Quantum spin systems and supersymmetric gauge theories. Part I

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the N site $$ \mathfrak{sl} $$ sl 2X X X spin chain (for infinite dimensional complex spin representations), as well as the SLN Gaudin system, which reduces, in a limiting case, to that of the N-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

On Asymptotic Dynamical Regimes of Manakov N-soliton Trains in Adiabatic Approximation

We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the Manakov model. The evolution of Manakov N-soliton trains is described by the complex Toda chain (CTC) which is a completely integrable dynamical model. Calculating the eigenvalues of its Lax matrix allows us to determine the asymptotic velocity of each soliton. So we describe sets of soliton parameters that ensure one of the two main types of asymptotic regimes: the bound state regime (BSR) and the free asymptotic regime (FAR). In particular we find explicit description of special symmetric configurations of N solitons that ensure BSR and FAR. We find excellent matches between the trajectories of the solitons predicted by CTC with the ones calculated numerically from the Manakov system for wide classes of soliton parameters. This confirms the validity of our model

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by $$N \gg 1$$ N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature $$\beta ^{-1}$$ β - 1 . Given a fixed $${1\le m \ll N}$$ 1 ≤ m ≪ N , we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order $$\beta $$ β , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

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 [Formula: see text]. Hence the SL(3) theory should also do not have a Gaussian random matrix spectrum.