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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2509.16997 |
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| _version_ | 1866916959152504832 |
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| author | Nassar, Ali |
| author_facet | Nassar, Ali |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_16997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Curves with Complex Multiplication in Hermitian Matrix Models Nassar, Ali High Energy Physics - Theory 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. |
| title | Spectral Curves with Complex Multiplication in Hermitian Matrix Models |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2509.16997 |