TL;DR
This paper introduces Joyce's motivic Hall algebra construction for coherent sheaves on a variety, focusing on Calabi-Yau threefolds, and discusses a semi-classical integration map relevant to Donaldson-Thomas invariants.
Contribution
It provides an accessible introduction to motivic Hall algebras and defines a semi-classical integration map for Calabi-Yau threefolds, linking algebraic structures to enumerative invariants.
Findings
Construction of motivic Hall algebra for coherent sheaves
Definition of semi-classical integration map on Calabi-Yau threefolds
Application to Donaldson-Thomas invariants
Abstract
We give an introduction to Joyce's 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 arxiv:1002.4374 to prove some basic properties of Donaldson-Thomas curve-counting invariants on Calabi-Yau threefolds.
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