Comment on "Geometry of the Grosse-Wulkenhaar model"

TL;DR

The paper revisits the geometric reinterpretation of the Grosse-Wulkenhaar model on a truncated Heisenberg space and identifies that the analysis in Section 6 was applied to a related mixed-product term rather than the actual $\Omega$-term. After correcting the operator identifications, the $\Omega$-term is shown to map to a curvature-coupled action with a revised kinetic coefficient and new parameter relations, preserving the core idea that the harmonic potential corresponds to background curvature. The corrected framework yields revised identifications $\kappa = 1 - \Omega^2$, $\frac{\Omega^2}{\epsilon^2} = 2 \kappa \xi$, $m^2 = \kappa\left(M^2 - \frac{15}{2} \xi \mu^2\right)$, and $\lambda = \kappa \Lambda$, and expresses the action in terms of the scalar curvature $R$. It also resolves prior discrepancies regarding vacuum solutions and clarifies the behavior near self-duality ($\kappa \to 0^+$), with implications for the matrix-model parameters and the relation between the triple-point shift and renormalizability.

Abstract

We clarify a key point in the geometric reinterpretation of the Grosse-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 applied not to the actual $Ω$-term in the GW action, but to a closely related term involving a mixed use of ordinary and star-products. Once corrected, the main conclusion -- relating the harmonic potential term to background curvature -- 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.