Threefold way to black hole entropy

Abstract

In this work we propose a correspondence between black hole entropy and a topological quantity defined for projective spaces based on the real, complex, and quaternion numbers. After interpreting Weinstein's integer as the normalized volume of the quantum phase space, whose logarithm gives place to the area law (in the real case) and to logarithmic corrections with $\ensuremath{-}\frac{1}{2}$ and $\ensuremath{-}\frac{3}{2}$ coefficients (in the complex and quaternionic cases, respectively), the exact Bekenstein-Hawking entropy is obtained when certain equally spaced spectrum for the event horizon area is imposed. Even more, the minimal area(s) which emerge from our model, are of the form $4\mathrm{log}k$, $k\ensuremath{\in}{2,4,16}$, in complete agreement with previous works. Finally, the role played by global (complex and quaternionic) phases in different descriptions of black hole entropy is clarified.