Structure of Loop Space at Finite $N$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909720056430592 |
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| author | Koch, Robert de Mello Jevicki, Antal |
| author_facet | Koch, Robert de Mello Jevicki, Antal |
| contents | The space of invariants for a single matrix is generated by traces containing at most $N$ matrices per trace. We extend this analysis to multi-matrix models at finite $N$. Using the Molien-Weyl formula, we compute partition functions for various multi-matrix models at different $N$ and interpret them through trace relations. This allows us to identify a complete set of invariants, naturally divided into two distinct classes: primary and secondary. The full invariant ring of the multi-matrix model is reconstructed via the Hironaka decomposition, where primary invariants act freely, while secondary invariants satisfy quadratic relations. Significantly, while traces with at most $N$ matrices are always present, we also find invariants involving more than $N$ matrices per trace. The primary invariants correspond to perturbative degrees of freedom, whereas the secondary invariants emerge as non-trivial background structures. The growth of secondary invariants aligns with expectations from black hole entropy, suggesting deep structural connections to gravitational systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_20097 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structure of Loop Space at Finite $N$ Koch, Robert de Mello Jevicki, Antal High Energy Physics - Theory The space of invariants for a single matrix is generated by traces containing at most $N$ matrices per trace. We extend this analysis to multi-matrix models at finite $N$. Using the Molien-Weyl formula, we compute partition functions for various multi-matrix models at different $N$ and interpret them through trace relations. This allows us to identify a complete set of invariants, naturally divided into two distinct classes: primary and secondary. The full invariant ring of the multi-matrix model is reconstructed via the Hironaka decomposition, where primary invariants act freely, while secondary invariants satisfy quadratic relations. Significantly, while traces with at most $N$ matrices are always present, we also find invariants involving more than $N$ matrices per trace. The primary invariants correspond to perturbative degrees of freedom, whereas the secondary invariants emerge as non-trivial background structures. The growth of secondary invariants aligns with expectations from black hole entropy, suggesting deep structural connections to gravitational systems. |
| title | Structure of Loop Space at Finite $N$ |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2503.20097 |