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PreprintCC BY 4.0arXiv

Loop Equations for Multi-Matrix Models

  • Aravinth Kulanthaivelu
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Abstract

We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.

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Cite

BibTeXRIS
BibTeX
@misc{kulanthaivelu2026loop,
  title = {{Loop Equations for Multi-Matrix Models}},
  author = {Aravinth Kulanthaivelu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25095},
  eprint = {2609.25095},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25095},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Loop Equations for Multi-Matrix Models
AU  - Aravinth Kulanthaivelu
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.25095
UR  - https://arxiv.org/abs/2609.25095
AB  - We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.
ER  -