Supersymmetric Origin of Four-Dimensional Space-time in the IIB Matrix Model

We investigate the constraints imposed by supersymmetry on the IIB matrix model (IKKT model) by requiring both the closure of the transformations and the satisfaction of the Ward identities at the leading order of the order expansion. Following the systematic methodology, we evaluate the most general forms of the effective action and supersymmetry transformations consistent with the $SU(N)$ algebra. In ten dimensions, we prove that these supersymmetric requirements lead to a non-renormalization theorem, which forces all coefficient functions to be constant. This result stems from the emergence of a 5-form tensor in the closure condition that cannot be absorbed by the $SU(N)$ algebra. This residual term strictly forbids non-trivial fluctuations at the leading order. While a similar non-renormalization theorem holds in four dimensions, we demonstrate that the four-dimensional Clifford algebra provides a unique exit through Hodge duality. This duality geometrically maps the anomalous high-rank tensor structures into absorbable lower-rank forms, allowing for non-trivial dynamical backgrounds prohibited in ten dimensions. We find that such non-trivial solutions are restricted to (anti-)self-dual configurations, which, through reality conditions, necessitate a Euclidean metric. Our results indicate that the emergence of a four-dimensional Euclidean space-time is a prerequisite for the theory to admit non-trivial backgrounds while preserving supersymmetry at the leading order.