Equivariant Enumerative Geometry and Donaldson-Thomas Theory

My research is on Equivariant Enumerative Geometry of moduli spaces of sheaves, in particular toric and/or Calabi-Yau varieties, in dimension 3 and 4. In particular, I focus on studying generating series of refined invariants coming from String Theory, in particular K-theoretic invariants à la Nekrasov-Okounkov. In my thesis I dealt with several topics and problems in this field. Chapter1: I discussed the relationship between Gromov-Witten/Pandharipande-Thomas invariants of local curves, (re)proving their conjectural relation using localization methods, providing new methods to compute refined invariants as well. Chapter2: I discussed the higher rank version of K-theoretic invariants on toric threefolds (in physical terms, these are counting BPS invariants of D0-D6 branes). This solves open conjectures of Awata-Kanno and Szabo. Chapter3: Donaldson-Thomas theory has been recently extended to Calabi-Yau 4-folds by the work of Cao-Leung and Borisov-Joyce. Motivated by the K-theoretic refinement of their invariants appeared in the physics literature, we conjectured a correspondence between Donaldson-Thomas and Pandharipande-Thomas invariants. We show that this correspondence contains (by taking suitable limits) the classical DT/PT correspondence for threefolds (and its K-theoretic analogue) and a DT/PT correspondence with tautological invariants on Calabi-Yau 4-folds. Chapter4: Gromov-Witten invariants of Calabi-Yau 4-folds can be expressed in terms of Gopakumar-Vafa type invariants by the work of Klemm-Pandharipande. Cao-Maulik-Toda gave a sheaf-theoretical interpretation of these GV invariants using stable pair invariants. Their checks are mainly limited to irreducible classes where the geometry is explicit. In this work, we develop techniques to compute such invariants in the case of local-surfaces