Superspace calculation of the four-loop spectrum in N=6 supersymmetric Chern-Simons theories

TL;DR

This work computes the four-loop spectrum in the SU(2)×SU(2) sector of ABJM/ABJ theories using ${\mathcal N}=2$ superspace, focusing on the interpolating magnon dispersion function $h^2(\bar{\lambda},\sigma)$. By evaluating a carefully organized set of supergraphs and applying finiteness and IR-cancellation constraints, the authors extract the four-loop coefficient $h_4(\sigma)$ and show it equals $h_4(\sigma)=-(4+\sigma^2)\zeta(2)$, in agreement with previous component-based results. They discuss plausible all-loop interpolations for $h^2(\bar{\lambda},\sigma)$, including a weak-to-strong coupling ansatz and potential links to matrix-model/Wilson-loop data, while also analyzing wrapping effects for short operators. The superspace approach significantly reduces diagrammatic complexity, paving the way for higher-loop checks of integrability and for extending the analysis beyond the SU(2)×SU(2) sector.

Abstract

Using N=2 superspace techniques we compute the four-loop spectrum of single trace operators in the SU(2) x SU(2) sector of ABJM and ABJ supersymmetric Chern-Simons theories. Our computation yields a four-loop contribution to the function h^2(λ) (and its ABJ generalization) in the magnon dispersion relation which has fixed maximum transcendentality and coincides with the findings in components given in the revised versions of arXiv:0908.2463 and arXiv:0912.3460. We also discuss possible scenarios for an all-loop function h^2(λ) that interpolates between weak and strong couplings.