Nonabelian Hodge Graphs

• Nonabelian Hodge Graphs are graph-based organizations that encode moduli spaces and correspondences in nonabelian Hodge theory, notably via the Betti, de Rham, and Dolbeault nodes.
• They integrate key geometric structures—such as symplectic, hyperkähler, and Poisson forms—to bridge analytic, algebraic, and gauge-theoretic perspectives across both tame and irregular settings.
• This framework facilitates practical classification and deformation analysis through enriched constructions like stack-theoretic CoHA, fission trees, and supernova graphs.

Nonabelian Hodge graphs are graph-like organizations of nonabelian Hodge theory in which vertices encode moduli spaces, stacks, irregular classes, Stokes data, or pro-algebraic fundamental groups, and edges encode correspondences such as Riemann–Hilbert, non-abelian Hodge, Fourier–Laplace, or $\mathbb{C}^\times$-equivariant morphisms. In the foundational formulation, the graph has three principal nodes—the Betti, de Rham, and Dolbeault moduli spaces—and canonical edges given by Riemann–Hilbert, Narasimhan–Seshadri, and the non-abelian Hodge correspondence, with Poisson or symplectic forms and a hyperkähler metric functioning as geometric weights (Thomas, 2022). Subsequent work extends the same language to moduli stacks with cohomological Hall algebra structures (Hennecart, 2023), to irregular and wild settings where graphs are extracted from Stokes circles and fission data (Douçot, 29 Sep 2025, Boalch et al., 2019), to genus zero deformation theory via enriched trees and supernova graphs (Douçot, 2024, Boalch, 2024), and to Hodge-theoretic anabelian constructions based on the pro-algebraic completion of the fundamental group equipped with a Simpson-induced $\mathbb{C}^\times$-action (Wang, 6 Mar 2026).

1. Foundational triangle of moduli spaces

For a compact Riemann surface $X$ and a complex reductive group $G$, the basic non-abelian Hodge graph has three nodes. The Betti node is the character variety

$M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$

the de Rham node parametrizes flat holomorphic $G$-connections $(E,\nabla)$ modulo holomorphic gauge, and the Dolbeault node is the moduli of polystable $G$-Higgs bundles $(E,\phi)$ with vanishing Chern classes, where

$\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$

with $\mathbb{C}^\times$0 and, on higher-dimensional Kähler manifolds, $\mathbb{C}^\times$1 (Thomas, 2022).

These nodes carry canonical geometric structures. At a reductive representation $\mathbb{C}^\times$2, the Zariski tangent space of the Betti moduli identifies with

$\mathbb{C}^\times$3

For closed surfaces, the smooth locus of reductive representations carries Goldman’s symplectic form, induced by the cup product and an Ad-invariant trace pairing; for $\mathbb{C}^\times$4, the Goldman bracket on trace functions is

$\mathbb{C}^\times$5

On the de Rham side, Atiyah–Bott reduction yields the symplectic form

$\mathbb{C}^\times$6

with curvature as moment map for gauge transformations. On the Dolbeault side, the moduli space is holomorphic symplectic and, globally, hyperkähler (Thomas, 2022).

The edges of the graph are correspondences. Riemann–Hilbert identifies $\mathbb{C}^\times$7 with $\mathbb{C}^\times$8 as complex analytic varieties in complex dimension one, matching Goldman’s symplectic form with the reduced Atiyah–Bott form. The non-abelian Hodge correspondence, in the Hitchin–Simpson–Corlette–Donaldson theorem, gives a diffeomorphism

$\mathbb{C}^\times$9

for compact Kähler $X$0, by solving the Hitchin–Simpson equations

$X$1

and forming the flat connection

$X$2

Narasimhan–Seshadri appears as the unitary edge, realized by the locus $X$3, where stable degree-zero bundles correspond to irreducible unitary representations (Thomas, 2022).

