Conjectural generating series for self-dual DT invariants of the \u007eA1 quiver with sign choices

Establish the proposed generating series identities for the motivic self-dual DT invariants of the self-dual \u007eA1 quiver under the three sign assignments specified by the authors. Concretely, prove that the series $\sum_{n=0}^\infty q^{n/2} \, \mathrm{DT}^{\mathrm{mot,sd}}_{(n,n)}(\tau)$ equals (i) $(1-q)^{1/2} / \bigl((1 - q^{1/2} \, \mathbb{L}^{-1/2})(1 - q^{1/2} \, \mathbb{L}^{1/2})\bigr)$ for $(+,++)$ and $(-,--)$, (ii) $\bigl((1+q^{1/2})/(1-q^{1/2})\bigr)^{1/2}$ for $(+,+-)$ and $(-,+-)$, and (iii) $(1-q)^{1/2}$ for $(+,--)$ and $(-,++)$.

Background

The authors apply their algorithm to compute motivic self-dual DT invariants for the self-dual \u007eA1 quiver with different sign assignments on vertices and arrows. Based on numerical evidence, they conjecture explicit closed-form generating functions depending on the sign data.

These identities, if proved, would provide strong structural insight into orthosymplectic DT invariants for quivers and connect them to simple rational functions of $q$ and the Lefschetz motive $\mathbb{L}$.