Abstract
Using methods of Hamiltonian dynamical systems, we show analytically that a dynamical system connected to the classical spinning string solution holographically dual to the principal Regge trajectory is non-integrable. The Regge trajectories themselves form an integrable island in the total phase space of the dynamical system. Our argument applies to any gravity background dual to confining field theories and we verify it explicitly in various supergravity backgrounds: Klebanov-Strassler, Maldacena-Núñez, Witten QCD and the AdS soliton. Having established non-integrability for this general class of supergravity backgrounds, we show explicitly by direct computation of the Poincaré sections and the largest Lyapunov exponent, that such strings have chaotic motion.
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G.F. Chew and S.C.Frautschi, Regge trajectories and the principle of maximum strength for strong interactions, Phys. Rev. Lett. 8 (1962) 41 [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].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
D.J. Gross and F. Wilczek, Asymptotically free gauge theories. 1, Phys. Rev. D 8 (1973) 3633 [INSPIRE].
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1133] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri and Y. Oz, Large-N field theories, string theory and gravity, Phys. Rept. 323 (2000) 183 [hep-th/9905111] [INSPIRE].
J. Sonnenschein, Stringy confining Wilson loops, hep-th/0009146 [INSPIRE].
Y. Kinar, E. Schreiber and J. Sonnenschein, \( Q\overline Q \) potential from strings in curved space-time: classical results, Nucl. Phys. B 566 (2000) 103 [hep-th/9811192] [INSPIRE].
L.A. Pando Zayas, J. Sonnenschein and D. Vaman, Regge trajectories revisited in the gauge/string correspondence, Nucl. Phys. B 682 (2004) 3 [hep-th/0311190] [INSPIRE].
F. Bigazzi, A.L. Cotrone, L. Martucci and L.A. Pando Zayas, Wilson loop, Regge trajectory and hadron masses in a Yang-Mills theory from semiclassical strings, Phys. Rev. D 71 (2005) 066002 [hep-th/0409205] [INSPIRE].
L.A. Pando Zayas and C.A. Terrero-Escalante, Chaos in the gauge/gravity correspondence, JHEP 09 (2010) 094 [arXiv:1007.0277] [INSPIRE].
P. Basu, D. Das and A. Ghosh, Integrability lost, Phys. Lett. B 699 (2011) 388 [arXiv:1103.4101] [INSPIRE].
P. Basu and L.A. Pando Zayas, Chaos rules out integrability of strings in AdS 5 × T 1,1, Phys. Lett. B 700 (2011) 243 [arXiv:1103.4107] [INSPIRE].
P. Basu and L.A. Pando Zayas, Analytic non-integrability in string theory, Phys. Rev. D 84 (2011) 046006 [arXiv:1105.2540] [INSPIRE].
A.T. Fomenko, Integrability and nonintegrability in geometry and mechanics, Kluwer Academic Publishers (1988).
J.J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, Birkhauser, Basel Switzerland (1999).
A. Goriely, Integrability and nonintegrability of dynamical systems, World Scientific (2001).
S.L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, Funct. Anal. Appl. 16 (1982) 181.
S.L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics II, Funct. Anal. Appl. 17 (1983) 617.
J.J. Morales-Ruiz and C. Simo, Picard-Vessiot theory and Ziglin’s theorem, J. Differ. Equations 107 (1994) 140.
J.J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrabitly of Hamiltonian systems I, Meth. Appl. Anal. 8 (2001) 33.
J.J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrabitly of Hamiltonian systems II, Meth. Appl. Anal. 8 (2001) 97.
J.J. Morales-Ruiz, J.-P. Ramis and C. Simò, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Normale Supérieure 40 (2007) 845.
J.J. Morales-Ruiz, Kovalevskaya, Liapounov, Painleve, Ziglin and the differential Galois theory, Regul. Chaotic Dyn. 5 (2000) 251.
J.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symb. Comput. 2 (1986) 3.
J.J. Morales-Ruiz and J.-P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Meth. Appl. Anal. 8 (2001) 113.
A.J Maciejewski and M. Szydlowski, Integrability and non-integrability of planar Hamiltonian systems of cosmological origin, in Proceedings of the 13th Workshop NEEDS’99: Nonlinear Evolution Equations and Dynamical Systems, Crete Greece, 20 June–30 July 1999, J. Nonlinear Math. Phys. 8 (2001) 200.
P.B. Acosta-Humanez, D. Blazquez-Sanz and C.A. Vargas Contreras, On Hamiltonian potentials with quartic polynomial normal variational equations, Nonlinear Stud. 16 (2009) 299 [arXiv:0809.0135].
R. Lakatos, On the nonintegrability of Hamiltonian systems with two degree of freedom with homogenous potential, Nonlinear J. 1 (1999) 64.
S.L. Ziglin, An analytic proof of the nonintegrability of the ABC-flow for A = B = C, Funct. Anal. Appl. 37 (2003) 225.
P. Acosta-Humanez and D. Blazquez-Sanz, Non-integrability of some Hamiltonians with rational potentials, Discr. Cont. Dyn. Syst. B 10 (2008) 265[math-ph/0610010].
E. Ott, Chaos in dynamical systems, second edition, Cambridge University Press, Cambridge U.K. (2002).
R.C. Hilborn, Chaos and nonlinear dynamics: an introduction for scientists and engineers, second edition, Oxford University Press, Oxford U.K. (2000).
J.C. Sprott, Chaos and time-series analysis, Oxford University Press, Oxford U.K. (2003).
I.R. Klebanov and M.J. Strassler, Supergravity and a confining gauge theory: duality cascades and χSB-resolution of naked singularities, JHEP 08 (2000) 052 [hep-th/0007191] [INSPIRE].
P. Candelas and X.C. de la Ossa, Comments on conifolds, Nucl. Phys. B 342 (1990) 246 [INSPIRE].
J.M. Maldacena and C. Núñez, Towards the large-N limit of pure N = 1 super Yang-Mills, Phys. Rev. Lett. 86 (2001) 588 [hep-th/0008001] [INSPIRE].
A.H. Chamseddine and M.S. Volkov, NonAbelian BPS monopoles in N = 4 gauged supergravity, Phys. Rev. Lett. 79 (1997) 3343 [hep-th/9707176] [INSPIRE].
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ArXiv ePrint: 1201.5634
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Basu, P., Das, D., Ghosh, A. et al. Chaos around holographic Regge trajectories. J. High Energ. Phys. 2012, 77 (2012). https://doi.org/10.1007/JHEP05(2012)077
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DOI: https://doi.org/10.1007/JHEP05(2012)077