The global geometry of this graph is organized by hyperkähler rotation and the twistor family. The common smooth manifold carries complex structures $X$4 satisfying quaternionic relations, and Simpson’s $X$5-connections are expressed by

$X$6

with $X$7 giving the Dolbeault fiber and $X$8 the flat connection. In this sense, a nonabelian Hodge graph is not merely a diagram of equivalences; it is a hyperkähler manifold with several distinguished complex incarnations (Thomas, 2022).

2. Stack-theoretic and CoHA enrichments

At the level of stacks, the nonabelian Hodge graph becomes a triangle of moduli stacks for a smooth projective complex curve $X$9: the Dolbeault, Betti, and de Rham stacks $G$0, $G$1, and $G$2, together with their good moduli spaces. The principal theorem in this setting gives canonical isomorphisms of Borel–Moore homology

$G$3

compatible with direct-sum grading, direct sum functoriality, and Jordan–Hölder maps. The Betti–de Rham comparison is obtained via derived Riemann–Hilbert, and the Dolbeault–de Rham comparison uses the Hodge–Deligne moduli stack of $G$4-connections (Hennecart, 2023).

Each node of this stack-level graph carries a cohomological Hall algebra. For $G$5, one considers

$G$6

with convolution product defined through the correspondence stack of short exact sequences

$G$7

The resulting compatibility theorem states that the pushforwards along the Riemann–Hilbert homeomorphism $G$8 and the nonabelian Hodge homeomorphism $G$9 identify these CoHA objects as algebra objects. Consequently,

$M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$0

and the corresponding BPS algebras and BPS Lie algebras also coincide (Hennecart, 2023).

The $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$1-family is the mechanism that turns the graph into a relative object. The Hodge–Deligne moduli stack

$M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$2

parametrizes $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$3-connections satisfying the $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$4-Leibniz rule, with fiber at $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$5 equal to the Dolbeault stack and fiber at $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$6 equal to the de Rham stack. The relative CoHA

$M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$7

specializes to $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$8 at $M_B(X,G)=\mathrm{Hom}(\pi_1(X),G)//G,$9 and to $G$0 at $G$1, and over $G$2 it is analytically trivialized by the de Rham side. This supplies a one-parameter path in the graph that is both geometric and homological (Hennecart, 2023).

The stack-theoretic formulation also clarifies the role of $G$3-Calabi–Yau structures. The categories of local systems and of connections on a curve have Euler form

$G$4

and the $G$5-CY duality

$G$6

underlies the symmetry and PBW phenomena in the CoHA. In genus $G$7, one obtains PBW-type descriptions and free-generator statements for the BPS structures; in genus $G$8, the CoHA becomes commutative; in genus $G$9, all three moduli stacks reduce to $(E,\nabla)$0 (Hennecart, 2023).

3. Irregular, wild, and Stokes-theoretic graph constructions

In the wild setting, nonabelian Hodge graphs become literal combinatorial invariants extracted from irregular formal data. For a meromorphic connection on the Riemann sphere, one associates a finite diagram $(E,\nabla)$1, called the core nonabelian Hodge diagram, encoding the global Cartan matrix. The vertices are Stokes circles of the irregular class at $(E,\nabla)$2, the off-diagonal integers $(E,\nabla)$3 are edge multiplicities, and the diagonal terms $(E,\nabla)$4 encode twice the number of loops. The associated symmetric matrix is

$(E,\nabla)$5

If there are no loops and no negative edges, then $(E,\nabla)$6 is a symmetric generalized Cartan matrix and the diagram is a nonabelian Hodge graph. The construction is invariant under the Fourier–Laplace transform, so one may reduce to the affine line with a single singularity at infinity (Douçot, 29 Sep 2025).

