The scattering-transform conjecture for noncommutative instantons

Let $\widetilde{M}(r)$ be the rank-$r$ moduli space, with matrices $(B_1,B_2,I,J)$, and let $\widetilde{M}(r)^{\mathbb{C}^\times}$ be the fixed locus of the $\mathbb{C}^\times$-action determined by a cocharacter $\mu$ satisfying $\mu_r\leq\dots\leq\mu_1=0$. Define an $r\times r$ matrix by

$$S\indices{_\alpha^\beta}(u):=\operatorname{Coeff}[w^{\mu_\beta}\zeta^{\mu_\alpha-1}]\left(G\indices{_\alpha^\beta}(\zeta;w)\big|_{\widetilde{M}(r)^{\mathbb{C}^\times}}\right).$$

Its entries are rational functions on $\mathbb{A}^1_u\times\widetilde{M}(r)^{\mathbb{C}^\times}$, with possible poles along $\det(u-B_2B_1)=0$. Scattering-transform conjecture. On each connected component of the fixed locus, the matrix $S\indices{_\alpha^\beta}(u)$ restricts to the monopole scattering matrix for $\widetilde{\mathscr{M}}^{ -\lambda}_{-\mu}$, equivalently the matrix description of the affine Grassmannian slice $\widetilde{\mathscr{W}}^{-\lambda}_{-\mu}$, under the stated isomorphism. This identifies the scattering transform of the noncommutative-instanton propagator with the monopole scattering data on each fixed-locus component; the source provides no resolution status, so the claim is recorded as open.