Lecture series on categorification of Donaldson-Thomas theory
Prof. Yukinobu Toda
2024-07-03 16:30-17:30
MCM110
Speaker: Prof. Yukinobu Toda (Kavli IPMU, The University of Tokyo) Time:
16:00-17:00 June 19, 2024 (Wednesday)
10:30-11:30 June 26, 2024 (Wednesday)
16:30-17:30 July 3, 2024 (Wednesday)
Venue: MCM110
Title: Lecture series on categorification of Donaldson-Thomas theory
Outline:
The 1st talk
Title: Introduction to Donaldson-Thomas theory and BPS invariants
Abstract: The DT invariants count stable coherent sheaves on Calabi-Yau 3-folds which were introduced by Thomas around 1998. Later Joyce-Song gave a definition of generalized DT invariants which also count semistable sheaves. The generalized DT invariants are rational numbers, but Davison-Meinhardt proved that they are described in terms of integer valued invariants, called BPS invaraints. The BPS invariants play important roles in modern enumerative geometry. In this talk, I give an introduction of DT and BPS invariants, and explain motivationof categorifying them.
The 2nd talk
Title: Quasi-BPS categories for symmetric quivers with potential
Abstract: In this talk, I introduce the notion of quasi-BPS categories for symmetric quivers with potential. The theory of quivers with potential is regarded as a local model of a theory of Calabi-Yau 3-folds, and our quasi-BPS category is regarded as a local model of categorified BPS invariant. We give several results on quasi-BPS categories; categorical PBW theorem, categorical support lemma and relation with BPS invariants via topological K theory. This is a joint work with Tudor Padurariu.
The 3rd talk
Title: Quasi-BPS categories for K3 surfaces and Higgs bundles
Abstract: In this talk, based on the construction of quasi-BPS categories for symmetric quivers with potential, we introduce quasi-BPS categories for K3 surfaces and Higgs bundles. In the case of K3 surfaces, when the weight is coprime to the Mukai vector, we show that the quasi-BPS category is regarded as a categorical twisted crepant resolutions of singularities for singular symplectic moduli spaces, which do not admit crepant resolutions except O'Grady's 10-dimensional example. In the case of Higgs bundles, we propose conjectural symmetry between quasi-BPS categories motivated by classical limit of geometric Langlands conjecture, and show it in low rank case. This is a joint work with Tudor Padurariu.
