[Paper Review] Log BPS numbers of log Calabi-Yau surfaces

Jinwon Choi, Michel van Garrel+2 more|arXiv (Cornell University)|Oct 4, 2018

Algebraic Geometry and Number Theory77 references4 citations

TL;DR

This paper introduces log BPS numbers as conjecturally integer-valued invariants that count $
$-curves in log Calabi-Yau surfaces $(S,E)$ with maximal tangency to the divisor $E$. Using a formula analogous to the multiple cover formula, these invariants are derived from genus 0 log Gromov-Witten invariants and are shown to match local BPS numbers and loop quiver Donaldson-Thomas invariants for del Pezzo surfaces with curve classes of arithmetic genus up to 2.

ABSTRACT

Let $(S,E)$ be a log Calabi-Yau surface pair with $E$ a smooth divisor. We define new conjecturally integer-valued counts of $\mathbb{A}^1$-curves in $(S,E)$. These log BPS numbers are derived from genus 0 log Gromov-Witten invariants of maximal tangency along $E$ via a formula analogous to the multiple cover formula for disk counts. A conjectural relationship to genus 0 local BPS numbers is described and verified for del Pezzo surfaces and curve classes of arithmetic genus up to 2. We state a number of conjectures and provide computational evidence.

Motivation & Objective

• To define new integer-valued invariants—log BPS numbers—for counting $
  $-curves in log Calabi-Yau surfaces with maximal tangency to the divisor $E$.
• To establish a formula analogous to the multiple cover formula that relates log Gromov-Witten invariants of maximal tangency to these log BPS numbers.
• To conjecture and verify a relationship between log BPS numbers and local BPS numbers, as well as loop quiver DT invariants, for del Pezzo surfaces.
• To investigate the independence of log BPS numbers from the point of contact on $E$, as conjectured in Conjecture 1.3.
• To provide computational evidence and prove conjectures for curve classes of arithmetic genus up to 2 in the del Pezzo case.

Proposed method

• Define log BPS numbers via a formula analogous to the multiple cover formula, using contributions from $l:1$ covers of irreducible curves and reducible configurations.
• Use the virtual fundamental class of the moduli space $
  \overline{\operatorname{M}}_{\beta}(S,E)$ of maximally tangent genus 0 stable log maps to define log Gromov-Witten invariants $\mathcal{N}_{\beta}(S,E)$.
• Relate log BPS numbers to generalized DT invariants of quivers, particularly loop quiver DT invariants, through contributions from irreducible and reducible curve components.
• Apply the Abel-Jacobi map and study fibers over line bundles in $\operatorname{Pic}^\beta(S)$ to analyze curve configurations in $|L|$.
• Use explicit calculations for $\mathbb{P}^2$ with $E$ a smooth cubic, computing contributions from flex tangent lines and conics to verify invariants.
• Verify the correspondence between log BPS numbers and local BPS numbers by comparing $m^k_{dh}$ and $\mathcal{N}^k_{dh}(\mathbb{P}^2,E)$ for degrees $d \leq 4$.

Experimental results

Research questions

• RQ1 Are the log BPS numbers, defined via a multiple cover-type formula, always integers as conjectured?
• RQ2 Does the log BPS number remain independent of the choice of contact point $P$ on $E$, as stated in Conjecture 1.3?
• RQ3 How do log BPS numbers relate to local BPS numbers and loop quiver DT invariants in the context of del Pezzo surfaces?
• RQ4 Can the contribution of each irreducible component of the moduli space $\overline{\operatorname{M}}^P_\beta(S,E)$ be expressed as a DT invariant of a quiver, as posed in Open Question 6.16?
• RQ5 What is the structure of the moduli space $\overline{\operatorname{M}}^P_\beta(S,E)$ for higher-degree curve classes, and how do its components contribute to invariants?

Key findings

• For $\mathbb{P}^2$ with $E$ a smooth cubic, the log BPS number $m^1_h = 1$ matches the log Gromov-Witten invariant $\mathcal{N}^1_h(\mathbb{P}^2,E) = 1$, confirming the formula for degree 1.
• For degree 2, $m^1_{2h} = 1$ and $\mathcal{N}^1_{2h}(\mathbb{P}^2,E) = \frac{3}{4}$, with the log BPS number arising from $\operatorname{DT}^{(2)}_2 = 1$, showing consistency with the multiple cover formula.
• For degree 3, $m^1_{3h} = 3$ and $\mathcal{N}^1_{3h}(\mathbb{P}^2,E) = 3 + \frac{1}{9}$, where $m^1_{3h}$ includes contributions from three flex tangent lines and two irreducible conics.
• For degree 4, $m^1_{4h} = 16$ and $\mathcal{N}^1_{4h}(\mathbb{P}^2,E) = 16 + \frac{3}{16}$, with contributions from $\operatorname{DT}^{(2)}_4 = 1$, minima of 3 and 9, and 8 irreducible components.
• The log BPS number $m^k_{dh}$ matches $\mathcal{N}^k_{dh}(\mathbb{P}^2,E)$ for all $d \leq 4$, confirming the conjectural formula in the del Pezzo case.
• The contribution of each isolated component of $\overline{\operatorname{M}}^P_\beta(S,E)$ to the invariants is shown to be expressible via DT invariants, particularly in the case of $\mathbb{P}^2$ with $E$ a cubic.