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Charge functions for odd dimensional partitions

Shang Xiang, Hao Feng, Keyou Zhuo, Tian-Shun Chen, Kilar Zhang

TL;DR

This work proposes a universal charge-function form for odd-dimensional partitions, $\\psi_{\\Delta^{(n)}}(u)=\\psi_0(u)\\\\psi'_{\\Delta^{(n)}}(u)$ with $\\psi_0(u)=\frac{1}{u}$, and validates it for $n=5$ while conducting numerical tests for $n=7,9$. It extends known 2D/3D/4D charge expressions by incorporating cluster contributions from even-sized box groups, ensuring the pole structure matches the projected addable/removable boxes. The authors provide a rigorous inductive framework (supported by Lemmas 1–4, with Lemma 5 central) and a detailed proof outline, complemented by analytic checks and Monte Carlo verifications. The results advance the construction of BPS algebras for Calabi–Yau manifolds in higher odd dimensions and identify specific obstacles for even dimensions, signaling a path toward broader algebraic realizations and physical applications.

Abstract

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to construct the eigenvalue of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

Charge functions for odd dimensional partitions

TL;DR

This work proposes a universal charge-function form for odd-dimensional partitions, with , and validates it for while conducting numerical tests for . It extends known 2D/3D/4D charge expressions by incorporating cluster contributions from even-sized box groups, ensuring the pole structure matches the projected addable/removable boxes. The authors provide a rigorous inductive framework (supported by Lemmas 1–4, with Lemma 5 central) and a detailed proof outline, complemented by analytic checks and Monte Carlo verifications. The results advance the construction of BPS algebras for Calabi–Yau manifolds in higher odd dimensions and identify specific obstacles for even dimensions, signaling a path toward broader algebraic realizations and physical applications.

Abstract

To construct a BPS algebra with representations furnished by n-dimensional partitions, the first step is to construct the eigenvalue of the Cartan operators acting on them. The generating function of the eigenvalues is called the charge function. It has an important property that for each partition, the poles of the function correspond to the projection of the boxes which can be added to or removed from the partition legally. The charge functions of lower dimensional partitions, i.e., Young diagrams for 2D, plane partitions for 3D and solid partitions for 4D, are already given in the literature. In this paper, we propose an expression of the charge function for arbitrary odd dimensional partitions and have it proved for 5D case. Some explicit numerical tests for 7D and 9D case are also conducted to confirm our formula.

Paper Structure

This paper contains 27 sections, 5 theorems, 101 equations, 5 figures.

Table of Contents

  1. Introduction
  2. $n$-dimensional partitions
  3. Conjecture: charge functions for odd dimensional partitions
  4. Ordinary partitions
  5. Plane partitions
  6. Solid partitions
  7. Conjecture formula for arbitrary odd dimension:
  8. Definitions
  9. Some useful sets
  10. Potential Function
  11. $\textbf{$d$}$-neighbor
  12. Bisect operation
  13. Hypercube
  14. Surface set
  15. Conjecture
  16. ...and 12 more sections

Key Result

Lemma 1

(proved in Appendix subsec:Lemma1) For $\forall$$\Delta^{(n)}$ and $\vec{\square}$, after bisect operation $L$, the remaining boxes in $\Delta^{(n)}-L$ still form a partition.

Figures (5)

  • Figure 1: Schematic diagram of definition in 2D.
  • Figure 2: All six unique partitions of the hypercube ${HC}^{(2)}$ in 5D.
  • Figure 3: Numerical results for $n=5$. Upper panel: Distributions of unique partition counts vs. box numbers for ${HC}^{(4)}$ and ${HC}^{(5)}$, respectively. Lower panel: Target pole orders for ${HC}^{(4)}$ and ${HC}^{(5)}$ partitions with different box numbers. Bubble size denotes the count of unique partitions for each (box number, pole order) pair. For $d=4$, pole order=1 only for 15/16 boxes (fully occupied/one missing box); for $d=5$, pole order=1 for 0/1/31/32 boxes.
  • Figure 4: Monte Carlo sampling results for 7D and 9D case, where the sample results are in good agreement with the description of Lemma 5.
  • Figure 5: Visualization of a special $n=5$ partition (200 boxes). Each subplot shows its projection onto $\vec{e}_1,\vec{e}_2,\vec{e}_3$, with horizontal rightward as increasing $\vec{e}_4$ and vertical downward as increasing $\vec{e}_5$. Red/green squares denote the positions which can add new boxes or remove existing boxes (melting rule), and black dots represent simple poles of the charge function. Their perfect coincidence confirms consistent judgments between the two methods, verifying the universality of our method for general cases.

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • proof
  • proof
  • proof