Universal optimality of the $E_8$ and Leech lattices and interpolation formulas (1902.05438v3)

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.

• The paper establishes universal optimality for the E8 and Leech lattices by proving they minimize potential energy for any completely monotonic squared-distance function.
• It employs advanced linear programming bounds and new interpolation formulas for radial Schwartz functions to extend sphere packing methodologies.
• The study uses rigorous computational techniques, including interval arithmetic and modular forms, to link discrete geometry with continuous analysis for future research applications.

Overview of Universal Optimality of the $E_8$ and Leech Lattices

The paper establishes the universal optimality of the $E_8$ lattice in eight-dimensional space and the Leech lattice in 24-dimensional space for potential energy minimization. These lattices are shown to minimize energy for any potential function that is a completely monotonic function of squared distance, including inverse power laws and Gaussian potentials. This discovery significantly extends their known optimality as sphere packings to broader classes of interactions.

To prove universal optimality, the authors employ linear programming bounds, building on previous work in which bounds for sphere packing were obtained using special function constructions. The authors construct new interpolation formulas for radial Schwartz functions, inspired by Viazovska's methods in solving the sphere packing problem. This interpolation reconstructs a function from the values and derivatives at specific points, namely the radii $\sqrt{2n}$ for integers $n \ge 1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$.

Numerical and Analytical Contributions

1. Sharper Bounds and Proof Techniques: The paper employs linear programming bounds established via analytic techniques, embracing functions of radial variables only. The $E_8$ and Leech lattices exhibit unique configurations that satisfy the conditions where bounds become equal, revealing unmatched minimal energy configurations for a multitude of potential functions.
2. Interpolative Framework: The interpolation basis, integral for solving these bounds, reveals a conceptual leap from finite-dimensional polynomial interpolation to infinite-dimensional function spaces. The functions are expressed using quasimodular forms and the Hauptmodul $\lambda$ for $\Gamma(2)$, extending the results of Viazovska and others.
3. Computational Considerations: Rigorous computational methods, including interval arithmetic, handle complex modular forms and elliptic integrals. These computational insights ensure accurate evaluations necessary for the exacting bounds demonstrated.

Theoretical Implications and Uniqueness

Their method attests to the rarity and precedence of universal optima, as these lattices remain optimal even when potential functions permit long-range interactions. The explicit relationship of energy minimization to number-theoretic special functions, such as Epstein zeta functions and theta series, shows the deep connection between discrete geometry and continuous analysis.

Furthermore, the uniqueness of these lattices as optimal configurations is established through a careful analysis of their structural properties, emphasizing that no other periodic configuration can achieve the same energy minimization across all monotonic potential functions.

Future Developments

The findings open a path for exploring universal optima in other dimensions and potential constructions, as well as applications in condensed matter physics and material science, where understanding particle arrangements subject to diverse interactions can be advantageous. It also hints at potential generalizations for other densely packed configurations and error-correcting codes.

Overall, this paper not only demonstrates significant progress in understanding high-dimensional lattice structures but also formulates a versatile approach to tackling such questions analytically and computationally, setting a stage for future explorations in mathematics and theoretical physics.