Toric non-archimedean $\mu$-entropy and thermodynamical structure

Abstract

We study non-archimedean $\mu$-entropy for toric variety as a further exploration of $\mu$K-stability. We show the existence of optimizer of toric non-archimedean ${\mu }^{\lambda }$-entropy for $\lambda \in \mathbb{R}$ and the uniqueness for $\lambda \le 0$. For the proof of existence, we establish a Rellich type compactness result for convex functions on simple polytope. We also reveal a thermodynamical structure on toric non-archimedean $\mu$-entropy. This observation allows us to interpret the enigmatic parameter $T=-\frac{\lambda }{2\pi }$ as temperature and non-archimedean $\mu$-entropy as entropy of an infinite dimensional composite system.

arXiv comment45 pages, comments are very welcome!