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 firstproposed 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.

Provides an introduction of DT theory for both mathematicians and physicists Emphasizes both the foundation and computations in the study of DT theory Contains a mathematical theory of GopakumarVafa invariants, a new subject not available in other survey works

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.

Klappentext

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.

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.