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

Abstract In this paper, we present a remarkably simple expression for the exact solution to the $$D = 5,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> $${\mathscr {N}}= 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> supergravity coupled to three vector multiplets with the pre-potential $$\mathscr {V} = STU -W^2U \equiv 1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:mi>T</mml:mi> <mml:mi>U</mml:mi> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>U</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which represents a five-dimensional static Kaluza–Klein black hole with squashed $$S^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> horizons and four independent electric charges. It is asymptotically locally flat and has a spatial infinity $$R \times S^1 \hookrightarrow S^2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>↪</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> 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. Besides, we have, for the first time, observed a novel black hole splitting phenomenon induced by the extra charge – namely, when the fourth Abelian vector field is added, the black hole solution to the usual STU model is split into two different branches in the new $$STU-W^2U$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>T</mml:mi> <mml:mi>U</mml:mi> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>U</mml:mi> </mml:mrow> </mml:math> model. This fact hints that the solution space of the $$STU-W^2U$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>T</mml:mi> <mml:mi>U</mml:mi> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>U</mml:mi> </mml:mrow> </mml:math> model has a much richer structure than that of the usual STU model. We also demonstrate that the special $$S^2U-W^2U$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>U</mml:mi> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>W</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>U</mml:mi> </mml:mrow> </mml:math> model can be converted into the usual STU model which suggests that one should coherently set $$\rho _2 = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in the general expression of the structure function: $$f(\rho ) = 1 -\rho _1/\rho +\rho _2/\rho ^2.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>2</mml:mn> </mml:msub>