Non-Abelian T-dualities in Two Dimensional $(0,2)$ Gauged Linear Sigma Models

Two dimensional gauged linear sigma models(GLSMs) with $(0,2)$ supersymmetry and $U(1)$ gauge group possesing global symmetries are considered. For the case obtained as a reduction from the $(2,2)$ supersymmetric GLSM, we find the Abelian T-dual, comparing with previous studies. Then, the Abelian T-dual model of the pure $(0,2)$ theory is found. Instanton corrections are also discussed in both situations. In the cases under study we explore the vacua for the scalar potential and we analyse the target space geometry of the dual model. A model with gauge symmetry $U(1)\times U(1)$ is also discussed as precursor of an interesting example with non-Abelian global symmetry. Non-Abelian T-dualization of $U(1)$ $(0,2)$ $2d$ GLSMs is implemented for models which arise as a reduction from $(2,2)$ supersymmetry; we study a concrete model with $U(1)$ gauge symmetry and $SU(2)$ global symmetry. It is shown that for a positive definite scalar potential, the dual vacua to $\mathbb{P}^1$ constitutes a disk. Instanton corrections to the superpotential are obtained and shown to be encoded in a shifting of the holomorphic function ${E}$. We conclude by analyzing an example with $SU(2)\times SU(2)$ global symmetry and we obtain that the space of dual vacua to $\mathbb{P}^1\times \mathbb{P}^1$ consists of two copies of the disk, also for the case of positive definite potential. Here we are able to fully integrate the equations of motion leading to non-Abelian duality, improving with respect to previous $(2,2)$ studies.