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Lattice points in polytope boundaries and formal geometric quantization of singular Calabi Yau hypersurfaces in toric varieties

Jonathan Weitsman

TL;DR

The paper establishes that the number of lattice points on the boundary of a positive integer dilation of a Delzant polytope, $R_{ abla}(k)$, is a polynomial in $k$ of degree $n-1$ with lacunarity, and provides an explicit formula via weighted Euler–Maclaurin operators using $ ext{A}$-hat functions. It connects these boundary counts to formal geometric quantization of singular Calabi–Yau hypersurfaces in toric varieties, yielding a KP-type operator expression for $ ext{dim }Q_k(Z)$ that mirrors the classical KP formula for interior lattice points. The work extends the theory to simple polytopes/orbifold toric varieties and discusses implications for mirror symmetry, providing concrete low-dimensional examples for unit simplices in dimensions 2–4. These results unify discrete boundary counting with toric CY geometry, offering a principled method to compute boundary lattice counts and highlighting lacunarity as a structural feature tied to quantization.

Abstract

We show that the number of lattice points in the boundary of a positive integer dilate of a Delzant integral polytope is a polynomial in the dilation parameter, analogous to the Ehrhart polynomial giving the number of lattice points in a lattice polytope. We give an explicit formula for this polynomial, analogous to the formula of Khovanskii-Pukhlikov for the Ehrhart polynomial. These counting polynomials satisfy a lacunarity principle, the vanishing of alternate coefficients, quite unlike the Ehrhart polynomial. We show that formal geometric quantization of singular Calabi Yau hypersurfaces in smooth toric varieties gives this polynomial, in analogy with the relation of the Khovanskii-Pukhlikov formula to the geometric quantization of toric varieties. The Atiyah-Singer theorem for the index of the Dirac operator gives a moral argument for the lacunarity of the counting polynomial. We conjecture that similar formulas should hold for arbitrary simple integral polytope boundaries.

Lattice points in polytope boundaries and formal geometric quantization of singular Calabi Yau hypersurfaces in toric varieties

TL;DR

The paper establishes that the number of lattice points on the boundary of a positive integer dilation of a Delzant polytope, , is a polynomial in of degree with lacunarity, and provides an explicit formula via weighted Euler–Maclaurin operators using -hat functions. It connects these boundary counts to formal geometric quantization of singular Calabi–Yau hypersurfaces in toric varieties, yielding a KP-type operator expression for that mirrors the classical KP formula for interior lattice points. The work extends the theory to simple polytopes/orbifold toric varieties and discusses implications for mirror symmetry, providing concrete low-dimensional examples for unit simplices in dimensions 2–4. These results unify discrete boundary counting with toric CY geometry, offering a principled method to compute boundary lattice counts and highlighting lacunarity as a structural feature tied to quantization.

Abstract

We show that the number of lattice points in the boundary of a positive integer dilate of a Delzant integral polytope is a polynomial in the dilation parameter, analogous to the Ehrhart polynomial giving the number of lattice points in a lattice polytope. We give an explicit formula for this polynomial, analogous to the formula of Khovanskii-Pukhlikov for the Ehrhart polynomial. These counting polynomials satisfy a lacunarity principle, the vanishing of alternate coefficients, quite unlike the Ehrhart polynomial. We show that formal geometric quantization of singular Calabi Yau hypersurfaces in smooth toric varieties gives this polynomial, in analogy with the relation of the Khovanskii-Pukhlikov formula to the geometric quantization of toric varieties. The Atiyah-Singer theorem for the index of the Dirac operator gives a moral argument for the lacunarity of the counting polynomial. We conjecture that similar formulas should hold for arbitrary simple integral polytope boundaries.
Paper Structure (12 sections, 5 theorems, 61 equations, 5 figures)

This paper contains 12 sections, 5 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The lattice point counting polynomial $R_{\partial\Delta}$ for polytope boundaries
  3. Relation to Geometric Quantization of singular Calabi Yau hypersurfaces in toric varieties
  4. Structure of the paper
  5. The weighted Euler-Maclaurin formulas and the counting polynomial $R_{\partial\Delta}(k)$
  6. Some remarks and computations
  7. Simple polytopes and orbifold toric varieties
  8. Mirror Symmetry
  9. Some explicit calculations
  10. The degeneration of a torus in $CP^2$
  11. Singular K3 surface in $CP^3$
  12. Singular quintic in $CP^4$

Key Result

Theorem 1.1

Let $\Delta$ be any lattice polytope. Then

Figures (5)

  • Figure 1: The polytope $\Delta_2$
  • Figure 2: The singular Calabi Yau $Z_1$ as a degeneration of a torus
  • Figure 3: The polytope $\Delta_3$
  • Figure 4: The polytope $2\Delta_2$
  • Figure 5: The polytope $\Delta_4$

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 2.3
  • Theorem 2.5