Theoretical studies on Integrable and Non-integrable strings in Gauge/String Duality

Abstract

In the present thesis, we plan to utilize the powerful analytical tool known as Kovacic’s Algorithm to determine the (non)-integrability of string sigma models within various supergravity backgrounds. The supergravity backgrounds we are considering are deformations of certain supergravity solutions. In addition to the analytical framework mentioned above, we use numerical techniques to identify two chaos indicators: Poincar´e sections and Lyapunov exponents, whenever necessary. It is important to emphasize that this constitutes a Hamiltonian approach, as contrary to the Lagrangian approach employed in the analytical computations. To begin with, we first investigate a novel class of Yang-Baxter deformed AdS4 × CP3 backgrounds, which exhibit non-chaotic dynamics for (super)strings propagating within them. To establish the non-chaotic nature of string σ-models over these deformed backgrounds, we explicitly use Kovacic’s algorithm. This analysis is complemented by numerical techniques, through which we probe the classical phase space of these (semi)classical strings and calculate various chaos indicators, such as Poincar´e sections and Lyapunov exponents. In our computations, using the standard Hamiltonian formulation, we explicitly checked that the shapes of the KAM tori remain undistorted as we increase the YB deformation parameters in all four cases. However, it is important to note that the nice foliations of the Poincar´e sections we observed do not necessarily guarantee that the system is nonchaotic across the entire range of parameter values, such as string energy (E) and various Yang-Baxter deformation parameters. To substantiate our claim, we examined additional sets of these parameters and found our results to be consistent. Furthermore, the Lyapunov exponent decays to zero over late time. These two findings lead us to conclude that the phase space of the propagating string does not exhibit chaotic behavior, thereby aligning with our analytical results. We find compatibility between the two approaches. However, our analysis does not ensure integrability; rather, it excludes the possibility of non-integrability for the given string embeddings.