The refined Gopakumar–Vafa/Pandharipande–Thomas correspondence

Let $P_n(X,\beta)$ be the moduli space of Pandharipande–Thomas stable pairs, let $[P_n(X,\beta)]_{\mathrm{vir}}$ be the refined virtual class, and define the motivic stable-pair generating series

$$Z_{PT}^r=\sum_{n\in\mathbb{Z}}\sum_{\beta\in H_2(X,\mathbb{Z})}[P_n(X,\beta)]_{\mathrm{vir}}q^nQ^\beta.$$

For half-integers $j_L,j_R$, let $m_L$ and $m_R$ range from $-j_L$ to $j_L$ and from $-j_R$ to $j_R$, respectively, and let $N_{j_L,j_R}^\beta$ be the refined Gopakumar–Vafa invariants.

Refined Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

$$Z_{PT}^r=\prod_{j_L,j_R\in\frac12\mathbb{Z}}\prod_{m_L=-j_L}^{j_L}\prod_{m_R=-j_R}^{j_R}\prod_{m=1}^\infty\prod_{j=0}^{m-1}\left(1-(-q)^{m-2m_L}\mathbb{L}^{-m/2+1/2+j-m_R}\right)^{(-1)^{2j_L+2j_R}N_{j_L,j_R}^\beta}.$$

This is the refined motivic version of the GV/PT correspondence. It is stated conditionally on the orientation and perverse-sheaf construction above, and the source gives no resolution.