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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.06301 |
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| _version_ | 1866916048859561984 |
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| author | Collier, Scott Eberhardt, Lorenz Rodriguez, Victor A. |
| author_facet | Collier, Scott Eberhardt, Lorenz Rodriguez, Victor A. |
| contents | We study the perturbative $S$-matrix of the $c=1$ string and show that it admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve $\mathsf{x}(z) = 2\sqrt{2}\cos(z)$, $\mathsf{y}(z)=\sin(z)$. Combined with the famous duality to matrix quantum mechanics, this establishes a triality between three formulations of the theory: the worldsheet description, matrix quantum mechanics, and a matrix integral.
Starting from the intersection number expressions for the complex Liouville string, we derive closed-form Feynman rule expressions for the $c=1$ amplitudes as intersection numbers on the moduli space of Riemann surfaces. The intersection theory naturally computes amplitudes corresponding to a discretized target space where momentum is conserved only modulo an integer. The physical $S$-matrix elements are recovered by restriction to the first `Brillouin zone' and analytic continuation to Lorentzian kinematics. We prove that these amplitudes satisfy perturbative spacetime unitarity directly from the intersection theory expressions, and show that they satisfy a Mirzakhani-type recursion relation. We show detailed agreement with the known matrix quantum mechanics results, providing strong evidence for the triality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $c=1$ strings as a matrix integral Collier, Scott Eberhardt, Lorenz Rodriguez, Victor A. High Energy Physics - Theory We study the perturbative $S$-matrix of the $c=1$ string and show that it admits a description in terms of a double-scaled (0+0)-dimensional matrix integral based on the spectral curve $\mathsf{x}(z) = 2\sqrt{2}\cos(z)$, $\mathsf{y}(z)=\sin(z)$. Combined with the famous duality to matrix quantum mechanics, this establishes a triality between three formulations of the theory: the worldsheet description, matrix quantum mechanics, and a matrix integral. Starting from the intersection number expressions for the complex Liouville string, we derive closed-form Feynman rule expressions for the $c=1$ amplitudes as intersection numbers on the moduli space of Riemann surfaces. The intersection theory naturally computes amplitudes corresponding to a discretized target space where momentum is conserved only modulo an integer. The physical $S$-matrix elements are recovered by restriction to the first `Brillouin zone' and analytic continuation to Lorentzian kinematics. We prove that these amplitudes satisfy perturbative spacetime unitarity directly from the intersection theory expressions, and show that they satisfy a Mirzakhani-type recursion relation. We show detailed agreement with the known matrix quantum mechanics results, providing strong evidence for the triality. |
| title | $c=1$ strings as a matrix integral |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2604.06301 |