Untwisted and twisted irregular types behave differently. In the untwisted case, every Stokes circle has ramification order $(E,\nabla)$7, the loop multiplicities vanish, and the off-diagonal multiplicities are nonnegative, so the core diagram is a graph. In the twisted case, loops and negative edges may appear, but twisted irregular classes can still produce genuine graphs. A basic example is the triangle with edge multiplicities $(E,\nabla)$8 arising from Stokes circles

$(E,\nabla)$9

for which all loop multiplicities vanish although the graph fails the ultrametric characterization of untwisted fission graphs (Douçot, 29 Sep 2025).

A central constraint is ultrametricity. For a decorated diagram $G$0 with ramification orders $G$1, the rescaled multiplicities

$G$2

satisfy the ultrametric condition

$G$3

together with the loop-edge inequality $G$4. In the untwisted case this becomes a characterization: a weighted graph is a fission graph if and only if every triangle is “acute isosceles,” meaning the two largest edge multiplicities coincide. The paper also proves that any simply-laced core nonabelian Hodge graph is a fission graph (Douçot, 29 Sep 2025).

For nonabelian Hodge spaces on the affine line, the graph is built from a core and legs. Given a formal diagonal irregular type $G$5 at $G$6, one takes one core node for each exponential factor circle $G$7, assigns multiplicity labels $G$8, and defines the adjacency matrix by

$G$9

where $(E,\phi)$0. In the untwisted case,

$(E,\phi)$1

Type $(E,\phi)$2 Dynkin legs are then attached using minimal markings of the formal monodromy conjugacy classes, and the dimension of the resulting wild character variety is read off from

$(E,\phi)$3

where $(E,\phi)$4 is the full dimension vector and $(E,\phi)$5 is the Cartan matrix (Boalch et al., 2019).

A complementary Betti-side construction uses embedded Stokes graphs on the real oriented blow-up of a punctured curve. Around each marked point, anti-Stokes directions determine punctures on halo circles, cilia connect these punctures to the boundary circles, and the annular sectors encode asymptotic regimes and filtration constraints. In the groupoid description, the loops $(E,\phi)$6 around halo punctures carry Stokes matrices $(E,\phi)$7, the loop $(E,\phi)$8 carries formal monodromy, and the total local monodromy factorizes as

$(E,\phi)$9

with global relation

$\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$0

This graph organizes filtered Stokes $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$1-local systems and the quasi-Hamiltonian construction of wild Betti moduli (Huang et al., 2024).

4. Fission trees, supernova graphs, and genus-zero representations

For genus zero nonabelian Hodge spaces, the basic discrete object is often not a graph directly but a fission tree. A local irregular class determines level data, common parts of Stokes circles, and pairwise fission exponents; these constitute the local fission datum. Two local irregular classes are admissible deformations of each other if and only if they have the same fission datum, and globally the admissible deformation class is determined by the fission forest together with conjugacy classes at the leaves. The resulting moduli are denoted $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$2 (Douçot, 2024).

The diagram invariant for a genus zero nonabelian Hodge space is built from modified formal data and consists of a core and legs. The core nodes are Stokes circles, the core edge multiplicities are defined by irregularities of $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$3 together with ramification data, and each node carries a leg encoding a conjugacy class. A key property is invariance under the full $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$4 action on connections: for any irreducible connection, the diagram $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$5 is preserved by the $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$6 orbit. The refined invariant is the enriched tree, or generic fission tree with its principal subtrees, and this extra fission data determines multiple “readings” of the same nonabelian Hodge space (Douçot, 2024).

If the generic fission tree has principal subtrees $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$7, then the pair $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$8 canonically determines $\phi\in H^0(X,\mathrm{End}(E)\otimes K_X),$9 nongeneric classes together with the generic class itself. The theorem states that these are exactly the $\mathbb{C}^\times$00 admissible deformation classes representing the same genus zero nonabelian Hodge space. In the simply-laced case, the partition of the core recovers the complete $\mathbb{C}^\times$01-partite core quiver of the supernova picture. Painlevé moduli furnish explicit examples: the framework organizes multiple Lax representations of Painlevé I–VI as different basic representations of one underlying nonabelian Hodge space (Douçot, 2024).

