Spectral Networks and Betti Lagrangians

TL;DR

This work develops a rigorous framework for spectral networks in real cotangent settings, extending GMN-inspired structures to Betti Lagrangians and higher rank via adiabatic Floer theory. It constructs a Family Floer functor and proves its equivalence with GMN non-abelianization in the adiabatic limit, embedding framed 2d-4d BPS states into (partially) wrapped 4d Fukaya categories. A new Morse-spectral-network theory is developed, including existence results, energy gaps, and a detailed link to D4^{-} flowtrees, walls, and BPS indices; Demazure weaves are used to generate explicit spectral networks and relate them to Legendrian augmentations. The results provide a robust bridge between Floer-theoretic operations and non-abelian parallel transport, with applications to cluster algebra structures on braid varieties and to the representations of irregular Stokes data in higher rank. Overall, the paper broadens the mathematical foundation of spectral networks beyond holomorphic curves, connecting real-Lagrangian Floer theory, wrapped and partially wrapped Fukaya categories, and combinatorial weave constructions.

Abstract

We introduce and develop the theory of spectral networks in real contact and symplectic topology. First, we establish the existence and pseudoholomorphic characterization of spectral networks for Lagrangian fillings in the cotangent bundle of a smooth surface. These are proven via analytic results on the adiabatic degeneration of Floer trajectories and the explicit computation of continuation strips. Second, we construct a Family Floer functor for Lagrangian fillings endowed with a spectral network and prove its equivalence to the non-abelianization functor. In particular, this implies that both the framed 2d-4d BPS states and the Gaiotto-Moore-Neitzke non-abelianized parallel transport are realized as part of the $A_\infty$-operations of the associated 4d partially wrapped Fukaya categories. To conclude, we present a new construction relating spectral networks and Lagrangian fillings using Demazure weaves, and show the precise relation between spectral networks and augmentations of the Legendrian contact dg-algebra.

Figures (27)

• Figure 1: A Betti surface ${\bf S}$ of genus $3$ and rank $3$, with 9 marked points $\boldsymbol{m}=\{m_1,\ldots,m_9\}$, in orange, and their associated Legendrian links in ${\bf \Lambda}$, in blue. The Legendrian links associated to marked points $m_1,m_5,m_8$ have concentric circles as their fronts, which corresponds to a regular singularity. A non-concentric Legendrian isotopy class corresponds to an irregular singularity.
• Figure 2: The local models for vertices in spectral networks: initial, interaction and non-interaction. There are two types of interaction vertices: creation and 6-valent type. The inconsistent vertex type is allowed for pre-spectral networks, but is not allowed for spectral networks.
• Figure 3: (1) A weave $\mathfrak{w}$ and the spectral network $\EuScript{W}_\mathfrak{w}$ constructed in \ref{['ssec:networks_Demazure_weaves']}. (2) The spectral network $\EuScript{W}_\mathfrak{w}$ in (1), drawn on its own, without the weave $\mathfrak{w}$. (3) Another weave with its associated spectral network. In this case, the spectral network recovers precisely the Berk-Nevins-Roberts network from BerkNevinsRoberts82_NewStokes by adiabatically degenerating the rigid pseudoholomorphic strips contributing to the augmentation associated to the Lagrangian filling of the weave.
• Figure 4: A smooth surface ${\bf S}$ of genus $3$ with 9 marked points $\boldsymbol{m}=\{m_1,\ldots,m_9\}$, in orange, and fronts for their associated Legendrian links in ${\bf \Lambda}=\{\Lambda_1,\ldots,\Lambda_9\}$, in blue. The annular neighborhoods $A_i\subset S$ for each marked point are shaded in pink. The positive braid words $\beta_1,\beta_5,\beta_8$ are the empty braid words in 3-strands and, for instance, we can choose $\beta_6=\sigma_2^2\sigma_1\sigma_2\sigma_1^2$ and $\beta_9=\sigma_1\sigma_2$.
• Figure 5: The asymptotics of an $ij$-flowline $\gamma(s)=(r(s),\theta(s))$ based on the sign of $(z^{i}-z^{j})$ at the Reeb chords. The case depicted in (1) occurs if $(z^{i}-z^{j})$ goes from negative to negative. Case (2) occurs if $(z^{i}-z^{j})$ goes from positive to negative. Case (3) if $(z^{i}-z^{j})$ goes from positive to positive. Case (4) never occurs, which would correspond to $(z^{i}-z^{j})$ going from negative to positive. The fact that the fourth case is excluded is the key ingredient behind the proof of \ref{['lemma:trappinglemma']} (the Trapping Lemma).

Theorems & Definitions (187)

• Theorem 1: Existence of Spectral Networks
• Theorem 2: Floer-theoretic characterization of Spectral Networks
• Theorem 3: Family Floer and Non-abelianization
• Theorem 4: Spectral curves and 4d Partially Wrapped Fukaya Categories
• Theorem 5: Augmentations, weaves and Spectral Networks
• Remark 2.1
• Remark 2.2
• Definition 2.3: Betti surfaces
• Remark 2.4
• Example 2.5: Betti surfaces from Stokes data