Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Charge functions for all dimensional partitions

Hao Feng, Tian-Shun Chen, Kilar Zhang

TL;DR

This work addresses the explicit construction of charge functions $\psi(u)$ for $n$-dimensional partitions that realize BPS Hilbert spaces on toric Calabi–Yau manifolds. It extends prior odd-dimensional results by proposing a universal conjecture for all even $n$ and proves the 6D case rigorously, with numerical verification in 8D. The authors provide detailed, dimension-dependent factorized formulas involving $p$-box clusters and $\varphi$-factors, and introduce a potential function to characterize pole structures, establishing precise criteria for addable/removable positions. By unifying the charge functions across all dimensions, the work enables systematic exploration of the associated BPS algebras in higher-dimensional Calabi–Yau geometries.

Abstract

The charge functions for n-dimensional partitions are known for n=2,3,4 in the literature. We give the expression for arbitrary odd dimension in a recent work, and now further conjecture a formula for all even dimensional cases. This conjecture is proved rigorously for 6D, and numerically verified for 8D.

Charge functions for all dimensional partitions

TL;DR

This work addresses the explicit construction of charge functions for -dimensional partitions that realize BPS Hilbert spaces on toric Calabi–Yau manifolds. It extends prior odd-dimensional results by proposing a universal conjecture for all even and proves the 6D case rigorously, with numerical verification in 8D. The authors provide detailed, dimension-dependent factorized formulas involving -box clusters and -factors, and introduce a potential function to characterize pole structures, establishing precise criteria for addable/removable positions. By unifying the charge functions across all dimensions, the work enables systematic exploration of the associated BPS algebras in higher-dimensional Calabi–Yau geometries.

Abstract

The charge functions for n-dimensional partitions are known for n=2,3,4 in the literature. We give the expression for arbitrary odd dimension in a recent work, and now further conjecture a formula for all even dimensional cases. This conjecture is proved rigorously for 6D, and numerically verified for 8D.
Paper Structure (14 sections, 1 theorem, 47 equations, 3 figures)

This paper contains 14 sections, 1 theorem, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. partitions
  3. charge functions
  4. Formula for arbitrary odd dimension
  5. Formula for arbitrary even dimension:
  6. Potential Function
  7. Proof
  8. Some analytically check of Lemma 1
  9. Numerical Verification
  10. Conclusion
  11. Lower Dimensions
  12. Case $n=2$: Young Diagrams
  13. Case $n=3$: Plane Partitions
  14. Case $n=4$: Solid Partitions

Key Result

Lemma A

(proved for 6D and discussed for higher n) For $\forall$$\Delta^{(n)}\subseteq HC^{(d)}(\vec{0},\{\vec{e}_{n_i}\})$,

Figures (3)

  • Figure 1: Schematic of a 4D partition. Green and red designate addable and removable box positions, respectively, while black dots indicate sites where our method predicts a box (either red or green) to be. The subfigures show slices at $l_4 = 0, 1, 2$ from left to right, all of which are consistent with the melting rule.
  • Figure 2: All six unique partitions of the hypercube ${HC}^{(2)}(n=6)$. Red star marks the target box position. Cyan color represents an occupied box.
  • Figure 3: Numerical verification of the pole order at the target position $\vec{\square}=(1,1,\dots,1)$ for even dimensional partitions within a hypercube. The red dashed line represents the theoretical threshold $\omega=1$. (a) The 6D case shows all possible partitions, forming a clear boundary. (b) The 8D case, obtained via sampling, shows a consistent distribution where all sampled states satisfy $\omega \leq 1$. Note that the pole order equals 1 only for the limiting cases ($N=0, 1, 2^n-1, 2^n$).

Theorems & Definitions (1)

  • Lemma A