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Title: Fundamental group actions on derived categories
Author: Kite, Alexandre
ISNI:       0000 0004 7970 0893
Awarding Body: King's College London
Current Institution: King's College London (University of London)
Date of Award: 2019
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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 auto-equivalences of the derived category corresponding to some \large radius" loops in the parameter space are well understood and are so-called \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 so-called \quasi-symmetric" 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 2-dimensional. This relies on a generalisation of windows called \fractional windows" which were introduced by Halpern-Leistner 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 3-dimensional 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 semi-orthogonal decompositions of derived categories and we prove that this relationship is at least well-de ned.
Supervisor: Rietsch, Konstanze Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available