Finite symmetry groups of two-dimensional toroidal conformal field theories

Two-dimensional toroidal models are among the simplest and best studied conformal field theories and they are one of the main building blocks in many string and superstring compactifications. Despite their "simplicity", apparently different toroidal models are related by an intricate web of dualities, i.e. non-trivial equivalences among field theories.

We focus on discrete groups of symmetries that are generated by self-dualities, i.e. non-trivial equivalences of a model with itself. The symmetry action of a self-duality group on the states of the theory will be called a lift of that symmetry: there are in general more possible lift choices, and they generically imply an increment in the group order associated to the symmetry. Outside the mere study of the symmetries of a given toroidal theory, this feature, sometimes overlooked in literature, is for example relevant in orbifold construction. In this presentation, we will focus on the cyclic self-duality groups case. In particular, finding general results on the conditions that ensure the existence of an order-preserving lift for self-duality groups will be our main task. We will then apply our results to some physically relevant examples, notably we draw some interesting conclusion on the lifts of the cyclic symmetries of heterotic string theories on T⁴ (that are dual to sigma models on K3).