Exceptional Mirror Symmetry and the String Worldsheet

Abstract

This thesis provides arguments to strengthen our understanding of mirror symmetry for manifolds with G2 holonomy, by providing worldsheet arguments to demonstrate the physical equivalence of topologically distinct geometries. In particular we investigate the worldsheet superconformal field theories corresponding to manifolds with G2 holonomy obtained by the quotient of the product of a Calabi-Yau threefold and a circle. The quotient acts on the Calabi-Yau as an antiholomorphic involution and on the circle by inversion. For such models, we argue that the Calabi-Yau mirror map implies a mirror map for the associated G2 varieties by examining how antiholomorphic involutions behave under Calabi-Yau mirror symmetry. The mirror geometries identified by the worldsheet CFT are consistent with earlier proposals for twisted connected sum G2 manifolds.

In order to be as self contained and pedagogical as possible, this thesis also provides a reasonably detailed review of Calabi-Yau manifolds and their associated CFTs. We also review details on the geometrical constructions of manifolds with G2 holonomy, in order to explain the geometrical equivalence of our CFT results.