Modularity of counting functions of convex planar polygons with rationality conditions

TL;DR

The paper links convex polygon counting to indefinite theta functions arising in homological mirror symmetry of elliptic curves. By imposing positivity and integrality (rationality) conditions on the quadratic form $Q$ and the cone data, it shows that regularized counting functions are mock modular, with completions built from generalized error functions that satisfy Vignères’ equation, yielding Jacobi-form–type behavior. The authors develop a comprehensive toolkit—generalized error functions, deformations $Q_t$, and cone decompositions—to prove modularity in both simplicial and non-simplicial settings, establishing weight and depth bounds that depend on polygon size and cone geometry. The work provides explicit modular structures for polygon-counting generating functions, connects degenerations to lower-dimensional theta functions on faces, and extends known results from small $N$ to wide classes of convex polygons, thereby enriching the interface between polygon enumeration, mock modularity, and HMS for elliptic curves.

Abstract

We study counting functions of planar polygons arising from homological mirror symmetry of elliptic curves. We first analyze the signature and rationality of the quadratic forms corresponding to the signed areas of planar polygons. Then we prove the convergence, meromorphicity, and mock modularity of the counting functions of convex planar polygons satisfying certain rationality conditions on the quadratic forms.

Figures (8)

• Figure 1: The outer angle at the vertex labelled by $k$ of the polygon is $\theta_{k}$, for $1\leq k \leq N$.
• Figure 2: From left to right: the polygons are simple and convex, complex, simple, complex.
• Figure 3: The shaded region is the cross section for $\mkern 1.5mu\overline{\mkern-1.5mu\Delta\mkern-1.5mu}\mkern 1.5mu$, the quadric is the cross section for $D_0$. The first case arises from consideration of counting functions of convex polytopes which involve no isotropic vectors, the second one from that of convex polytopes which involve finitely many isotropic vectors, the final one from that of concave polytopes which can involve infinitely many isotropic vectors.
• Figure 4: Slicing $\mathbb{R}^{N-2}$ using $\bm{\rho}$ and $\bm{\eta}^{\perp}$.
• Figure 5: A cross section of the polyhedron $R_{t}(C_{T^{c}\perp C_{T}})$.

Theorems & Definitions (79)

• Definition 1.1
• Theorem 1.2
• Example 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• Remark 2.4
• Corollary 2.5
• Remark 2.6