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Abstract: I will start by reviewing gauge theories on various dimensions, especially on dimension 3 and 4. Then we complexify them on Calabi-Yau manifolds following Donaldson-Thomas' proposal of higher dimensional gauge theory. Specifically, on CY 4-folds, we will discuss a complexification of Donaldson theory in detail. The corresponding counting invariants are called DT4 invariants as they are the 4-fold analogue of DT invariants originally defined on CY 3-folds. Applications and examples will also be discussed.
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Abstract: TBA
Material: Lecture notes
Starts on: TBA
Abstract: The aim of the course is to define refined Vafa-Witten invariants of projective surfaces, and calculate them in some cases. This constitutes different projects joint with Yuuji Tanaka, Davesh Maulik and Amin Gholampour. 1. Virtual cycles and localisation in cohomology. 2. Virtual structure sheaves and localisation in K-theory. 3. Vafa-Witten invariants: stable case, semistable case, and refinement. 4. Degeneracy loci and Carlsson-Okounkov operators in cohomology and K-theory.
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Abstract: I will talk about sheaf theoretic definitions of Gopakumar-Vafa invariants for Calabi-Yau 3-folds and 4-folds. In the case of CY 3-folds, I will define GV invariants using perverse sheaves of vanishing cycles on moduli spaces of one dimensional stable sheaves. This definition is a modification of earlier definitions by Kiem-Li and Hosono-Saito-Takahashi. Then I will show that our GV invarinats agree with Pandharipande-Thomas invariants for local surfaces with irreducible curve classes. I will also give a counter-example to the Kiem-Li conjectures, where our invariants match the predicted answer. In the case of CY 4-folds, I will define genus zero GV invariants as Donaldson-Thomas invariants for CY 4-folds recently defined by Cao-Leung and Borisov-Joyce, and conjecture that they are related to genus zero Gromov-Witten invariants via multiple cover formula. This is a joint work with D. Maulik (for CY 3-folds), with Y. Cao and D. Maulik (for CY 4-folds).