Lecture notes on current-current deformations

These are pedagogical lecture notes discussing current-current deformations of 2-dimensional field theories. The deformations that are considered here are generated infinitesimally by bilinears of Noether currents corresponding to internal global symmetries of the "seed" theory. When the seed theory is conformal, these deformations are marginal and are often known as $J\bar J$-deformations. In this context, we review the criterion for marginal operators due to Chaudhuri and Schwartz. When the seed theory is an integrable $\sigma$-model (in the sense that it possesses a Lax connection), these deformations preserve the integrability. Here we review this fact by viewing the deformations as maps that leave the equations of motion and the Poisson brackets of the 2-dimensional $\sigma$-models invariant. The reinterpretation as undeformed theories with twisted boundary conditions is also discussed, as well as the effect of the deformation at the level of the S-matrix of the quantum theory. The finite (or integrated) form of the deformations is equivalent to sequences of T-duality--shift--T-duality transformations (TsT's), and here we review the $O(d,d)$-covariant formalism that is useful to describe them. The presentation starts with pedagogical examples of deformations of free massless scalars in 2 dimensions, and minimal prerequisites on conformal field theories or integrability are needed to understand later sections. Moreover, guided exercises are proposed to the reader. These notes were prepared for the Young Researchers Integrability School and Workshop (YRISW) held in Durham from 17 to 21 July 2023.