Localization of the four-dimensional $ \mathcal{N}=4 $ SYM to a two-sphere and 1/8 BPS Wilson loops

Abstract

We localize the four-dimensional N = 4 super Yang-Mills theory on S 4 to the two-dimensional constrained Hitchin/Higgs-Yang-Mills (cHYM) theory on S 2 . We show that expectation values of certain 1/8 BPS supersymmetric Wilson loops on S 2 in the 4d N = 4 SYM is captured by the 2d cHYM theory. We further argue that expectation values of Wilson loops in the cHYM theory agree with the prescription "two-dimensional bosonic Yang-Mills excluding instanton contributions". Hence, we support the recent conjecture by Drukker, Giombi, Ricci and Trancanelli on the 1/8 BPS Wilson loops on S 2 in the 4d N = 4 SYM. * On leave of absence from ITEP, 117218, Moscow, Russia

A supermatrix model for $$ \mathcal{N} $$ = 6 super Chern-Simons-matter theory

“…The matrix model calculates the contribution of a single saddle point in a perturbative expansion of the Chern-Simons theory. This is reminiscent of the situation for the Wilson loops on S 2 in N = 4 supersymmetric Yang-Mills in four dimensions and their relation to two-dimensional Yang-Mills [37,38]. They are not given by the full answer in the 2-dimensional theory, but rather by a semiclassical expansion around the zero-instanton sector [39][40][41].…”

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

“…The subscript ν will indicate the appearance of ν-phases e ±i πν 2 . 4 Localization usually reduces the path-integral to a sum over discrete or continuous constant field configurations, but, in general, it could also lead to a lower dimensional field theory [46].…”

Perturbative and Non-perturbative Approaches to String Sigma-Models in AdS/CFT

“…The proposal was extended in [354] to all correlators of 1/8-BPS DGRT loops and chiral primary operators (CPOs) of the form When also Wilson loops are present, mixed correlators are no longer protected from radiative corrections, but retain two supercharges [354,356] which allow to set up the machinery of supersymmetric localization. The authors of [354] supported the conjecture that mixed correlation functions…”

“…A generic loop preserves a SU (2) × U (1) × SU (2) bosonic subgroup of P SU (2, 2 4) and 8 supercharges, which can be organized to form a SU (2 2) supergroup [298]. Strong evidences [384], later substantially 9 proved using supersymmetric localization [356], relate these operators to the matrix model of the circular loop (5.13)-(5.15) by a rescaling of the YM coupling g YM → g YM cos θ 0 (and λ → λ cos 2 θ 0 in planar limit). 9 The proof is not completed at the same level of rigour of [151] for the circle: one still needs to compute the one-loop determinants around the localization locus and effectively prove that the Wilson loops localizes in YM2 on S 2 .…”