Recent Progress on the Donaldson-Thomas Theory

Wall-Crossing and Refined Invariants
1. Auflage 2021
ISBN: 978-981-16-7837-0
Verlag: Springer Nature Singapore

Buch, Englisch, Band 43, 104 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 184 g

Reihe: SpringerBriefs in Mathematical Physics

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 firstproposed 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.

Zielgruppe

Research

Autoren/Hrsg.

Toda, Yukinobu

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie
• Mathematik | Informatik Mathematik Algebra Homologische Algebra
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie

Inhaltsverzeichnis

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.

Über Autor(innen)

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.