Note on the motivic DT/PT correspondence and the motivic Flop formula

TL;DR

This paper establishes the motivic Donaldson-Thomas/Pandharipande-Thomas correspondence and the motivic flop formula for curve counting invariants on Calabi-Yau threefold DM stacks, using Hall algebra identities and motivic integration.

Contribution

It proves the motivic versions of the DT/PT correspondence and flop formula, extending previous results to the motivic setting for Calabi-Yau threefold DM stacks.

Findings

Motivic DT/PT correspondence proved for Calabi-Yau threefold DM stacks.

Motivic flop formula established for curve counting invariants.

Uses Bridgeland's Hall algebra identities and motivic integration techniques.

Abstract

We prove the motivic version of the DT/PT-correspondence in \cite{PT} and the motivic flop formula of the curve counting invariants in the derived category of smooth Calabi-Yau threefold DM stacks. The main method we use is Bridgeland's Hall algebra identities and the motivic integration map of Bridgeland and Joyce from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold DM stack $Y$ to the motivic quantum torus, which is a Poisson algebra homomorphism.