Çarks: Modular Graphs & Their Applications

• Çarks are combinatorial and topological graphs that encode moduli spaces, algebraic curves, binary quadratic forms, and subgroup data for the modular group.
• Stable modular graphs stratify the Deligne–Mumford moduli spaces and can be enumerated computationally, reflecting the complex degeneration of algebraic curves.
• Infinite modular çarks represent equivalence classes of indefinite binary quadratic forms and connect arithmetic reduction theory with geometric group theory.

Çarks, also known as modular or stable graphs, are combinatorial and topological objects that encode the structure of moduli spaces, the classification of algebraic curves, the arithmetic of binary quadratic forms, and subgroup data for the modular group. Multiple distinct definitions, all under the umbrella term “çark,” arise in the literature: finite stable modular graphs model degenerations of algebraic curves and stratify the Deligne–Mumford moduli spaces, while infinite dessins-like modular graphs represent equivalence classes of indefinite binary quadratic forms and surface group quotients. Both finite and infinite variants play central roles in algebraic geometry, geometric group theory, arithmetic, and mathematical physics (Maggiolo et al., 2010, Uludağ et al., 2015, &&&2&&&).

1. Stable Modular Graphs and Their Role in Moduli of Curves

A stable modular graph of type $(g,n)$ is a colored, undirected multigraph $\Gamma = (V, E)$ equipped with the following vertex data: an integer genus $g_v \geq 0$ and a non-negative integer $n_v$ of marked legs for each $v \in V$, and a multiset $E \subset (V \times V)/\Sigma_2$ of edges. The degree of $v$ is defined as $$\deg(v) = 2\,\mathrm{mult}(v,v) + \sum_{w \neq v}\mathrm{mult}(v,w)$$. Stability requires each genus-zero vertex to carry at least three “special points” (nodes or marked points): $2g_v - 2 + \deg(v) + n_v > 0$. Global invariants are given by $g = \sum_{v\in V} g_v + |E| - |V| + 1$ and $n = \sum_{v \in V} n_v$. The existence of a stable graph for $(g,n)$ demands $2g + n - 2 > 0$.

These graphs provide the combinatorial encoding of the boundary strata of the Deligne–Mumford compactification $\Mbar_{g,n}$, where each stratum corresponds to a locus of stable, nodal curves with prescribed topological type. The dual graph of a stable curve $C$ mirrors its irreducible components, nodes, and markings: vertices represent components (weighted by their genera), edges correspond to nodes, and legs to markings. Every stable curve gives rise to a stable graph, and every stable graph arises as the dual graph of some stable curve of type $(g,n)$ (Maggiolo et al., 2010).

2. Enumeration and Representation of Stable Graphs: The Boundary Program

Enumeration of stable graphs up to isomorphism is highly nontrivial due to the combinatorial explosion as $g$ and $n$ increase. The program boundary formalizes this enumeration. Its workflow:

• Fixes $K = |V|$, the number of vertices, within $1 \leq K \leq 3g-3+n$.
• Recursively assigns ordered sequences for genera, number of legs, and loops, bounded by sum-constraints and early stability checks.
• Constructs symmetric adjacency matrices $a_{ij} = \mathrm{mult}(i,j)$ to specify edge multiplicities, enforcing connectedness and stability, and organizes columns in lex order to avoid redundant cases.
• Prunes isomorphic graphs using nauty.

Empirically, the number of stable graphs grows exponentially with $g+n$, and the method produces exactly one representative per isomorphism class (Maggiolo et al., 2010).

Table: Key Structural Data for Stable Modular Graphs

Invariant	Formula	Condition / Role
Global genus	$g = \sum_v g_v + |E| - |V| + 1$	Overall topological type
Markings	$n = \sum_v n_v$	Number of unordered markings
Stability	$2g_v-2+\deg(v)+n_v>0$	Allows only stable components
Nonemptiness	$2g + n - 2 > 0$	Ensures nontrivial moduli space
Stratum dimension	$\dim(\Delta_\Gamma) = 3g-3 + n - |E|$	Codimension by number of nodes

Implementationally, the graph is encoded by integer vectors and an adjacency matrix, with sharp pruning strategies yielding fast generation and low memory cost (Maggiolo et al., 2010).

3. Çarks as Infinite Modular Graphs: Arithmetic and the Modular Group

A structurally distinct incarnation of the “çark” arises in the theory of indefinite binary quadratic forms: each primitive equivalence class of forms of discriminant $\Delta > 0$ corresponds to an infinite ribbon graph, the çark $C_M = \langle M \rangle \backslash F$, where $F$ is the Farey tree and $M \in \mathrm{PSL}_2(\mathbb{Z})$ is the fundamental automorphism of the class. The çark is a bipartite infinite graph embedded in an annulus, containing a unique cycle called the spine. Each edge of the çark corresponds to a binary quadratic form in the class, with reduced forms marking distinguished edges on the spine. Gauss reduction on quadratic forms is realized as moving along the spine of the çark; periodicity of this traversal recovers the classical reduction cycle (Uludağ et al., 2015).

