An introduction to motivic Hall algebras

We give an introduction to Joyceʼs construction of the motivic Hall algebra of coherent sheaves on a variety M. When M is a Calabi–Yau threefold we define a semi-classical integration map from a Poisson subalgebra of this Hall algebra to the ring of functions on a symplectic torus. This material will be used in Bridgeland (2011) [3] to prove some basic properties of Donaldson–Thomas curve-counting invariants on Calabi–Yau threefolds.