Critical even unimodular lattices in the Gaussian core model

TL;DR

This paper investigates the critical points of even unimodular lattices under Gaussian potential energy, identifying their Morse indices and discovering local maxima and non-critical lattices in higher dimensions.

Contribution

It provides a comprehensive analysis of criticality and Morse indices of even unimodular lattices up to dimension 24 and extends findings to higher dimensions, revealing new lattice behaviors.

Findings

All even unimodular lattices up to dimension 24 are critical.

Lattices in dimension 32 include local maxima.

Beyond dimension 32, non-critical lattices appear.

Abstract

We consider even unimodular lattices which are critical for potential energy with respect to Gaussian potential functions in the manifold of lattices having point density 1. All even unimodular lattices up to dimension 24 are critical. We show how to determine the Morse index in these cases. While all these lattices are either local minima or saddle points, we find lattices in dimension 32 which are local maxima. Also starting from dimension 32 there are non-critical even unimodular lattices.