Title:

Fundamental group actions on derived categories

The study of the gauged linear sigma model in physics has led to a prediction that the fundamental groupoid of a space of physically meaningful parameters (the FI parameters) acts on the derived categories of certain Calabi{Yau varieties. These varieties occur as GIT quotients of a linear space by a torus action. The autoequivalences of the derived category corresponding to some \large radius" loops in the parameter space are well understood and are socalled \window shifts". These arise naturally from the representation theory and we can try to use them to construct the conjectural representation. This has been carried out successfully for certain toric examples by Donovan and Segal in [18] and for all socalled \quasisymmetric" examples by Halpern Leistner and Sam in [24]. In both cases, the authors rely on the existence of special con gurations of line bundles called \magic windows" (introduced in [18]) to prove relations between the various window shifts. In this thesis, we move beyond these examples and construct a representation of the fundamental groupoid on two basepoints of an open subset of the FI parameter space whenever this space is 2dimensional. This relies on a generalisation of windows called \fractional windows" which were introduced by HalpernLeistner and Shipman in [25]. Moreover, we describe several examples where we can extend this representation over the whole parameter space. When the dimension of this space becomes larger, constructing the representation becomes more complicated. Nonetheless, we construct such a representation in a new example whose parameter space is 3dimensional using the Lefschetz hyperplane theorem. We also discuss an approach to the same problem using nite covers of the parameter space (based on [18]). Finally, we recall a conjecture of Aspinwall, Plesser and Wang [4] about how to construct a representation more generally. This leads us to conjecture a relationship between some intersection multiplicities and semiorthogonal decompositions of derived categories and we prove that this relationship is at least wellde ned.
