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Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, Maryna Viazovska

Abstract

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

Universal optimality of the $E_8$ and Leech lattices and interpolation formulas

Abstract

We prove that the root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions and , respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function from the values and radial derivatives of and its Fourier transform at the radii for integers in and in . To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.

Paper Structure

This paper contains 40 sections, 38 theorems, 266 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Potential energy minimization
  3. Universal optimality
  4. Linear programming bounds
  5. Interpolation
  6. Proof techniques
  7. Origins of the argument
  8. Organization of the paper
  9. Preliminaries and background on modular forms
  10. Modular and quasimodular forms
  11. Modular forms for $\Gamma(1)=\mathop{\textup{SL}}\nolimits_2({\mathbb Z})$
  12. Modular forms for $\Gamma(2)$
  13. The modular function $\lambda$ and its logarithm
  14. Elliptic integrals
  15. Radial Schwartz functions
  16. ...and 25 more sections

Key Result

Theorem 1.4

The $E_8$ root lattice and the Leech lattice are universally optimal in ${\mathbb R}^8$ and ${\mathbb R}^{24}$, respectively. Furthermore, they are unique among periodic configurations, in the following sense. Let ${\mathcal{C}}$ be $E_8$ or the Leech lattice, and let ${\mathcal{C}}'$ be any periodi

Figures (3)

  • Figure 4.1: The fundamental domain $\mathcal{D}$ for $\Gamma(2)$ defined in \ref{['domainD']} is shaded, with the fundamental domain $\mathcal{F}$ for $\mathrm{SL}_2(\mathbb{Z})$ defined in \ref{['domainF']} shaded darker (note that $\mathcal{F} \subseteq \mathcal{D}$). The six marked points in the interior of $\mathcal{D}$ are the images of a point $\tau \in \mathcal{F}$.
  • Figure 5.3: The contours $\alpha_{-1}$, $\alpha_0$, $\alpha_1$, $\alpha_\infty$, $ST\alpha_{-1}$, and $ST^{-1}\alpha_{1}$ superimposed on the fundamental domain $\mathcal{D}$ shown in Figure \ref{['fig:domainD']}. All four contours on the left share a common terminus at $6i/5$, with $\alpha_{-1}$, $\alpha_0$, and $\alpha_1$ ending there and $\alpha_\infty$ starting there, while those on the right go from $i \infty$ to $-1/(\pm1+6i/5)$.
  • Figure 5.4: The contours $C_1$, $C_2$, and $C_3$ (shown here for $ST\alpha_{-1}$) keep distance at least $1/4$ from any ${\mathbb Z}$-translate of $\tau$.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 33 more