  In this paper, we apply recent methods of localized GLSMs to make predictions
for Gromov-Witten invariants of noncommutative resolutions, as defined by e.g.
Kontsevich, and use those predictions to examine the connectivity of the SCFT
moduli space. Noncommutative spaces, in the present sense, are defined by their
sheaves, their B-branes. Examples of abstract CFT's whose B-branes correspond
with those defining noncommutative spaces arose in examples of abelian GLSMs
describing branched double covers, in which the double cover structure arises
nonperturbatively. This note will examine the GLSM for P^7[2,2,2,2], which
realizes this phenomenon. Its Landau-Ginzburg point is a noncommutative
resolution of a (singular) branched double cover of P^3. Regardless of the
complex structure of the large-radius P^7[2,2,2,2], the Landau-Ginzburg point
is always a noncommutative resolution of a singular space, which begs the
question of whether the noncommutative resolution is connected in SCFT moduli
space by a complex structure deformation to a smooth branched double cover.
Using recent localization techniques, we make a prediction for the
Gromov-Witten invariants of the noncommutative resolution, and find that they
do not match those of a smooth branched double cover, telling us that these
abstract CFT's are not continuously connected to sigma models on smooth
branched double covers.
