On Rational Points in CFT Moduli Spaces

TL;DR

This paper investigates the distribution of rational points with enhanced symmetry in CFT moduli spaces, focusing on bosonic and supersymmetric sigma-models, revealing complex behaviors and challenging assumptions about symmetry point density.

Contribution

It provides new insights into the distribution of enhanced symmetry points in CFT moduli spaces, especially in the context of bosonic and supersymmetric sigma-models, and challenges previous assumptions about their density.

Findings

Enhanced symmetry points are not necessarily dense despite vanishing twist gaps.

Discovered new features in the $S^1$ bosonic sigma-model.

Made observations on the emergence and disappearance of chiral currents under perturbations.

Abstract

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 $S^1$ target space in detail and uncover hitherto unknown features. We find for instance that the vanishing of the twist gap, though true for the $S^1$ 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.