Ambiguous and reciprocal classes admit symmetries (reflections and involutions) in their çarks, tying group-theoretic structure to quadratic form arithmetic. The length of the spine equals the period of the continued fraction expansion of a root of the quadratic form, illuminating deep classical correspondences (Uludağ et al., 2015).

4. Modular Coset Graphs, Congruence Subgroups, and Dessins

In a further generalization, finite modular coset graphs—often called çarks in mathematical physics and number theory—arise as Schreier–Cayley graphs of finite-index subgroups $\Gamma$ of $\mathrm{PSL}_2(\mathbb{Z})$. Vertices correspond to the right-cosets $g_i\Gamma$, and edges to the action of generators $S$ and $T$. By collapsing each $T$-triangle to a trivalent vertex, one obtains a finite cubic (trivalent) graph whose structure reflects the modular data of $\Gamma$. In the torsion-free, genus-zero case, these trivalent graphs correspond bijectively to dessins d’enfant with explicit ramification data, and serve as skeletons for physical quiver gauge theories (He et al., 2012).

For example, the nine torsion-free genus-zero congruence subgroups of index $24$ correspond to cubic graphs with $8$ vertices and $12$ edges, closely linked to extremal elliptic K3 surfaces. The ramification structure of the modular $J$-function matches the local vertex and edge degrees of the coset çark, and the dessin decorations encode arithmetic invariants (He et al., 2012).

5. Çarks, Moduli Space Stratification, and Geometric-Topological Connections

Stable modular graphs play a central role in the stratification of the boundary $\partial \overline{\mathcal{M}}_{g,n}$ of the moduli space: each stable graph $\Gamma$ defines a stratum

$$\Delta_\Gamma \cong \left(\prod_{v \in V} \overline{\mathcal{M}}_{g_v, n_v + \deg(v)}\right)/\mathrm{Aut}(\Gamma)$$

of complex dimension $3g-3+n - |E|$. The union of these strata covers the entire boundary. Graph isomorphisms correspond to automorphisms of the stratum, and the dimension lowers by one for each node (edge) added to the graph.

Conversely, the infinite modular çark encapsulates the combinatorics of class group dynamics and geodesic flow on the modular surface, providing a unified framework for both geometric and arithmetic reduction theories (Maggiolo et al., 2010, Uludağ et al., 2015, He et al., 2012).

6. Examples, Symmetry, and Enumeration

Examples for small values illustrate the diversity and constraints:

• For $(g,n) = (1,2)$ the stable modular graphs are:
  • ($a$) A single vertex ($g_v=1$) with one loop and two legs.
  • ($b$) Two genus-zero vertices joined at one node, each with one marking, stabilized by virtual half-edges.

Enumeration of connected cubic modular coset graphs is asymptotically $d_n \sim (6n)!/((3n)! 288^n) e^{-2}$ for $2n$ vertices without spikes. Allowing graphs with univalent “spikes” increases the combinatorial count, described by explicit generating functions involving character sums mod $2,3$ (He et al., 2012).

Symmetries in çarks correspond to ambiguous and reciprocal classes:

• Reflection symmetry in the conformal annulus corresponds to ambiguous quadratic form classes.
• Dihedral symmetries reflect reciprocal classes, realized by conjugating the fundamental automorphism.

Periodic çarks (those arising from non-primitive forms $M = N^k$) correspond to periodic necklaces in combinatorial models and encode coverings between çarks of differing periodicity.

7. Applications and Interrelations

Çarks unify combinatorial, topological, and arithmetic perspectives across several domains:

• In algebraic geometry, they encode the stratification of the boundary of $\overline{\mathcal{M}}_{g,n}$, specifying loci of stable curves by combinatorial type (Maggiolo et al., 2010).
• In number theory, they provide a combinatorial model for the arithmetic of binary quadratic forms, continued fractions, and class group dynamics (Uludağ et al., 2015).
• In mathematical physics, trivalent modular coset çarks serve as skeletons for quivers in four-dimensional $\mathcal{N}=2$ gauge theories, relate to extremal K3 surfaces, and describe ramification of modular functions (He et al., 2012).

These correspondences allow the translation of classical reduction theory, modular geometry, and modern gauge-theoretic constructions into the universal language of modular (çark) graphs.

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Key References:

• "Generating stable modular graphs" (Maggiolo et al., 2010)
• "Binary quadratic forms as dessins" (Uludağ et al., 2015)
• "N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces" (He et al., 2012)