Noncritical Dimensions for Critical String Theory: Life beyond the Calabi-Yau Frontier

TL;DR

This paper reviews a new framework for string compactification using noncritical manifolds of higher dimension, which can produce critical string vacua and relate to Calabi-Yau spaces, expanding the understanding of string theory landscapes.

Contribution

It introduces a class of higher-dimensional noncritical manifolds that generalize Calabi-Yau spaces and can generate critical string groundstates, offering new insights into string compactification and mirror symmetry.

Findings

Higher-dimensional noncritical manifolds can produce critical string spectra.

Some noncritical theories can explicitly construct Calabi-Yau manifolds.

The framework suggests modifications to Gepner's conjecture and questions about Kaluza-Klein compactification.

Abstract

A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension $2k+D_{crit}$, $k\geq 1$, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string groundstates. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi--Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi--Yau manifolds because it contains spaces which correspond to critical string vacua with no K\"ahler deformations, i.e. no antigenerations, hence providing mirrors of rigid Calabi--Yau manifolds. The constructions reviewed here lead to new insight into the relation between exactly solvable models and their mean field theories on the one hand and Calabi--Yau manifolds on the other, leading, for instance, to a modification of Gepner's conjecture. They also raise fundamental questions about the Kaluza--Klein concept of string compactification, in particular regarding the r\^{o}le played by the dimension of the internal theories.