Roger Casals, Yoon Jae Nho (2025), arXiv preprint — ARXIV-2504.08144
Key takeaways
- Spectral networks, introduced by Gaiotto-Moore-Neitzke for meromorphic spectral curves, can be characterised through Floer theory of real Lagrangian fillings in T*C for a smooth surface C, in any rank and without algebraicity; a Family Floer functor built this way is equivalent to non-abelianization.
Seeded from the abstract and a skim, before the reading in NOTE-022. What follows is what the work says about itself.
The authors develop spectral networks within real contact and symplectic topology. They prove that spectral networks exist for Lagrangian fillings in the cotangent bundle of a smooth surface and characterise them pseudo-holomorphically, via adiabatic degeneration of Floer trajectories and explicit continuation strips. They build a Family Floer functor for fillings with a spectral network and show it is equivalent to the non-abelianization functor. As a result, framed 2d-4d BPS states and GMN non-abelianized parallel transport appear inside the A∞-operations of 4d partially wrapped Fukaya categories. They also relate spectral networks to Lagrangian fillings through Demazure weaves and to augmentations of the Legendrian contact dg-algebra.
Standing in the record
Filed by the reading-time triage of 2026-09-25: 1,000 seconds of active reading over 3 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-022 is the close reading of 2026-09-25, and it placed the work: Proposed — an unrefereed v1 preprint that is carefully motivated, but whose analytic core (§4 adiabatic convergence, §5 continuation strips) leans on Ekholm 2007 and on the second author’s 2024 preprint, and whose Theorem 5 has no written proof. What would settle it: a refereed version or independent expert check of §§4–5, a written proof of Proposition 7.5, and theorem statements narrowed to the hypotheses the proofs use (creative networks, exactness, trimmed limits). Only §1 is useful to a non-specialist here.
When seeded, before the reading in NOTE-022, the entry gave a priority for a deeper reading: low — despite substantial owner time (t=1000 s), it is a 150-page research-level symplectic geometry preprint far from the record’s focus; the introduction as skimmed captures what a non-specialist can extract.
What the seeding asked a deeper reading to check (NOTE-022 is that reading):
- A specialist research monograph. Its correctness rests on the analytic adiabatic-limit results (§4), which only a symplectic topologist can check.
- ML link: none. It is filed in the quantum cluster via the physics (BPS states, WKB), not quantum foundations.
- A deeper reading would first need the GMN papers and background on Fukaya categories; the payoff for this reading record is unclear.
Access when seeded: arXiv abs page and full PDF (150 pp.) retrieved; read outline and §1 Introduction / §1.1 Scientific context (pp. 2-4). A Crossref search found no journal version, so it is treated as a preprint as of this check. The main theorems’ statements (§1.2) and the proofs were not read.