A Scattering Transform for Noncommutative Instantons

TL;DR

This paper develops a geometric framework for calculating R-matrices of affine Yangians using path integrals of chiral fermions in noncommutative instanton backgrounds, unifying and extending previous results.

Contribution

It introduces a novel geometric method to compute R-matrices of shifted affine Yangians via path integrals, linking instanton theory with algebraic structures.

Findings

Provides a new geometric interpretation of R-matrices.

Unifies previous results in instanton and Yangian theories.

Suggests new conjectures relating double affine Grassmannian slices to matrix models.

Abstract

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.