Lattice invariants from the heat kernel

TL;DR

This paper introduces new modular form invariants for integer lattices derived from heat kernel analysis, capturing geometric properties like lengths and angles, and providing a novel approach to lattice classification.

Contribution

It develops a method to derive lattice invariants from heat flux using harmonic polynomials, resulting in modular form invariants that depend only on geometric features.

Findings

New lattice invariants as modular forms from heat flux

Invariants depend solely on lengths and angles in the lattice

Provides a novel approach to lattice classification

Abstract

We derive lattice invariants from the heat flux of a lattice. Using systems of harmonic polynomials, we obtain sums of products of spherical theta functions which give new invariants of integer lattices which are modular forms. In particular, we show that the modular forms $\Theta_{nn,\Lambda}$ depend only from lengths and angles in the lattice.