The relativistic avatars of giant magnons and their S-matrix

Abstract

The motion of strings on symmetric space target spaces under lies the integrability of the AdS/CFT correspondence. Although these theories, whose excitations are giant magnons, are non-relativistic they are classically equivalent, via the Polhmeyer reduction, to a relativistic integrable field theory known as a symmetric space sine-Gordon theory. These theories can be formulated as integrable deformation s of gauged WZW models. In this work we consider the class of symmetric spaces \( \mathbb{C}{P^{n + 1}} \) and solve the corresponding generalized sine-Gordon theories at the quantum level by finding the exact spectrum of topological solitons, or kinks, and their S-matrix. The latter involves a trignometric solution of the Yang-Baxer equation which exhibits a quantum group symmetry with a tower of states that is bounded, unlike for magnons, as a result of the quantum group deformation parameter q being a root of unity. We test the S-matrix by taking the semi-classical limit and comparing with the time delays for the scattering of clas sical solitons. We argue that the internal \( \mathbb{C}{P^{n - 1}} \) moduli space of collective coordinates of the solitons in the classical theory can be interpreted as a q-deformed fuzzy space in the quantum theory. We analyse the n = 1 case separately and provide a further test of the S-matrix conjecture in this case by calculating the central charge of the UV CFT using the ther modynamic Bethe Ansatz.