The Fuzzy-Sphere as a Black Hole in the IKKT Matrix Model: An Assessment
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, within which we investigate this correspondence. 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, and compare against the D0 and D2 black holes as the closest available reference points, since no gravitational dual is known for this specific background. 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. Holding the physical radius fixed instead yields a genuine $O(N^2)$ scaling, showing that the power of $N$ depends on the large-$N$ prescription and is therefore not by itself an unambiguous test of a black-hole interpretation. We further find no intrinsic Hawking temperature -- only a characteristic scale -- and a heat capacity that is negative at low temperature but positive at high temperature. The absence of an intrinsic Hawking temperature, tied to the missing gravitational dual rather than to the choice of large-$N$ prescription, persists in either scaling limit. 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.