Not Doomed to Fail

TL;DR

This paper clarifies that geometric symmetries in toroidal conformal field theories are not hindered by the mod two obstruction, ensuring their lifts to symmetries in the theory are always possible, which impacts symmetry analysis in K3 theories.

Contribution

It demonstrates, using elementary linear algebra, that geometric symmetries are unaffected by the mod two obstruction in toroidal conformal field theories.

Findings

Geometric symmetries are not doomed to fail due to the mod two condition.

Symmetry groups in K3 theories are unaffected by this obstruction.

Lifts of geometric symmetries do not require doubling the order.

Abstract

In their recent manuscript "An Uplifting Discussion of T-Duality", arXiv:1707.08888, J. Harvey and G. Moore have reevaluated a mod two condition appearing in asymmetric orbifold constructions as an obstruction to the description of certain symmetries of toroidal conformal field theories by means of automorphisms of the underlying charge lattice. The relevant "doomed to fail" condition determines whether or not such a lattice automorphism g may lift to a symmetry in the corresponding toroidal conformal field theory without introducing extra phases. If doomed to fail, then in some cases, the lift of g must have double the order of g. It is an interesting question, whether or not "geometric" symmetries are affected by these findings. In the present note, we answer this question in the negative, by means of elementary linear algebra: "geometric" symmetries of toroidal conformal field theories are not doomed to fail. Consequently, and in particular, the symmetry groups involved in symmetry surfing the moduli space of K3 theories do not differ from their lifts.