In the untwisted multiplicity-$\mathbb{C}^\times$02 case, fission trees admit an explicit enumeration. Let $\mathbb{C}^\times$03 be the number of fission trees of slope $\mathbb{C}^\times$04 with $\mathbb{C}^\times$05 leaves, and let $\mathbb{C}^\times$06 count trees of slope at most $\mathbb{C}^\times$07 with $\mathbb{C}^\times$08 leaves. Then

$\mathbb{C}^\times$09

and the generating functions satisfy the Euler-transform recursion

$\mathbb{C}^\times$10

The base cases are $\mathbb{C}^\times$11 for $\mathbb{C}^\times$12 and

$\mathbb{C}^\times$13

where $\mathbb{C}^\times$14 is the partition function. For example, the slope-$\mathbb{C}^\times$15 sequence begins

$\mathbb{C}^\times$16

for $\mathbb{C}^\times$17 (Boalch, 2024).

From a fission tree $\mathbb{C}^\times$18 of slope $\mathbb{C}^\times$19, one obtains a fission graph $\mathbb{C}^\times$20 whose vertices are the leaves and whose multiplicity between distinct leaves is $\mathbb{C}^\times$21, where $\mathbb{C}^\times$22 is the height of the nearest common ancestor. Supernova graphs are obtained by attaching legs to the core vertices of such a fission graph. In this formalism, a nonabelian Hodge graph is any supernova graph $\mathbb{C}^\times$23, and every supernova graph occurs as the diagram of a wild nonabelian Hodge moduli space. In the simply-laced case, the fission graphs are exactly the complete multipartite graphs (Boalch, 2024).

5. Analytic, gauge-theoretic, and anabelian reformulations

Analytic non-abelian Hodge theory recasts the graph in operator-algebraic terms. For a compact Kähler manifold $\mathbb{C}^\times$24, the Betti representation functor on Banach algebras is represented by a Fréchet completion $\mathbb{C}^\times$25, while the pluriharmonic functor on $\mathbb{C}^\times$26-algebras is represented by a pro-$\mathbb{C}^\times$27-bialgebra $\mathbb{C}^\times$28. The latter carries a canonical continuous $\mathbb{C}^\times$29-action

$\mathbb{C}^\times$30

making $\mathbb{C}^\times$31 a pro-$\mathbb{C}^\times$32-dynamical system. The graph then has nodes given by analytic completions, de Rham and Dolbeault representation spaces, pluriharmonic local systems, and twistor families, with edges given by Riemann–Hilbert, the de Rham and Dolbeault projections, and the twistor map

$\mathbb{C}^\times$33

In this setting, the graph records a pure Hodge structure of weight $\mathbb{C}^\times$34 and its twistor splitting (Pridham, 2012).

A gauge-theoretic version arises from the Kapustin–Witten equations on a compact Kähler surface. The node at $\mathbb{C}^\times$35 is the moduli of solutions of the Simpson system

$\mathbb{C}^\times$36

while the node at $\mathbb{C}^\times$37 is the moduli of pluri-harmonic systems equivalent to semisimple local systems. On compact Kähler surfaces with $\mathbb{C}^\times$38, the paper proves that the $\mathbb{C}^\times$39 Kapustin–Witten moduli are real-analytically isomorphic, away from singular loci, to the $\mathbb{C}^\times$40 Kapustin–Witten moduli, with the nonabelian Hodge correspondence furnishing the intermediate edge between Higgs and flat moduli. On closed four-manifolds with structure group $\mathbb{C}^\times$41 or $\mathbb{C}^\times$42, the expected dimensions are controlled by index formulas; under $\mathbb{C}^\times$43, both sides have expected dimension $\mathbb{C}^\times$44 (Liu et al., 2020).

