The Fuzzy-Sphere as a Black Hole in the IKKT Matrix Model: An Assessment

Abstract

Black-hole thermodynamics has been reproduced with remarkable success from the BFSS matrix model, but the corresponding test in the closely related IKKT matrix model has awaited its recently proposed finite-temperature formulation. We investigate this correspondence within that framework. Introducing a cubic Myers term, we compute the complete one-loop thermodynamics -- free energy, internal energy, entropy, and heat capacity -- of a fuzzy $S^2$ background. No gravitational dual is known for this specific background, so we compare these quantities against the D0 and D2 black holes as the closest available reference points. At fixed Myers coupling, the leading entropy exhibits an $O(N^3)$ scaling rather than the $O(N^2)$ behavior characteristic of conventional black holes; this scaling mismatch is the central result of our analysis. We further find no intrinsic Hawking temperature -- only a characteristic scale -- and a heat capacity that is negative at both the low- and high-temperature endpoints. Holding the physical radius fixed instead yields a genuine $O(N^2)$ scaling, so the power of $N$ is convention-dependent rather than an unambiguous prediction -- though the sign and temperature-scale mismatches persist either way. Rather than establishing a black-hole interpretation of the finite-temperature IKKT matrix model, these results provide a quantitative benchmark against which future analytical, numerical, and holographic investigations can be tested.