Beta deformed sigma model and strong deformation coupling limit

We study the beta deformation of the superstring in $AdS_5\times S^5$ at all orders in the deformation parameter, using the pure spinor formalism. This is necessary to study the regime of strong deformation parameter, which in the field side is related to fishnet theories. We compare the pure spinor sigma model approach to the previously known supergravity description. We find a complete agreement. Moreover, the BRST structure of the worldsheet model provides a natural explanation of the peculiar features of the worldsheet model in the fishnet limit. In particular, we study the degeneracy of the sigma model Lagrangian. We show that the BRST structure is responsible for a particularly "tame" degeneration of the fishnet sigma-model.