Recent Progress on the DonaldsonThomas Theory

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.