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.

Autorentext
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.

Zusammenfassung
This book is an exposition of recent progress on the DonaldsonThomas (DT) theory. The DT invariant was introduced by R. Thomas in 1998 as a virtual counting of stable coherent sheaves on CalabiYau 3-folds. Later, it turned out that the DT invariants have many interesting properties and appear in several contexts such as the GromovWitten/DonaldsonThomas 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 CalabiYau 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 GopakumarVafa invariant, which was first proposed by GopakumarVafa 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.

Inhalt
1DonaldsonThomas invariants on CalabiYau 3-folds.- 2Generalized DonaldsonThomas invariants.- 3DonaldsonThomas invariants for quivers with super-potentials.- 4DonaldsonThomas invariants for Bridgeland semistable objects.- 5Wall-crossing formulas of DonaldsonThomas invariants.- 6Cohomological DonaldsonThomas invariants.- 7GopakumarVafa invariants.- 8Some future directions.