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A Scattering Transform for Noncommutative Instantons

We give a detailed and mathematically rigorous analysis of the path integrals of chiral fermions supported on holomorphic curves on $T^* \mathbb{C}$ in a general noncommutative instanton background. It is shown that such path integrals can be interpreted as computing instanton analogs of matrix coefficients of monopole scattering matrices. Generalizing the known relation between monopole scattering matrices and $R$-matrices of (shifted) Yangians $\mathsf{Y}(\mathfrak{gl}_r)$, our formalism gives rise to a novel geometric method to calculate $R$-matrices of (shifted) affine Yangians $\mathsf{Y}(\widehat{\mathfrak{gl}}_r)$. This may also be viewed as an explicit description of double affine Grassmannian slices by $\infty \times \infty$ matrices, compatible with factorization. Our approach unifies a number of earlier results in the literature, and also leads to interesting new results and conjectures.