Integrability and non-Integrability in N=2 SCFTs and their Holographic Backgrounds

TL;DR

The paper addresses whether the worldsheet theory of strings in holographic backgrounds dual to $N=2$ SCFTs is classically integrable. It combines analytic techniques (Normal Variational Equation and Kovacic’s Liouvillian criteria) with numerical probes (Lyapunov exponents, Poincaré sections, Fourier spectra) applied to Gaiotto–Maldacena backgrounds, including the Sfetsos–Thompson and Maldacena–Nuñez solutions. The main finding is that generic GM backgrounds are non-integrable, while the Sfetsos–Thompson background remains an integrable exception due to a canonical transformation linking its Lax structure to that of the $AdS_5\times S^5$ string. The results illuminate how background data, such as flavor content and quiver structure, correlate with chaotic dynamics on the string worldsheet and have implications for understanding integrable subsectors in holographic $N=2$ theories.

Abstract

We show that the string worldsheet theory of Gaiotto-Maldacena holographic duals to N=2 superconformal field theories generically fails to be classically integrable. We demonstrate numerically that the dynamics of a winding string configuration possesses a non-vanishing Lyapunov exponent. Furthermore an analytic study of the Normal Variational Equation fails to yield a Liouvillian solution. An exception to the generic non-integrability of such backgrounds is provided by the non-Abelian T-dual of $AdS_5 \times S^5$. Here by virtue of the canonical transformation nature of the T-duality classical integrability is known to be present.

Figures (8)

• Figure 1: For various values of the parameter $\epsilon$ defining the deformation of the Sfetsos-Thompson solution given by eq. \ref{['vdef']} we show the evolution of the Lyapunov coefficient (whose $t\to \infty$ limit is the Lyapunov exponent). The initial conditions for our analysis are, $\chi(0)=0.5$, $\eta(0) =0$, $\dot{\chi}(0)=0.1$, $\dot{\eta}(0)=0.1$ with the parameter $E$ fixed such that the Hamiltonian vanishes.
• Figure 2: Charge density $\lambda(\eta)$ of a "one-kink" spacetime (left) and an "Uluru" spacetime (right) with the values used for the numerical analysis.
• Figure 3: Plots of example trajectories in the $\eta(\tau), \cos(\chi(\tau))$ plane in the one-kink space time. Left we have $E=0.05$ and on the right $E=5.0$
• Figure 4: Power spectra associated to the trajectories in the $\eta(\tau), \cos(\chi(\tau))$ plane displayed in fig. \ref{['fig:TrajectoryOneKink']} for the one-kink space time. Left we have $E=0.05$ and on the right $E=5.0$
• Figure 5: $\eta-P_\eta$ plane projections of the Poincare section at $\chi = 0$ for the one-kink spacetime. Clockwise from top left we vary the parameter $E = \{ 0.1, 0.5, 1, 2.5\}$. The plots fill an area bounded by maximal value of $P_\eta$ compatible with the constraint that $H=0$ indicated with a grey dashed line. The 100 different seed initial conditions that are numerically evolved to generate these sections are indicated by colour. For small values of $E$ we have a perturbation around an integrable Hamiltonian (for $E=0$ the Hamiltonian is trivial and vanishing) and one sees vestiges of KAM tori which as $E$ is increased dissolve away.