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In this paper we compute the motivic Donaldson--Thomas invariants for thequiver with one loop and any potential. As the presence of arbitrary potentialsrequires the full machinery of \hat(\mu)-equivariant motives, we give adetailed account of them. In particular, we will prove two results for themotivic vanishing cycle which might be of importance not only inDonaldson--Thomas theory.

Motivic DT-invariants for the one loop quiver with potential