Mirror Symmetry and String Vacua from a Special Class of Fano Varieties

Abstract

Because of the existence of rigid Calabi--Yau manifolds, mirror symmetry cannot be understood as an operation on the space of manifolds with vanishing first Chern class. In this article I continue to investigate a particular type of Kähler manifolds with positive first Chern class which generalize Calabi--Yau manifolds in a natural way and which provide a framework for mirrors of rigid string vacua. This class is comprised of a special type of Fano manifolds which encode crucial information about ground states of the superstring. It is shown in particular that the massless spectra of $(2,2)$--supersymmetric vacua of central charge $\hat{c}=D_{crit}$ can be derived from special Fano varieties of complex dimension $(D_{crit}+2(Q-1))$, $Q>1$, and that in certain circumstances it is even possible to embed Calabi--Yau manifolds into such higher dimensional spaces. The constructions described here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and their corresponding Calabi--Yau manifolds on the other. It is furthermore shown that Witten's formulation of the Landau--Ginzburg/Calabi--Yau relation can be applied to the present framework as well.

54 pp, 2 epsf figures

Cited by in corpus (9)

• (0,2) Mirror Symmetry
• Unification of M- and F- Theory Calabi-Yau Fourfold Vacua
• Emergent spacetime from modular motives
• String Modular Phases in Calabi-Yau Families
• Fano hypersurfaces and Calabi-Yau supermanifolds
• Super Landau-Ginzburg mirrors and algebraic cycles
• Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-Hübsch-Krawitz duality
• Gradings of Lie algebras, magical spin geometries and matrix factorizations
• Scaling Behavior in String Theory