaldi et al., Toward holographic reconstruction of bulk geometry from lattice simulations, JHEP 02 (2018) 042 [arXiv:1709.01932] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    Y. Asano, D. Kawai and K. Yoshida, Chaos in the BMN matrix model, JHEP 06 (2015) 191 [arXiv:1503.04594] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    I.Ya. Aref’eva, P.B. Medvedev, O.A. Rytchkov and I.V. Volovich, Chaos in M(atrix) theory, Chaos Solitons Fractals 10 (1999) 213 [hep-th/9710032] [INSPIRE].
  24. [24]
    R. Hübener, Y. Sekino and J. Eisert, Equilibration in low-dimensional quantum matrix models, JHEP 04 (2015) 166 [arXiv:1403.1392] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J. Castelino, S. Lee and W. Taylor, Longitudinal five-branes as four spheres in matrix theory, Nucl. Phys. B 526 (1998) 334 [hep-th/9712105] [INSPIRE].
  26. [26]
    Y. Kimura, Noncommutative gauge theory on fuzzy four sphere and matrix model, Nucl. Phys. B 637 (2002) 177 [hep-th/0204256] [INSPIRE].
  27. [27]
    H.C. Steinacker, One-loop stabilization of the fuzzy four-sphere via softly broken SUSY, JHEP 12 (2015) 115 [arXiv:1510.05779] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  28. [28]
    J. Maldacena and A. Milekhin, To gauge or not to gauge?, JHEP 04 (2018) 084 [arXiv:1802.00428] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    E. Berkowitz, M. Hanada, E. Rinaldi and P. Vranas, Gauged and ungauged: a nonperturbative test, JHEP 06 (2018) 124 [arXiv:1802.02985] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  30. [30]
    T. Azuma, S. Bal, K. Nagao and J. Nishimura, Absence of a fuzzy S 4 phase in the dimensionally reduced 5D Yang-Mills-Chern-Simons model, JHEP 07 (2004) 066 [hep-th/0405096] [INSPIRE].
  31. [31]
    H. Grosse, C. Klimčík and P. Prešnajder, On finite 4D quantum field theory in noncommutative geometry, Commun. Math. Phys. 180 (1996) 429 [hep-th/9602115] [INSPIRE].
  32. [32]
    S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461 [hep-th/0105006] [INSPIRE].
  33. [33]
    Y. Abe, Construction of fuzzy S 4, Phys. Rev. D 70 (2004) 126004 [hep-th/0406135] [INSPIRE].
  34. [34]
    A.P. Balachandran, S. Kurkcuoglu and S. Vaidya, Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114 [INSPIRE].
  35. [35]
    B. Ydri, A. Rouag and K. Ramda, Emergent fuzzy geometry and fuzzy physics in four dimensions, Nucl. Phys. B 916 (2017) 567 [arXiv:1607.08296] [INSPIRE].
  36. [36]
    D. Harland and S. Kurkcuoglu, Equivariant reduction of Yang-Mills theory over the fuzzy sphere and the emergent vortices, Nucl. Phys. B 821 (2009) 380 [arXiv:0905.2338] [INSPIRE].
  37. [37]
    S. Kurkcuoglu, Noncommutative Vortices and Flux-Tubes from Yang-Mills Theories with Spontaneously Generated Fuzzy Extra Dimensions, Phys. Rev. D 82 (2010) 105010 [arXiv:1009.1880] [INSPIRE].
  38. [38]
    S. Kurkcuoglu, Equivariant reduction of U(4) gauge theory over S F2 × S F2 and the emergent vortices, Phys. Rev. D 85 (2012) 105004 [arXiv:1201.0728] [INSPIRE].
  39. [39]
    S. Kurkcuoglu, New fuzzy extra dimensions from \( SU\left(\mathcal{N}\right) \) gauge theories, Phys. Rev. D 92 (2015) 025022 [arXiv:1504.02524] [INSPIRE].
  40. [40]
    S. Kürkçüoğlu and G. Ünal, Equivariant fields in an \( \mathrm{S}\mathrm{U}\left(\mathcal{N}\right) \) gauge theory with new spontaneously generated fuzzy extra dimensions, Phys. Rev. D 93 (2016) 105019 [arXiv:1506.04335] [INSPIRE].
  41. [41]
    S. Kürkçüoğlu and G. Ünal, U (3) gauge theory on fuzzy extra dimensions, Phys. Rev. D 94 (2016) 036003 [arXiv:1607.00075] [INSPIRE].
  42. [42]
    P. Aschieri, T. Grammatikopoulos, H. Steinacker and G. Zoupanos, Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking, JHEP 09 (2006) 026 [hep-th/0606021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  43. [43]
    A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes, Fortsch. Phys. 58 (2010) 537 [arXiv:0909.5559].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    E. Ott, Chaos in dynamical systems, Cambridge University Press, Cambridge U.K. (2002).Google Scholar
  45. [45]
    R. Hilborn, Chaos and nonlinear dynamics: an introduction for scientists and engineers, Oxford University Press, Oxford U.K. (2000).CrossRefzbMATHGoogle Scholar
  46. [46]
    S. Coleman, Aspects of symmetry: selected erice lectures, Cambridge University Press, Cambridge U.K. (1988).Google Scholar
  47. [47]
    R. Rajaraman, Solitons and instantons. An introduction to solitons and instantons in quatum field theory, Elsevier, The Netherlands (1982).Google Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Ü. H. Coşkun
    • 1
    • 2
  • S. Kürkçüoğlu
    • 1
  • G. C. Toga
    • 1
    • 3
  • G. Ünal
    • 1
  1. 1.Middle East Technical University, Department of PhysicsAnkaraTurkey
  2. 2.University of Kentucky, Department of Physics and AstronomyLexingtonU.S.A.
  3. 3.Gazi University, Department of PhysicsAnkaraTurkey

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