Spectral Curves with Complex Multiplication in Hermitian Matrix Models
High Energy Physics - Theory
2025-09-23 v1
Abstract
We show that elliptic curves with complex multiplication (CM) naturally emerge in the spectral geometry of Hermitian one-matrix models in the two-cut phase. Focusing on a symmetric quartic potential, we derive the corresponding genus-one spectral curve and compute its modular -invariant in closed form as a function of the quartic coupling . We identify specific values of for which the elliptic curve exhibits , i.e., its endomorphism ring is larger than . This establishes a direct connection between number-theoretic structures and the spectral data of random matrix ensembles.
Cite
@article{arxiv.2509.16997,
title = {Spectral Curves with Complex Multiplication in Hermitian Matrix Models},
author = {Ali Nassar},
journal= {arXiv preprint arXiv:2509.16997},
year = {2025}
}