Predictions for Gromov–Witten invariants of noncommutative resolutions

In this paper, we apply recent methods of localized GLSM’s 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 through complex structure deformations. 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 arise in examples of abelian GLSM’s describing branched double covers, in which the double cover structure arises nonperturbatively. This note will examine the GLSM for P7[2,2,2,2]P7[2,2,2,2], which realizes this phenomenon. Its Landau–Ginzburg point is a noncommutative resolution of a (singular) branched double cover of P3P3. Regardless of the complex structure of the large-radius P7[2,2,2,2]P7[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 through complex structure deformations.