Exact approaches on the string worldsheet

TL;DR

This review consolidates exact worldsheet approaches to string theory, centering on Green-Schwarz strings in curved AdS3 backgrounds with mixed RR/NSNS flux. It surveys the supercoset sigma-model framework and its classical integrability, analyzes the spectrum in plane-wave limits, and discusses how Wess-Zumino-type terms encode NSNS flux while preserving integrability. The text also addresses the limitations of coset constructions, the necessity of full GS action in complete gauge choices, and the construction of the worldsheet S-matrix—both perturbatively and via integrability—for AdS3xS3xT4. Together, these perspectives illuminate how integrability and holography interlock in low-dimensional AdS backgrounds and set the stage for quantitative spectral and S-matrix analyses. The synthesis highlights how flux mixtures, gauge choices, and coset vs GS formalisms shape our understanding of string dynamics on AdS3 geometries, with implications for AdS3/CFT2 and non-relativistic deformations.

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 behaviours, highlighting the connection with integrable $T\bar{T}$ 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.

Figures (9)

• Figure 1: The D1-D5-D5' brane configuration. A dash $-$ means that the brane is extended in that direction, while a dot $\cdot$ means that the brane is perpendicular to the direction, correspondingly to Neumann and Dirichlet boundary conditions respectively. For completeness, the tilde $\sim$ indicates that the brane can be smeared or delocalised in that direction. From the diagram it is apparent that the original $SO(1,9)$ symmetry is broken down to a $SO(4)_{2345}\times SO(4)_{6789}$ symmetry, where the subscript indicates the relevant directions. Note however that the last factor, $SO(4)_{6789}$ remains only unbroken at low energy, e.g. at the supergravity regime, when the compactified manifold in these directions is small. At larger energies the four dimensional manifold $T^4$ further breaks down the symmetry group to $SO(4)_{6789}\rightarrow U(1)^4$. Including the last row (adding an additional set of D5 branes) realises the $AdS_3\times S^3\times S^3\times S^1$-background.
• Figure 2: A summary of the string configurations construction when $L\rightarrow \infty$. In the figure $Y$ means "Yes" and $N$ means "No"
• Figure 3: Q-Strings for $Q=1,2,3,4,5$, centered at different values of $u\in \mathbb{R}$.
• Figure 4: We can interpret this as the time evolution of the system happening in two different cycles on the torus. In particular, the Euclidean partition function can be computed from two different theories (with Lorentzian signature): the original and the mirror theories.
• Figure 5: The geometry of radial quantisation: we let the coordinate $z\in\mathbb{C}$ parametrize our worldsheet. Circles of constant radius are thought of as circles of constant time, and $z=0$ corresponds to the infinite past, where asymptotic states are prepared.

Theorems & Definitions (1)

• Remark 4.1