The central-charge threshold for wall-crossing in two-variable open FJRW theory

Abstract

For the genus-zero open FJRW theory of Gross, Kelly and Tessler attached to a two-variable Fermat polynomial, we prove, as they predicted, that the primary invariants do not depend on the choice of symmetric canonical boundary conditions exactly for the simple singularities, of types $A$, $D_4$, $E_6$ and $E_8$, that is, if and only if the central charge is less than one. The proof bounds the critical graphs, which carry the jumps, by the central charge and passes to the invariants through the wall-crossing group. Whenever a critical graph has one boundary marking of each type, this group acts simply transitively only after the vector field of the anti-diagonal torus action is adjoined. Beyond the simple range, dependence first appears with two interior insertions when both exponents are at least four and with three otherwise, while one insertion of descendent degree one produces wall-crossing for every pair. At central charge one every primary wall generator is a multiple of the torus vector field. A B-model analogue, for open Saito potentials, is due to Maher.