Pidstrigach–Tyurin reduction formula for PU(2) monopoles

Let $X$ be a four-manifold, let $E$ be a rank-two bundle, and let $z\in\mathbb A(X)$. Let $n_a$, $d_a$, $\overline{\operatorname{Red}}(W,E)$, $\overline V(z)$, $\overline W(x^{n_a-1})$, and $\mathbf L_{W,E_{-\ell},L_1}$ denote the quantities associated in the source to the $\operatorname{PU}(2)$ monopole compactification, its reducible strata, geometric representatives, and links. Pidstrigach–Tyurin reduction formula.

$$2^{n_a-1}D_X^{c_1(E)}(z)=\sum_{L_1\in\overline{\operatorname{Red}}(W,E)}\overline V(z)\cap\overline W(x^{n_a-1})\cap\mathbf L_{W,E_{-\ell},L_1},\qquad \deg z=2d_a,$$

and

$$0=\sum_{L_1\in\overline{\operatorname{Red}}(W,E)}\overline V(z)\cap\overline W(x^{n_a-1})\cap\mathbf L_{W,E_{-\ell},L_1},\qquad \deg z>2d_a.$$

Here $\ell$ is determined by the reduction $E_{-\ell}=L_1\oplus((\det E)\otimes L_1)$, with $\det E_{-\ell}=\det E$ and $c_2(E_{-\ell})=c_2(E)-\ell$. The formula is intended to express Donaldson invariants through pairings over links of Seiberg–Witten moduli spaces; analytical and compactification issues make the general assertion open.