Abstract
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 4F 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 4F 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 4F 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
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].
B. Ydri, Review of M(atrix)-theory, type IIB matrix model and matrix string theory, arXiv:1708.00734 [INSPIRE].
E. Kiritsis, String theory in a nutshell, Princeton University Press, Princeton U.S.A. (2011).
B. Ydri, Lectures on matrix field theory, Lect. Notes Phys. 929 (2017) 1 [arXiv:1603.00924].
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].
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].
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].
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].
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].
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].
S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [arXiv:1306.0622] [INSPIRE].
G. Gur-Ari, M. Hanada and S.H. Shenker, Chaos in classical D0-brane mechanics, JHEP 02 (2016) 091 [arXiv:1512.00019] [INSPIRE].
D. Berenstein and D. Kawai, Smallest matrix black hole model in the classical limit, Phys. Rev. D 95 (2017) 106004 [arXiv:1608.08972] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
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].
Y. Sekino and L. Susskind, Fast scramblers, JHEP 10 (2008) 065 [arXiv:0808.2096] [INSPIRE].
V.G. Filev and D. O’Connor, The BFSS model on the lattice, JHEP 05 (2016) 167 [arXiv:1506.01366] [INSPIRE].
V.G. Filev and D. O’Connor, A computer test of holographic flavour dynamics, JHEP 05 (2016) 122 [arXiv:1512.02536] [INSPIRE].
E. Rinaldi et al., Toward holographic reconstruction of bulk geometry from lattice simulations, JHEP 02 (2018) 042 [arXiv:1709.01932] [INSPIRE].
Y. Asano, D. Kawai and K. Yoshida, Chaos in the BMN matrix model, JHEP 06 (2015) 191 [arXiv:1503.04594] [INSPIRE].
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].
R. Hübener, Y. Sekino and J. Eisert, Equilibration in low-dimensional quantum matrix models, JHEP 04 (2015) 166 [arXiv:1403.1392] [INSPIRE].
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].
Y. Kimura, Noncommutative gauge theory on fuzzy four sphere and matrix model, Nucl. Phys. B 637 (2002) 177 [hep-th/0204256] [INSPIRE].
H.C. Steinacker, One-loop stabilization of the fuzzy four-sphere via softly broken SUSY, JHEP 12 (2015) 115 [arXiv:1510.05779] [INSPIRE].
J. Maldacena and A. Milekhin, To gauge or not to gauge?, JHEP 04 (2018) 084 [arXiv:1802.00428] [INSPIRE].
E. Berkowitz, M. Hanada, E. Rinaldi and P. Vranas, Gauged and ungauged: a nonperturbative test, JHEP 06 (2018) 124 [arXiv:1802.02985] [INSPIRE].
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].
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].
S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461 [hep-th/0105006] [INSPIRE].
Y. Abe, Construction of fuzzy S 4, Phys. Rev. D 70 (2004) 126004 [hep-th/0406135] [INSPIRE].
A.P. Balachandran, S. Kurkcuoglu and S. Vaidya, Lectures on fuzzy and fuzzy SUSY physics, hep-th/0511114 [INSPIRE].
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].
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].
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].
S. Kurkcuoglu, Equivariant reduction of U(4) gauge theory over S 2F × S 2F and the emergent vortices, Phys. Rev. D 85 (2012) 105004 [arXiv:1201.0728] [INSPIRE].
S. Kurkcuoglu, New fuzzy extra dimensions from \( SU\left(\mathcal{N}\right) \) gauge theories, Phys. Rev. D 92 (2015) 025022 [arXiv:1504.02524] [INSPIRE].
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].
S. Kürkçüoğlu and G. Ünal, U (3) gauge theory on fuzzy extra dimensions, Phys. Rev. D 94 (2016) 036003 [arXiv:1607.00075] [INSPIRE].
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].
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].
E. Ott, Chaos in dynamical systems, Cambridge University Press, Cambridge U.K. (2002).
R. Hilborn, Chaos and nonlinear dynamics: an introduction for scientists and engineers, Oxford University Press, Oxford U.K. (2000).
S. Coleman, Aspects of symmetry: selected erice lectures, Cambridge University Press, Cambridge U.K. (1988).
R. Rajaraman, Solitons and instantons. An introduction to solitons and instantons in quatum field theory, Elsevier, The Netherlands (1982).
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: 1806.10524
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Coşkun, Ü.H., Kürkçüoğlu, S., Toga, G.C. et al. Chaos from equivariant fields on fuzzy S4. J. High Energ. Phys. 2018, 15 (2018). https://doi.org/10.1007/JHEP12(2018)015
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2018)015