Topological Gauging and Double Current Deformations

We study solvable deformations of two-dimensional quantum field theories driven by a bilinear operator constructed from a pair of conserved $U(1)$ currents $J^a$. We propose a quantum formulation of these deformations, based on the gauging of the corresponding symmetries in a path integral. This formalism leads to an exact dressing of the $S$-matrix of the system, similarly as what happens in the case of a $\mathrm{T}\overline{\mathrm{T}}$ deformation. For conformal theories the deformations under study are expected to be exactly marginal. Still, a peculiar situation might arise when the conserved currents $J^a$ are not well-defined local operators in the original theory. A simple example of this kind of system is provided by rotation currents in a theory of multiple free, massless, non-compact bosons. We verify that, somewhat unexpectedly, such a theory is indeed still conformal after deformation and that it coincides with a TsT transformation of the original system. We then extend our formalism to the case in which the conserved currents are non-Abelian and point out its connection with Deformed T-dual Models and homogeneous Yang-Baxter deformations. In this case as well the deformation is based on a gauging of the symmetries involved and it turns out to be non-trivial only if the symmetry group admits a non-trivial central extension. Finally we apply what we learned by relating the $\mathrm{T}\overline{\mathrm{T}}$ deformation to the central extension of the two-dimensional Poincar\'{e} algebra.