Spectral curves with complex multiplication in Hermitian matrix models
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 $j$-invariant in closed form as a function of the quartic coupling $g$. We identify specific values of $g$ for which the elliptic curve exhibits $CM$, i.e., its endomorphism ring is larger than $\mathbb{Z}$. This establishes a direct connection between number-theoretic structures and the spectral data of random matrix ensembles.
Fields of Study & Subjects
Advanced Algebra and Geometry
Advanced Topics in Algebra
Algebraic structures and combinatorial models
Publication details
- Journal / source
- Nuclear Physics B
- Volume
- 1029
- Pages
- 117582–117582
- Publication date
- 8 July 2026
- Language
- EN