标题： 黑洞热力学自由能作为$A$-判别式 显示英文标题

标题： Black hole thermodynamic free energy as $A$-discriminants

摘要： 我们证明了黑洞的自由能$\mathcal{F}$和温度$T$，作为热力学系统，可以看作是适当定义的多项式的$A$-判别式。 因此，关于$A$-判别式的数学结果可能会对黑洞热力学产生影响。 特别是，对于具有球对称、平面或双曲对称的静态时空，不同热力学相的数量取决于度规分量$g_{tt}$中不同项的数量。 我们证明，如果$g_{tt}$包含$N_f$个不同的项，则$\mathcal{F}$-$T$曲线包含$N_f-2$个尖点，这进而导致$N_f-1$个不同的热力学相。 该结果被应用于施瓦茨希尔德-反德西特、雷斯纳-诺德斯特伦、幂律麦克斯韦和欧拉-海森堡黑洞的显式例子。 显示英文摘要

摘要： We show that the free energy $\mathcal{F}$ and temperature $T$ of black holes, considered as a thermodynamic system, can be viewed as an $A$-discriminant of an appropriately-defined polynomial. As such, mathematical results about $A$-discriminants may lead to implications about black hole thermodynamics. In particular, for static spacetimes with spherical, planar, or hyperbolic symmetry, the number of distinct thermodynamic phases depend on the number of distinct terms in the metric component $g_{tt}$. We prove that if $g_{tt}$ consists of $N_f$ distinct terms, then the $\mathcal{F}$-$T$ curve consists of $N_f-2$ cusps, which in turn leads to $N_f-1$ distinct thermodynamic phases. This result is applied to explicit examples of the Schwarzschild-AdS, Reissner--Nordstr\"{o}m, power-law Maxwell, and Euler--Heisenberg black holes.