A further extension replaces moduli of bundles by the pro-algebraic completion of the fundamental group. For a connected compact Kähler manifold $\mathbb{C}^\times$45, the Tannakian group

$\mathbb{C}^\times$46

carries a Simpson-induced outer $\mathbb{C}^\times$47-action

$\mathbb{C}^\times$48

whose derivative gives a nonnegative grading

$\mathbb{C}^\times$49

In this Hodge-theoretic anabelian picture, a vertex is a triple $\mathbb{C}^\times$50, and an edge is a $\mathbb{C}^\times$51-equivariant open homomorphism of pro-algebraic fundamental groups. For smooth projective hyperbolic curves over $\mathbb{C}^\times$52, dominant morphisms $\mathbb{C}^\times$53 are in bijection with such equivariant open homomorphisms, and analogous statements are proved for compact ball quotients (Wang, 6 Mar 2026).

6. Geometry, applications, and classification themes

The graph language is effective because it packages multiple geometric structures simultaneously. On the foundational Dolbeault side, the Hitchin map

$\mathbb{C}^\times$54

makes the Higgs moduli into an algebraically completely integrable system, with generic fibers given by torsors for abelian varieties via spectral curves. Through the non-abelian Hodge diffeomorphism, this integrable structure is transported to de Rham and Betti moduli, and for split real groups the resulting Higgs sections produce Hitchin components generalizing Teichmüller space (Thomas, 2022).

In the irregular case, graph invariants control quiver-type realizations of wild character varieties. For a meromorphic connection on $\mathbb{C}^\times$55 that is tame at finite points and untwisted at $\mathbb{C}^\times$56, Boalch and Hiroe–Yamakawa construct an explicit graph such that an open subset of the wild character variety, or of the corresponding nonabelian Hodge moduli space, is isomorphic to the Nakajima quiver variety of the doubled quiver of that graph, while the full moduli space is a generalized multiplicative quiver variety. Twisting at infinity enlarges the class of allowable graphs, but ultrametric constraints and Fourier–Laplace invariance remain decisive classification tools (Douçot, 29 Sep 2025).

A computational incarnation appears in the study of opers on $\mathbb{C}^\times$57. There, Stokes graphs in rank $\mathbb{C}^\times$58 and spectral networks in rank $\mathbb{C}^\times$59 function as nonabelian Hodge graphs whose combinatorics determine spectral coordinates $\mathbb{C}^\times$60 from determinant invariants of subdominant solutions. These coordinates satisfy the Gaiotto–Moore–Neitzke integral equations, encode Stokes data of opers and Hitchin-section flat connections, and serve as Darboux coordinates for Hitchin’s hyperkähler metric. Numerical experiments on polynomial differentials support the conjectural formulas for both Stokes data and the restriction of Hitchin’s metric to the Hitchin section (Dumas et al., 2020).

A distinct surface-theoretic use of the term occurs in Reider’s nonabelian Jacobian of a smooth projective surface. There, weighted trivalent graphs $\mathbb{C}^\times$61 encode the orthogonal decomposition of $\mathbb{C}^\times$62, the triangular pieces $\mathbb{C}^\times$63 of the Higgs or multiplication operator, and the toric Fano variety $\mathbb{C}^\times$64 defined by

$\mathbb{C}^\times$65

These graphs simultaneously record period-map data, reductive Lie algebra structure, and Calabi–Yau hyperplane sections, showing that “nonabelian Hodge graph” can also denote a graph governing Hodge-like and Higgs-theoretic structures on moduli attached to surfaces (Reider, 2011).

Across these settings, the phrase denotes a family of compatible constructions rather than a single rigid object. What remains common is the use of graph-theoretic data to organize nonabelian Hodge correspondences, to encode symplectic and hyperkähler structures, to control deformation classes and quiver modularity, and to translate between holomorphic, flat, Stokes-theoretic, homological, and representation-theoretic descriptions of the same underlying geometry (Thomas, 2022).