Dualities Arising from Borcea-Voisin Threefolds

In the early 1990s, Borcea-Voisin orbifolds were some of the earliest examples of Calabi-Yau threefolds shown to exhibit mirror symmetry, but at the quantum level this has been poorly understood. Here the enumerative geometry of this family is placed in the context of a gauged linear sigma model which encompasses the threefolds’ Gromov-Witten theory and three companion theories (FJRW theory and two mixed theories). For certain Borcea-Voisin orbifolds of Fermat type, all four genus zero theories are calculated explicitly. Furthermore, the I-functions of these theories are related by analytic continuation and symplectic transformation. In particular, it is shown that 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. For certain mirror families, the corresponding Picard-Fuchs systems are then derived and the I-functions are shown to solve them, thus demonstrating that the mirror symmetry of Borcea-Voisin orbifolds extends to the quantum level.PHDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138597/1/trygve_1.pd