Graßmannian integrals as matrix models for non-compact Yangian invariants

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.