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$Δ$ Invariants of Plumbed Manifolds

Shimal Harichurn, András Némethi, Josef Svoboda

TL;DR

We introduce and analyze the Delta invariants $\\Delta_b$ as the smallest q-exponents in the BPS q-series $\\widehat{Z}_b(Y,q)$ for negative definite plumbed 3-manifolds with spin$^c$ structures, tied to a topological invariant $\\gamma(Y)$. For Seifert manifolds, we establish the explicit formula $\\Delta_{\\mathrm{can}} = -\\gamma(Y)/4 + 1/2$ and show that it is minimal among all $\\Delta_b$ unless $Y$ is a lens space, with cancellations and two-variable refinements leading to nuanced behavior beyond Seifert cases. Beyond Seifert manifolds, splice diagrams are employed to understand minimizing vectors, revealing rich structure and examples where $\\Delta$ can be larger or smaller than naive bounds; cancellations persist even in refined two-variable series. Finally, we compare $\\Delta_b$ with Heegaard–Floer correction terms $d_b(Y)$, showing that while they relate through quadratic-form frameworks in the Seifert setting, they generally diverge in Brieskorn spheres, highlighting distinct arithmetic and geometric information carried by these invariants.

Abstract

We study the minimal $q$-exponent $Δ$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^{\rm c}$-structure. We express $Δ$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $Δ$ for non-Seifert manifolds. Finally, we compare $Δ$ invariants with correction terms in Heegaard-Floer homology.

$Δ$ Invariants of Plumbed Manifolds

TL;DR

We introduce and analyze the Delta invariants as the smallest q-exponents in the BPS q-series for negative definite plumbed 3-manifolds with spin structures, tied to a topological invariant . For Seifert manifolds, we establish the explicit formula and show that it is minimal among all unless is a lens space, with cancellations and two-variable refinements leading to nuanced behavior beyond Seifert cases. Beyond Seifert manifolds, splice diagrams are employed to understand minimizing vectors, revealing rich structure and examples where can be larger or smaller than naive bounds; cancellations persist even in refined two-variable series. Finally, we compare with Heegaard–Floer correction terms , showing that while they relate through quadratic-form frameworks in the Seifert setting, they generally diverge in Brieskorn spheres, highlighting distinct arithmetic and geometric information carried by these invariants.

Abstract

We study the minimal -exponent in the BPS -series of negative definite plumbed 3-manifolds equipped with a spin-structure. We express of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of for non-Seifert manifolds. Finally, we compare invariants with correction terms in Heegaard-Floer homology.

Paper Structure

This paper contains 25 sections, 13 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Plumbed 3-manifolds
  4. $\boldsymbol{\mathop{\mathrm{Spin^c}}\nolimits}$ structures
  5. The $\boldsymbol{\widehat{Z}_b}$ invariants
  6. $\boldsymbol{\Delta_b}$ invariants
  7. The exponents of $\boldsymbol{\widehat{Z}_b}$
  8. Cancellations
  9. Delta for Seifert manifolds
  10. Seifert manifolds
  11. $\boldsymbol{\Delta_{\mathrm{can}}}$ of Seifert manifolds
  12. Lens spaces
  13. Beyond Seifert manifolds
  14. Splice diagrams
  15. $\boldsymbol{\Delta}$ for homology spheres with H-shaped splice diagrams
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ be a Seifert manifold associated with a negative definite plumbing graph. Let $\mathrm{can}$ be the canonical $\mathop{\mathrm{spin^c}}\nolimits$ structure of $Y$. Then $\Delta_{\mathrm{can}}$ satisfies If $Y$ is not a lens space, then $\Delta_{\mathrm{can}}$ is minimal among all $\Delta_b$, $b \in \mathop{\mathrm{spin^c}}\nolimits(Y)$.

Figures (3)

  • Figure 1: Plumbing graph of a Seifert manifold.
  • Figure 2: H-shaped splice diagram $\Omega$.
  • Figure 3: The plumbing graph associated to the splice diagram in Example \ref{['ex:original']}.

Theorems & Definitions (27)

  • Theorem 1.1
  • Proposition 2.1: NeNi02
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 4.1
  • Remark 4.2
  • ...and 17 more