Static four-charge squashed black hole in five-dimensional $STU-W^2U$ supergravity and its thermodynamics

TL;DR

The paper constructs a simple four-charge static squashed black hole in five-dimensional $\mathcal{N}=2$ STU-W^2U supergravity with prepotential $\mathcal{V}=STU - W^2 U = 1$, yielding a metric built from harmonic functions $Z_I$ and an ALF asymptotic with a squashed $S^3$ horizon. It then computes conserved charges and thermodynamic quantities via the counterterm method, showing that the first law and the Bekenstein–Smarr relation hold when the length of the extra dimension $L_{\infty}$ is treated as a thermodynamic variable. The results provide concise expressions for mass, tension, entropy, temperature, charges, and potentials, and demonstrate a consistent thermodynamic framework for squashed KK black holes in the STU-W^2U model. The work suggests extensions to rotating cases and a supersymmetric version, with supersymmetric branches discussed in the Appendix.

Abstract

In this paper, we present a remarkably simple expression for the exact solution to the $D = 5$, $\mathcal{N} = 2$ supergravity coupled to three vector multiplets with the prepotential $\mathcal{V} = STU -W^2U \equiv 1$, which represents a five-dimensional static Kaluza-Klein black hole with squashed $S^3$ horizons and four independent electric charges. It is asymptotically locally flat and has a spatial infinity $R \times S^1 \hookrightarrow S^2$. We compute its conserved charges via the counterterm method and demonstrate that the thermodynamic quantities satisfy both the first law and Bekenstein-Smarr mass formula, provided the length of the compact extra dimension is treated as a thermodynamic variable.