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Peer-reviewedCC BY 4.0SciPost Physics

c=1 strings as a matrix integral

  • Scott Collier1
  • Lorenz Eberhardt2
  • Victor A. Rodriguez3
  1. 1Syracuse University
  2. 2Institute for Theoretical Physics Amsterdam
  3. 3University of California, Santa Barbara
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Abstract

We study the perturbative S S -matrix of the c=1 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) 𝗑 ( z ) = 2 2 cos ⁑ ( z ) , \mathsf{y}(z)=\sin(z) 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 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 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.

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Cite

BibTeXRIS
BibTeX
@article{collier2026strings,
  title = {{$c=1$ strings as a matrix integral}},
  author = {Scott Collier and Lorenz Eberhardt and Victor A. Rodriguez},
  journal = {SciPost Physics},
  year = {2026},
  month = oct,
  doi = {10.21468/scipostphys.21.4.085},
  url = {https://scipost.org/10.21468/SciPostPhys.21.4.085},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - c=1 strings as a matrix integral
AU  - Scott Collier
AU  - Lorenz Eberhardt
AU  - Victor A. Rodriguez
PY  - 2026
DA  - 2026/10/08
JO  - SciPost Physics
DO  - 10.21468/scipostphys.21.4.085
UR  - https://scipost.org/10.21468/SciPostPhys.21.4.085
AB  - We study the perturbative S S -matrix of the c=1 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) 𝗑 ( z ) = 2 2 cos ⁑ ( z ) , \mathsf{y}(z)=\sin(z) 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 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 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.
SN  - 2542-4653
ER  -