A Twistorial Description of the IKKT-Matrix Model

Abstract: We consider the fuzzy 4-sphere $S_N^4$ as a background in the IKKT matrix model and explore the relation between $S_N^4$ and fuzzy twistor space in the semi-classical limit. A novel description for the IKKT-matrix model in terms of spinorial indices is given, which is reminiscent of $\mathcal{N}=4$ super-symmetric Yang-Mills (SYM) in $4d$. On fuzzy twistor space, the interactions of the IKKT model are of gravitational type. The higher-spin (HS) gauge theory emerging in this limit from the IKKT model, denoted as HS-IKKT, on fuzzy twistor space is shown to be a higher-spin extension of $\mathcal{N}=4$ SYM, with vertices that have more than two derivatives. We obtain its (Euclidean) spacetime action using the Penrose transform. Although this is a gravitational theory, it shares many features with the higher-spin extensions of Yang-Mills in $4d$ flat space obtained in arXiv:2105.12782, arXiv:2107.04500. The tree-level amplitudes of the HS-IKKT are studied in the semi-classical flat limit. The self-dual sector of the IKKT model is obtained by dropping some parts of the cubic- and the quartic interactions, which is shown to reduce to a BF-type action on commutative deformed projective twistor space.