Sparse Modular Forms, Lattices, and Codes

TL;DR

The paper develops a unified sparseness framework for holography-inspired objects by analyzing the large-$n$ limits of modular forms, lattices, and codes via the scaled free energy ${\cal F}_n(y)$. It shows extremal modular forms and Eisenstein constructions yield sparse families, and introduces a generalized sparseness notion for Construction A lattices through $\theta_3$-sparseness, linking lattice theta functions to code weight enumerators. By examining self-dual Reed–Muller codes and their lattices, the authors derive bounds and evidence suggesting subexponential light-state growth, supporting potential sparsity in large-$c$ chiral lattice CFTs. The results connect Siegel–Weil averages, BEC capacity bounds, and code-based lattice constructions to identify concrete pathways toward sparse lattice CFTs and related holographic structures.

Abstract

Motivated by sparseness conditions for holographic CFTs, we investigate sparseness of modular forms, lattices, and codes. For this we investigate the free energy of such objects as their weight, dimension or size goes to infinity. We construct families of modular forms that are sparse, such as the Eisenstein series $E_{2k}(τ)$. We then investigate lattices that come from codes and introduce a sparseness condition for such lattices. We investigate the limit of lattices constructed from self-dual Reed-Muller codes and provide evidence that they are sparse in this sense.

Theorems & Definitions (11)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Example 3.1
• Lemma 3.1
• proof
• Lemma 3.2