Reflections in the mirror: Studying infinite Coxeter symmetries of GV-invariants

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa.