Critical Strings from Noncritical Dimensions: A Framework for Mirrors of Rigid Vacau

TL;DR

This paper explores the role of certain complex manifolds with positive first Chern class in string theory, revealing their cohomological structure and implications for string modes beyond Calabi-Yau spectra.

Contribution

It introduces a framework for understanding noncritical dimensional manifolds in string theory, highlighting their cohomology and potential for additional string modes.

Findings

Higher dimensional manifolds lack a unique holomorphic form.

These manifolds exhibit extra modes beyond standard Calabi-Yau spectra.

Implications for string modes include relevance of the antisymmetric torsion field.

Abstract

The role in string theory of manifolds of complex dimension $D_{crit} + 2(Q-1)$ and positive first Chern class is described. In order to be useful for string theory, the first Chern class of these spaces has to satisfy a certain relation. Because of this condition the cohomology groups of such manifolds show a specific structure. A group that is particularly important is described by $(D_{crit} + Q-1, Q-1)$--forms because it is this group which contains the higher dimensional counterpart of the holomorphic $(D_{crit}, 0)$--form that figures so prominently in Calabi--Yau manifolds. It is shown that the higher dimensional manifolds do not, in general, have a unique counterpart of this holomorphic form of rank $D_{crit}$. It is also shown that these manifolds lead, in general, to a number of additional modes beyond the standard Calabi--Yau spectrum. This suggests that not only the dilaton but also the other massless string modes, such as the antisymmetric torsion field, might be relevant for a possible stringy interpretation.