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PreprintCC BY 4.0arXiv

Tropicalization of Fock's inverse spectral transform

  • Terrence George
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Abstract

The dimer spectral transform gives algebro-geometric action-angle coordinates for cluster integrable systems. Fock gave an explicit formula for its inverse. We study Fock's inverse map in the tropical limit where Harnack spectral curves degenerate to nodal curves. We determine the leading order asymptotics of the classical objects appearing in Fock's formula and show that they are governed by their tropical counterparts. This yields a tropical inverse spectral transform on tropical spectral data. We show that Fock's inverse spectral transform commutes with tropicalization, and also derive a tropical version of Fay's trisecant identity.

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Cite

BibTeXRIS
BibTeX
@misc{george2026tropicalization,
  title = {{Tropicalization of Fock's inverse spectral transform}},
  author = {Terrence George},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06241},
  eprint = {2609.06241},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06241},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Tropicalization of Fock's inverse spectral transform
AU  - Terrence George
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.06241
UR  - https://arxiv.org/abs/2609.06241
AB  - The dimer spectral transform gives algebro-geometric action-angle coordinates for cluster integrable systems. Fock gave an explicit formula for its inverse. We study Fock's inverse map in the tropical limit where Harnack spectral curves degenerate to nodal curves. We determine the leading order asymptotics of the classical objects appearing in Fock's formula and show that they are governed by their tropical counterparts. This yields a tropical inverse spectral transform on tropical spectral data. We show that Fock's inverse spectral transform commutes with tropicalization, and also derive a tropical version of Fay's trisecant identity.
ER  -