New vortex-string worldsheet theories from supersymmetric localization

Abstract We use supersymmetric localization techniques to study the low-energy dynamics of BPS vortex-strings in four-dimensional N = 2 $$ \mathcal{N}=2 $$ theories. We focus on theories with SU(N c ) × U(1) gauge group and N f hypermultiplets, all in the fundamental representation of SU(N c ) but with general U(1) charges. Recently, we proposed a condition that determines whether the low-energy string dynamics is captured by a two-dimensional worldsheet theory that decouples from the bulk [1]. For strings for which this decoupling applies, we propose a prescription for extracting the two-sphere partition function of the string worldsheet theory from the four-ellipsoid partition function of the parent theory. We obtain a general formula for the worldsheet two-sphere partition function in terms of the parameters of the four-dimensional theory and identify N = 2 2 $$ \mathcal{N}=\left(2,2\right) $$ GLSMs that possess these partition functions in a large class of examples. In these examples, the weak coupling regime of the four-dimensional theory is mapped to the weak coupling regime of the worldsheet theory. In addition, we study the classical string zero-modes in flat space and obtain predictions for the worldsheet spectra, which agree with the low-energy spectra of the GLSMs obtained in the localization analysis. For N f = 2N c = 4, we discuss the map between string worldsheet theories under four-dimensional S-duality and use our prescription to study examples in which the weak coupling regime of the four-dimensional theory is mapped to the strong coupling regime of the worldsheet theory