Critical Even Unimodular Lattices in the Gaussian Core Model

Abstract

Abstract We consider even unimodular lattices that 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.