Project description
Novel mathematical approaches enhance our description of string theory
String theory attempts to unify currently incompatible theories of quantum mechanics and general relativity. Matter and force particles are described as a collection of 1D strings rather than 0D points, vibrating in a 10-dimensional space – 9 dimensions of space and 1 of time. The additional 6D space beyond our classic 4D one is described by so-called Calabi–Yau manifolds. There are many, perhaps infinite, possible Calabi–Yau threefolds (three complex dimensions), confounded by mirror symmetry in which some of these may look different geometrically but are essentially equivalent in the context of string theory. The EU-funded MMiMMa project is delving deep into Calabi–Yau threefolds and mirror symmetry for new insight into string theory.
Objective
Geometrically, this proposal is concerned primarily with Calabi--Yau threefolds, their (local) classification, their homological properties, various associated structures such as stability conditions and Frobenius manifolds, and the resulting predictions across mirror symmetry. Our approach to these problems is through noncommutative algebra, and necessarily so. We will use techniques from contraction algebras and noncommutative resolutions to classify, using both theoretical and constructive methods, and in the process verify an amended version of a string theory prediction. We will use this to push forward curve-counting and derived category consequences and obstructions, and will work towards building a full database of 3-fold flops. On a parallel track, we will treat fundamental problems in noncommutative resolutions and their variants, and approach some of the founding conjectures in the area. We will tackle problems such as existence of MMAs through to more specific problems such as faithful actions and K(pi,1) through stability manifolds and tilting theory on preprojective algebras. We will furthermore merge all this into an emerging theory of Frobenius manifolds, SKMS, and schobers, and through this expand on recent work constructing mirrors to various flopping contractions.
Fields of science
Programme(s)
Funding Scheme
ERC-COG - Consolidator GrantHost institution
G12 8QQ Glasgow
United Kingdom