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.
Highlights
The maximally supersymmetric Yang-Mills theory in four-dimensions, for short N = 4 SYM, is a remarkably rich mathematical model
We propose that changing variables to oscillator representations of psu(2, 2|4) turns the deformed Graßmannian integrals into certain matrix models
In this work we showed that the Graßmannian integral, commonly used in the realm of N = 4 SYM scattering amplitudes, can be applied to construct Yangian invariants for oscillator representations of the non-compact algebra u(p, q)
Summary
It was observed that the tree-level amplitudes allow for multi-parameter deformations while maintaining Yangian invariance. We establish a connection between the Graßmannian integral formulation of Yangian invariants and unitary matrix models. This motivates us to conjecture that the “unitary contour” works as well for general deformation parameters This would mean that the Graßmannian integrals can be considered as novel types of unitary matrix models. We provide a nontrivial example of this conjecture by investigating the invariant with (N, K) = (4, 2) In this example the Graßmannian integral becomes a U (2) matrix model that correctly evaluates to the u(p, q) R-matrix, which is known to be Yangian invariant. This R-matrix generates non-compact integrable spin chains. There are further fascinating prospects which we elaborate on in the outlook of section 7
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