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

TL;DR

This paper investigates the Gromov-Witten theory of Borcea-Voisin orbifolds, explicitly computes genus zero theories for certain cases, and explores their relations via analytic continuation, advancing understanding of mirror symmetry and Landau-Ginzburg/Calabi-Yau correspondence.

Contribution

It provides explicit calculations of all four genus zero theories for specific Borcea-Voisin orbifolds and relates their I-functions through analytic continuation and symplectic transformation.

Findings

Explicit genus zero Gromov-Witten, FJRW, and mixed theories computed for Fermat type orbifolds.

I-functions related via analytic continuation and symplectic transformation.

Gromov-Witten and FJRW theories exemplify Landau-Ginzburg/Calabi-Yau correspondence.

Abstract

In the early 1990s, Borcea-Voisin orbifolds were some of the ear- liest 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 encom- passes 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 transfor- mation. 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.