Calabi-Yau categories and quivers with superpotential
Keywords: McKay Quivers
Derived Equivalences
Koszul Functor
Tilting
Quivers with Superpotential
Calabi-Yau Categories

This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers.
A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds.
One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.
Let π : V &rightarrow; X be an anti-semiample vector bundle over a smooth projective variety X, i.e., SkVV is globally generated for k » 0. Given a full exceptional sequence E on Db(Coh (X)), under some cohomological vanishing conditions, we construct a dg-quiver QE in terms of the dual exceptional sequence of E such that Db(Coh (V)) ≅ Dbfg(Rep(QE )). Moreover, this equivalence restricts to an equivalence between Dbcs(Coh (V)), the full subcategory containing complexes of compact support, and Dbfd(Rep(QE )), the full subcategory containing complexes with finite dimensional cohomology. If V is non-compact Calabi- Yau, we show that QE is equipped with a superpotential Φ, i.e., the differential on QE is determined by the "function" Φ. In this case, the triangulated categories Dbcs(Coh (V)) and Dbfd(Rep(QE )) are both Calabi-Yau.
We can also construct derived equivalences equivariantly. Suppose a finite group G acts on X and this action lifts to V , endowing π : V &rightarrow; X the structure of an equivariant vector bundle. Suppose further that each object in the exceptional sequence E is equipped with a G-linearization. Then we can construct a quotient dg-quiver QE/G from QE , generalizing the construction of the McKay quiver, such that Db(Coh G(V )) ≅ Db(Repfg(QE/G)). If V is non-compact Calabi-Yau equivariantly, then QE/G is also equipped with a superpotential.
We also give a product construction for derived equivalences. Suppose we have vector bundles πV : V &rightarrow; X and πW : W &rightarrow; Y , with full exceptional sequences E on Db(Coh (V )) (resp. F on Db(Coh (W))), then we can construct a product dg-quiver QE x QF such that Db(Coh (V x W)) ≅ Db(Repfg(QE x QF)). If both V and W are Calabi-Yau, then QE x QF is also equipped with a superpotential. Using these constructions, we can produce a lot of beautiful pictures of quivers with superpotential derived equivalent to the total spaces of vector bundles which are Calabi-Yau. Examples include TVℙ2 , Kℙn, and Oℙ2 (-1) ⊕ Oℙ2 (-2) etc.
Finally, we try to connect quivers with superpotential to the recent work by Pantev, Toën, Vaquiée and Vezzosi [58] and Ben-Bassat, Brav, Bussi and Joyce [4] on shifted symplectic structures. We outline a strategy of proof for the existence of shifted symplectic structures in a standard 'Darboux form' on the derived moduli stack of representations of quivers with superpotential.