The Gromov-Witten Theory of Borcea-Voisin Orbifolds and Its Analytic Continuations

Abstract

In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry. However, their quantum theory has been poorly investigated. We study this in the context of the gauged linear sigma model, which in their case encompasses Gromov-Witten theory and its three companions (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, we calculate all four genus zero theories explicitly. Furthermore, we relate the I-functions of these theories by analytic continuation and symplectic transformation. In particular, the relation between the Gromov-Witten and FJRW theories can be viewed as an example of the Landau-Ginzburg/Calabi-Yau correspondence for complete intersections of toric varieties.

Gromov-Witten theory of toroidal orbifolds and GIT wall-crossing

“…The first genus-zero LG/CY correspondence was proved for the quintic threefold by Chiodo and Ruan ([10]). It has since been proven for several other classes of targets, including Calabi-Yau hypersurfaces in weighted projective spaces ( [8]), many classes of Calabi-Yau complete intersections in weighted projective spaces ( [14,15]), and some other examples ( [28,31]). (Acosta ([3]) also developed a similar correspondence for non-Calabi-Yau hypersurfaces in weighted projective spaces.)…”

“…(Acosta ([3]) also developed a similar correspondence for non-Calabi-Yau hypersurfaces in weighted projective spaces.) All of these used techniques quite different from those presented here; [8], [14], [28], and [31] used a direct computational method, and [15] (following previous work for hypersurfaces in [26]) used a reduction to the crepant transformation conjecture, previously established in the relevant cases in [17].…”