Exceptional Mirror Symmetry and the String Worldsheet

The way strings see geometry is fundamentally different from the way that particles experience geometry. One the one hand, strings can detect features that are hard to grasp from a particle point of view due to their ability to wind around compact cycles. On the other hand, geometries that are different classically can lead to indistinguishable physics from the point of view of string theory. Mirror symmetry for type II string theories on Calabi-Yau manifolds is an instance of such a phenomenon which far-reaching mathematical and physical implications. Whereas many features can already be anticipated from a semi-classical perspective, it was only when the world-sheet perspective on mirror symmetry was sufficiently developed that the richness of this subject began to be uncovered.

Calabi-Yau manifolds are not the only types of geometries with close ties to string theory. Another class with much richer physics is given by manifolds with the exceptional holonomy groups G2 and Spin(7). Due to the absence of the algebraic geometry technics available for Calabi-Yau manifolds, these are much more challenging to construct and study. From the point of view of physics, the richness of these geometries can be expressed by noting that they preserve less supersymmetry.

Although there have been early indications that there is a phenomenon similar to mirror symmetry for G2 and Spin(7) manifolds, not much progress has been made due to the absence of large classes of examples and sufficiently general constructions. Recently, this topic has received renewed interest and concrete proposals for how to construct instances of mirror G2 and Spin(7) manifolds have been formulated. So far, these proposals have mostly studied from a space-time point of view and no detailed understanding of the world-sheet aspects has been developed.

Starting from the large set of conjectured instances of G2 and Spin(7) mirrors, the central aim of this project is to prove the equivalence of type II strings on these geometries from the world-sheet point of view. A first target of such an undertaking is the formulation of the world-sheet superconformal field theory (SCFT) in terms of minimal models. Such a formulation was a crucial step in the development of mirror symmetry for Calabi-Yau manifolds, and prior studies indicate that similar methods will be useful in the context of exceptional holonomy. Once such a description has been sufficiently developed, it is possible to derive the mathematical consequences of the statement of string dualities. For Calabi-Yau manifolds, this has famously led to surprising computational techniques, and we expect that similar methods can be developed in the context of exceptional holonomy manifolds. This is particular interesting in the light of the sparseness of mathematical results on exceptional holonomy manifolds that can be naively anticipated to generalise Calabi-Yau mirror symmetry.