\( \mathfrak{hs} \)-extended gravity from the IKKT matrix model

We elaborate further on the one-loop effective action of the IKKT model on 3 + 1 dimensional covariant quantum spacetime in the presence of fuzzy extra dimensions. In particular, we describe the one-loop effective action in terms of a remarkable SO(1, 9) character, which allows to evaluate the pertinent traces over the internal modes explicitly. This also allows to estimate the higher-order contributions (in the internal flux \( {\mathcal{F}}_{\textrm{IJ}} \)) to the one-loop effective action in a systematic way. We show that all higher-order contributions are generally suppressed and UV finite, which justifies the previous treatment of the induced gravitational action. We also obtain explicit expressions for the effective Newton constant, and determine the dynamics of the Kaluza-Klein scale \( {\Delta }_{\mathcal{K}} \) of the fuzzy extra dimensions \( \mathcal{K} \).