On rational points in CFT moduli spaces

Motivated by the search for rational points in moduli spaces of two-dimensional conformal field theories, we investigate how points with enhanced symmetry algebras are distributed there. We first study the bosonic sigma-model with S1 target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the S1 example, does not automatically follow from enhanced symmetry points being dense in the moduli space. We then explore the supersymmetric sigma-model on K3 by perturbing away from the torus orbifold locus. Though we do not reach a definite conclusion on the distribution of enhanced symmetry points in the K3 moduli space, we make several observations on how chiral currents can emerge and disappear under conformal perturbation theory.