Integrability and Spectral Form Factor in Chern-Simons Formulation

We study the integrability from the spectral form factor in the Chern-Simons formulation. The effective action in the higher spin sector was not derived so far. Therefore, we begin from the SL(3) Chern-Simons higher spin theory. Then the dimensional reduction in this Chern-Simons theory gives the SL(3) reparametrization invariant Schwarzian theory, which is the boundary theory of an interacting theory between the spin-2 and spin-3 fields at the infrared or massless limit. We show that the Lorentzian SL(3) Schwarzian theory is dual to the integrable model, SL(3) open Toda chain theory. Finally, we demonstrate the application of open Toda chain theory from the SL(2) case. The numerical result shows that the spectral form factor loses the dip-ramp-plateau behavior, consistent with integrability. The spectrum is not a Gaussian random matrix spectrum. We also give an exact solution of the spectral form factor for the SL(3) theory. This solution provides a similar form to the SL(2) case for $\beta\neq 0$. Hence the SL(3) theory should also do not have a Gaussian random matrix spectrum.