A one-loop test for construction of 4D N=4 SYM from 2D SYM via fuzzy sphere geometry

TL;DR

This paper performs a one-loop quantum check on constructing 4D ${ m N}=4$ SYM from 2D SYM with fuzzy sphere geometry, confirming the equivalence at the quantum level in certain limits.

Contribution

It provides a perturbative one-loop analysis validating the emergence of 4D ${ m N}=4$ SYM from 2D SYM via fuzzy sphere geometry, including quantum effects and noncommutative limits.

Findings

One-loop effective action matches 4D ${ m N}=4$ SYM results.

Noncommutative anomaly appears in the $U(1)$ sector but is likely a gauge artifact.

Classical limits reproduce ordinary ${ m N}=4$ SYM on ${ m R}^4$.

Abstract

As a perturbative check of the construction of four-dimensional (4D) ${\cal N}=4$ supersymmetric Yang-Mills theory (SYM) from mass deformed ${\cal N}=(8,8)$ SYM on the two-dimensional (2D) lattice, the one-loop effective action for scalar kinetic terms is computed in ${\cal N}=4$ $U(k)$ SYM on ${\mathbb R}^2 \times$ (fuzzy $S^2$), which is obtained by expanding 2D ${\cal N}=(8,8)$ $U(N)$ SYM with mass deformation around its fuzzy sphere classical solution. The radius of the fuzzy sphere is proportional to the inverse of the mass. We consider two successive limits: (1) decompactify the fuzzy sphere to a noncommutative (Moyal) plane and (2) turn off the noncommutativity of the Moyal plane. It is straightforward at the classical level to obtain the ordinary ${\cal N}=4$ SYM on ${\mathbb R}^4$ in the limits, while it is nontrivial at the quantum level. The one-loop effective action for $SU(k)$ sector of the gauge group $U(k)$ coincides with that of the ordinary 4D ${\cal N}=4$ SYM in the above limits. Although "noncommutative anomaly" appears in the overall $U(1)$ sector of the $U(k)$ gauge group, this can be expected to be a gauge artifact not affecting gauge invariant observables.