Comment on "Geometry of the Grosse-Wulkenhaar model"

TL;DR

This paper clarifies a key geometric aspect of the Grosse-Wulkenhaar model, correcting previous analysis and confirming the relation between the harmonic potential and background curvature.

Contribution

It corrects the analysis of the $$-term in the GW model, ensuring the main geometric interpretation remains valid with revised parameter identification.

Findings

The analysis was performed on a related term, not the actual $$-term involving both products.

The main geometric relation between harmonic potential and background curvature remains valid after correction.

The correction resolves discrepancies about vacuum solutions in the self-dual limit.

Abstract

We clarify a key point in the geometric reinterpretation of the Grosse$\unicode{x2013}$Wulkenhaar (GW) model proposed in "Geometry of the Grosse-Wulkenhaar model" [JHEP 03 (2010) 053]. Specifically, we show that the analysis in Section 6 was performed not for the actual $\Omega$-term in the GW action, which involves both ordinary and star-products, but for a closely related term containing only star-products. Once corrected, the main conclusion$\unicode{x2014}$relating the harmonic potential term to background curvature$\unicode{x2014}$remains valid, though the parameter identification must be revised. This also resolves a discrepancy concerning the emergence of certain vacuum solutions in the self-dual limit of the model.