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Any Topological Recursion on a Rational Spectral Curve is KP Integrable
A Alexandrov1, B Bychkov2, P Dunin-Barkowski3,4,5
1Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673 Korea.
Communications in Mathematical Physics
|March 12, 2026
Summary
We demonstrate that correlation differentials from topological recursion are KP integrable for genus zero spectral curves. This confirms KP integrability for partition functions linked to log canonical bundles via ELSV-type formulas.
Area of Science:
- Mathematics
- Mathematical Physics
Background:
- Topological recursion is a powerful tool for studying discrete structures in quantum field theory and string theory.
- KP integrability is a significant property in soliton theory and mathematical physics, indicating deep underlying structures.
Purpose of the Study:
- To prove the KP integrability of correlation differentials in topological recursion for genus zero spectral curves.
- To establish the KP integrability of partition functions associated with log canonical bundles using ELSV-type formulas.
Main Methods:
- Utilizing properties of spectral curves and topological recursion.
- Applying ELSV-type formulas to relate partition functions to correlation differentials.
Main Results:
- We prove that correlation differentials of topological recursion are KP integrable for any initial data on a genus zero spectral curve.
- We establish the KP integrability of partition functions associated with the r-th roots of twisted powers of log canonical bundles.
Conclusions:
- The findings confirm a deep connection between topological recursion and KP integrability.
- This work opens new avenues for exploring integrable structures in mathematical physics and related fields.
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