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

Abstract

As a perturbative check of the construction of four-dimensional (4D) $\mathcal{N}=4$ supersymmetric Yang-Mills theory (SYM) from mass deformed $\mathcal{N}=(8,8)$ SYM on the two-dimensional (2D) lattice, the one-loop effective action for scalar kinetic terms is computed in $\mathcal{N}=4$ $U(k)$ SYM on ${\mathbb{R}}^2\times$ (fuzzy $S^2$), which is obtained by expanding 2D $\mathcal{N}=(8,8)$ $U(N)$ 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 $\mathcal{N}=4$ 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 $\mathcal{N}=4$ 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.

Comments: 109 pages, 2 figures, comments added, appendix F added, final version to be published PTEP