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 AdS 3 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 AdS 3 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 AdS 3 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 behaviors, highlighting the connection with integrable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mrow> <mml:mover> <mml:mi>T</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> 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.