Recent Progress on the Donaldson-Thomas Theory: Wall-Crossing and Refined Invariants

Yukinobu Toda - Collection Yellow Sale 2023

104 pages, parution le 15/12/2021

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Résumé

This book is an exposition of recent progress on the Donaldson-Thomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on Calabi-Yau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the Gromov-Witten/Donaldson-Thomas conjecture on curve-counting theories, wall-crossing in derived categories with respect to Bridgeland stability conditions, BPS state counting in string theory, and others.
Recently, a deeper structure of the moduli spaces of coherent sheaves on Calabi-Yau 3-folds was found through derived algebraic geometry. These moduli spaces admit shifted symplectic structures and the associated d-critical structures, which lead to refined versions of DT invariants such as cohomological DT invariants. The idea of cohomological DT invariants led to a mathematical definition of the Gopakumar-Vafa invariant, which was first proposed by Gopakumar-Vafa in 1998, but its precise mathematical definition has not been available until recently.
This book surveys the recent progress on DT invariants and related topics, with a focus on applications to curve-counting theories.
1Donaldson-Thomas invariants on Calabi-Yau 3-folds.- 2Generalized Donaldson-Thomas invariants.- 3Donaldson-Thomas invariants for quivers with super-potentials.- 4Donaldson-Thomas invariants for Bridgeland semistable objects.- 5Wall-crossing formulas of Donaldson-Thomas invariants.- 6Cohomological Donaldson-Thomas invariants.- 7Gopakumar-Vafa invariants.- 8Some future directions.
The author is currently Professor and Principal investigator at Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo. He was an invited speaker at the ICM 2014.