Grassmannian Integrals as Matrix Models for Non-Compact Yangian Invariants

TL;DR

This paper introduces a novel approach to evaluate deformed Grassmannian integrals in planar N=4 super Yang-Mills theory by transforming them into matrix models using oscillator representations, connecting to non-compact integrable systems.

Contribution

It proposes transforming Grassmannian integrals into matrix models via oscillator representations, enabling new evaluation techniques and linking to non-compact integrable spin chains.

Findings

Formulated Yangian invariants as matrix models for u(p,q)

Generalized Brezin-Gross-Witten and Leutwyler-Smilga models

Included a matrix model for the u(p,q) R-matrix

Abstract

In the past years, there have been tremendous advances in the field of planar N=4 super Yang-Mills scattering amplitudes. At tree-level they were formulated as Grassmannian integrals and were shown to be invariant under the Yangian of the superconformal algebra psu(2,2|4). Recently, Yangian invariant deformations of these integrals were introduced as a step towards regulated loop-amplitudes. However, in most cases it is still unclear how to evaluate these deformed integrals. In this work, we propose that changing variables to oscillator representations of psu(2,2|4) turns the deformed Grassmannian integrals into certain matrix models. We exemplify our proposal by formulating Yangian invariants with oscillator representations of the non-compact algebra u(p,q) as Grassmannian integrals. These generalize the Brezin-Gross-Witten and Leutwyler-Smilga matrix models. This approach might make elaborate matrix model technology available for the evaluation of Grassmannian integrals. Our invariants also include a matrix model formulation of the u(p,q) R-matrix, which generates non-compact integrable spin chains.