FJRW theory is a formulation of physical Landau-Ginzburg models with a rich algebraic structure, rooted in enumerative geometry. As a consequence of a major physical conjecture, called the Landau-Ginzburg Calabi-Yau correspondence, several birational morphisms of Calabi-Yau orbifolds should correspond to isomorphisms in FJRW theory. In this paper it is shown that not only does this claim prove to be the case, but is a special case of a wider FJRW isomorphism theorem, which in turn allows for a proof of mirror symmetry for a new class of cases in the Landau-Ginzburg setting. We also obtain several interesting geometric applications regarding the Chen-Ruan cohomology of certain Calabi-Yau orbifolds.
In @cite_0 , the last author has considered exactly the form of mirror symmetry we propose here with the restriction that the defining polynomials must be Fermat type. In fact, he was able to show that for the mirror pairs we consider here, there is a mirror map relating the FJRW invariants of the A--model to the Picard--Fuchs equations of the B--model. In @cite_33 he also gave an LG CY correspondence relating the FJRW invariants of the pair @math to the corresponding Gromov--Witten invariants of the corresponding Borcea--Voisin orbifold. Although these results are broader in scope, the restriction to Fermat type polynomials is significant, reducing the number of weight systems from which one can select a K3 surface to 10 (from the 48 mentioned above). Furthermore, there is no general method of proof for a state space isomorphism provided there. However, we expect results regarding the FJRW invariants, Picard--Fuchs equations and GW invariants to hold in general, and the state space isomorphism we establish here is the first step to such results. This will be the topic of future work.
@cite_0 @cite_33
2265209761 2265209761
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. 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.