Orbifolds, Cohen-Macaulay modules and cluster categories

Abstract

We investigate certain Frobenius structures arising on categories Vect(X) of vector bundles over one-dimensional orbifolds. These we classify by coloured ane Dynkin diagrams ∆.

One can view these Frobenius categories as categories MFΓ(R, f ) of equivariant matrix factorizations of curve singularities (or more generally, categories of p-cycles). When the orbifold X is Fano, we establish a derived equivalence between the stable category MFΓ(R, f ) = CMΓ(R/f ) and the black part ∆b of the diagram ∆. We show that CMΓ(R/f ) carries a natural automorphism which is equivalent to the cluster auto-morphism on Db(∆b). This allows us to construct finite-type cluster categories as stable categories of Cohen-Macaulay modules equivariant with respect to a finite group.

Date of Award	29 Feb 2016
Original language	English
Awarding Institution	University of Bath
Supervisor	Alastair King (Supervisor)