Exact approaches on the string worldsheet

Abstract

We review different exact approaches to string theory. In the context of the Green-Schwarz superstring, we discuss the action in curved backgrounds and its supercoset formulation, with particular attention to superstring backgrounds of the AdS3 type supported by both Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. This is the basis for the discussion of classical integrability, of worldsheet-scattering factorisation in the uniform lightcone gauge, and eventually of the string spectrum through the mirror thermodynamic Bethe ansatz, which for AdS3​ backgrounds was only derived and analysed very recently. We then illustrate some aspects of the Ramond-Neveu-Schwarz string, and introduce the formalism of Berkovits-Vafa-Witten, which has seen very recent applications to AdS3 physics, which we also briefly review. Finally, we present the relation between M-theory in the discrete lightcone quantisation and decoupling limits of string theory that exhibit non-relativistic behaviours, highlighting the connection with integrable T-Tbar deformations, as well as the relation between spin-matrix theory and Landau-Lifshitz models. This review is based on lectures given at the Young Researchers Integrability School and Workshop 2022 "Taming the string worldsheet" at NORDITA, Stockholm.