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

TL;DR

This paper explores a special class of Fano varieties that generalize Calabi--Yau manifolds, providing a new framework for understanding mirror symmetry and string vacua, including their spectra and relations to solvable models.

Contribution

It introduces a class of Fano manifolds that encode string ground states and demonstrates their role in deriving spectra and embedding Calabi--Yau manifolds, extending mirror symmetry concepts.

Findings

Massless spectra derived from special Fano varieties.

Embedding Calabi--Yau manifolds into higher-dimensional Fano spaces.

Application of Landau--Ginzburg/Calabi--Yau relation to this framework.

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 K\"ahler 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.