Construct dictionaries from thermodynamic curvature to topological and Riemann-surface invariants

Construct explicit mappings from the normalized thermodynamic curvature scalar R_N to the topological number W of the global topological framework and to the Riemann-surface foliation number of the complex-analysis framework, using the local geometric framework as an intermediary.

Background

The paper proves that Ruppeiner and Weinhold thermodynamic geometry are exactly connected to the local geometric framework through the correspondence between curvature-scalar divergences and solutions of T'(r_h)=0. The local geometric framework is itself connected to the global topological and complex-analysis frameworks, allowing the authors to infer that corresponding mappings should exist between R_N and the topological number W or the Riemann-surface foliation number.

However, the paper does not explicitly construct these dictionaries. Establishing them would complete the direct correspondence among the characteristic quantities used by the four analytical frameworks for black hole first-order phase transitions.

References

Although in this work we have not established a dictionary between the curvature scalar and the topological number $W$ (the core characteristic quantity of the global topological framework) or the Riemann surface foliation number (the core characteristic quantity of the complex analysis framework), through the local geometric framework as an intermediary, the correspondence between $R_N$ and the topological number and the Riemann surface foliation number is straightforward, but the explicit construction of these mappings will be left for future work.

— Thermodynamic geometry as the missing link: toward a unified framework for black hole first-order phase transitions  (2609.31144 - Zhang et al., 25 Sep 2026) in Section 5, “The unified framework for black hole first-order phase transitions”