Chaos in Matrix Gauge Theories with Massive Deformations

Starting from an $SU(N)$ matrix quantum mechanics model with massive deformation terms and by introducing an ansatz configuration involving fuzzy four- and two-spheres with collective time dependence, we obtain a family of effective Hamiltonians, $H_n \,, (N = \frac{1}{6}(n+1)(n+2)(n+3))$ and examine their emerging chaotic dynamics. Through numerical work, we model the variation of the largest Lyapunov exponents as a function of the energy and find that they vary either as $\propto (E-(E_n)_F)^{1/4}$ or $\propto E^{1/4}$, where $(E_n)_F$ stand for the energies of the unstable fixed points of the phase space. We use our results to put upper bounds on the temperature above which the Lyapunov exponents comply with the Maldacena-Shenker-Stanford (MSS) bound, $2 \pi T$, and below which it will eventually be violated.Comment: 17 pages, 2 figures, 5 tables, Talk given by S. K\"{u}rk\c{c}\"{u}o\v{g}lu at the workshop on "Quantum Geometry, Field Theory and Gravity", Corfu Summer Institute 202

Chaotic Dynamics of the Mass Deformed ABJM Model

We explore the chaotic dynamics of the mass deformed ABJM model. To do so, we first perform a dimensional reduction of this model from $2+1$- to $0+1$-dimensions, considering that the fields are spatially uniform. Working in the 't Hooft limit and tracing over ansatz configurations involving fuzzy two spheres, which are described in terms of the GRVV matrices with collective time dependence, we obtain a family of reduced effective Lagrangians and demonstrate that they have chaotic dynamics by computing the associated Lyapunov spectrum. In particular, we analyze in detail, how the largest Lyapunov exponent, $\lambda_L$, changes as a function of $E/N^2$. Depending on the structure of the effective potentials, we find either $\lambda_L \propto (E/N^2)^{1/3}$ or $\lambda_L \propto (E/N^2 - \gamma_N)^{1/3}$, where $\gamma_N(k, \mu)$ is a constant determined in terms of the Chern-Simons coupling $k$, the mass $\mu$, and the matrix level $N$. Using our results, we investigate the temperature dependence of the largest Lyapunov exponents and give upper bounds on the temperature above which $\lambda_L$ values comply with the MSS bound, $\lambda_L \leq 2 \pi T$, and below which it will eventually be violated.Comment: 35 pages, 8 figure