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.