Chaotic dynamics of the mass deformed ABJM model

We explore the chaotic dynamics of the mass-deformed Aharony-Bergman-Jafferis-Maldacena 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 2-spheres, which are described in terms of the Gomis–Rodriguez-Gomez–Van Raamsdonk–Verlinde 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 exponents. In particular, we focus on 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 ${\lambda }_{L}\propto \left(E/{N}^{2}{\right)}^{1/3}$ or ${\lambda }_{L}\propto \left(E/{N}^{2}-{\gamma }_{N}{\right)}^{1/3}$, where ${\gamma }_{N}\left(k,\mu \right)$ are constants determined in terms of the Chern-Simons coupling $k$, the mass $\mu $, and the matrix level $N$. Noting that the classical dynamics approximates the quantum theory only in the high-temperature regime, 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 Maldacena-Shenker-Stanford bound, ${\lambda }_{L}\le 2\pi T$, and below which it will eventually be not obeyed.