The absence of poles conjecture for motivic Donaldson–Thomas invariants

Let $\Gamma$ be the lattice of a $3d$ Calabi–Yau category with a stability condition, let $V$ be a strict sector, and let $A_{V,q}$ be the corresponding element of the completed quantum torus. Write

$$C_0(V)=C_0(V,Z,Q),\qquad D_q^+:={\bf Z}[q^{\pm 1/2}].$$

Absence of poles conjecture. For any $3d$ Calabi–Yau category with a stability condition and any strict sector $V$, the automorphism $x\mapsto A_{V,q}xA_{V,q}^{-1}$ preserves the subring

$$\prod_{\gamma\in C_0(V)\cap \Gamma}D_q^+\hat{e}_{\gamma}.$$

The conjecture asserts that the quantum wall-crossing automorphism has no poles at roots of unity; it is still open in general, although partial results confirm it.