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A Scattering Transform for Noncommutative Instantons

Spencer Tamagni

TL;DR

The work constructs a rigorous bridge between monopole scattering and noncommutative instantons, embedding both in a unified framework tied to infinite Grassmannians and affine Yangians. It introduces instanton scattering matrices $\mathbb{S}(u)$ as semiclassical $R$-matrices whose matrix elements are global functions on instanton moduli, obtained from Ext^1 data and Plücker coordinates, and links them to KP tau-functions. A top-down string/M-theory perspective clarifies the origin of these matrices as brane intersections and Chern-Simons defect data, while a bottom-up fermionic path-integral approach yields explicit determinant formulas and Miura-factorizations. The paper further argues that monopole data can be recovered from the instanton matrices, offering a new nonperturbative route to the $R$-matrix theory of shifted affine Yangians and a geometric realization of double affine Grassmannian slices. Together, these results provide new tools and conjectures for noncommutative gauge theories, integrable systems, and geometric representation theory, with broad implications for the interplay between physics and advanced algebraic structures.

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.

A Scattering Transform for Noncommutative Instantons

TL;DR

The work constructs a rigorous bridge between monopole scattering and noncommutative instantons, embedding both in a unified framework tied to infinite Grassmannians and affine Yangians. It introduces instanton scattering matrices as semiclassical -matrices whose matrix elements are global functions on instanton moduli, obtained from Ext^1 data and Plücker coordinates, and links them to KP tau-functions. A top-down string/M-theory perspective clarifies the origin of these matrices as brane intersections and Chern-Simons defect data, while a bottom-up fermionic path-integral approach yields explicit determinant formulas and Miura-factorizations. The paper further argues that monopole data can be recovered from the instanton matrices, offering a new nonperturbative route to the -matrix theory of shifted affine Yangians and a geometric realization of double affine Grassmannian slices. Together, these results provide new tools and conjectures for noncommutative gauge theories, integrable systems, and geometric representation theory, with broad implications for the interplay between physics and advanced algebraic structures.

Abstract

We give a detailed and mathematically rigorous analysis of the path integrals of chiral fermions supported on holomorphic curves on 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 -matrices of (shifted) Yangians , our formalism gives rise to a novel geometric method to calculate -matrices of (shifted) affine Yangians . This may also be viewed as an explicit description of double affine Grassmannian slices by matrices, compatible with factorization. Our approach unifies a number of earlier results in the literature, and also leads to interesting new results and conjectures.
Paper Structure (102 sections, 26 theorems, 294 equations)

This paper contains 102 sections, 26 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Notations and conventions
  4. Statement of mathematical results
  5. Statement of physics results
  6. Instanton scattering matrices and $R$-matrices
  7. Acknowledgements
  8. Outline of paper
  9. Monopole moduli spaces and Yangians
  10. Scattering transform and monopole moduli
  11. General setup
  12. Scattering matrix
  13. Explicit scattering matrices in rank 2
  14. Integrable structures of monopole moduli
  15. Symplectic structure of the monopole moduli space
  16. ...and 87 more sections

Key Result

Proposition 3.1

The map $\textnormal{Ext}^1_{\mathscr{D}_\hbar}(\widehat{\mathscr{O}}_{\mathbb{C}}, \mathscr{E}) \simeq \mathbb{W} \to \mathscr{W}((z^{-1}))$ written in Winclude is an inclusion of vector bundles, moreover we have (via Proposition explicitext) isomorphisms of sheaves over $\widetilde{M}(r)$, where $B_2: \mathscr{V} \to \mathscr{V}$ is the tautological section corresponding to the ADHM matrix.

Theorems & Definitions (52)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 42 more