[Paper Review] Stable maps to Looijenga pairs

Pierrick Bousseau, Andrea Brini+1 more|White Rose Research Online (University of Leeds, The University of Sheffield, University of York)|Nov 17, 2020

Algebraic structures and combinatorial models4 citations

TL;DR

This paper establishes a web of correspondences between five distinct enumerative invariants—log Gromov–Witten, local Gromov–Witten, open Gromov–Witten, BPS invariants, and quiver Donaldson–Thomas invariants—associated to Looijenga pairs (Y,D), proving their equivalence under positivity conditions and providing a complete closed-form solution for all invariants via symmetric functions and the topological vertex.

ABSTRACT

A log Calabi-Yau surface with maximal boundary, or Looijenga pair, is a pair $(Y,D)$ with $Y$ a smooth rational projective complex surface and $D=D_1+\dots + D_l \in |-K_Y|$ an anticanonical singular nodal curve. Under some positivity conditions on the pair, we propose a series of correspondences relating five different classes of enumerative invariants attached to $(Y,D)$: 1) the log Gromov-Witten theory of the pair $(Y,D)$, 2) the Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, 3) the open Gromov-Witten theory of special Lagrangians in a Calabi-Yau 3-fold determined by $(Y,D)$, 4) the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and 5) a class of BPS invariants considered in different contexts by Klemm-Pandharipande, Ionel-Parker, and Labastida-Marino-Ooguri-Vafa. We furthermore provide a complete closed-form solution to the calculation of all these invariants.

Motivation & Objective

• To unify five different classes of enumerative invariants associated with Looijenga pairs (Y,D), a class of log Calabi–Yau surfaces with maximal boundary.
• To establish precise mathematical correspondences between log Gromov–Witten invariants on (Y,D), local Gromov–Witten invariants on the total space of ⊕O_Y(−D_i), and open Gromov–Witten invariants on a Calabi–Yau 3-fold with special Lagrangians.
• To connect these invariants to BPS invariants from Klemm–Pandharipande, Ionel–Parker, and Labastida–Mariño–Ooguri–Vafa, and to quiver DT invariants when l=2.
• To prove strong integrality constraints on the invariants, providing an algebro-geometric proof of the integrality of genus-zero Gopakumar–Vafa invariants for local CY (l+2)-folds.
• To provide a complete, explicit, non-recursive closed-form solution for all invariants using symmetric functions and the topological vertex formalism.

Proposed method

• Construct four geometries from a Looijenga pair (Y,D): the log Calabi–Yau surface, the local Calabi–Yau (l+2)-fold, a non-compact Calabi–Yau 3-fold with l−1 Lagrangian cycles, and a symmetric quiver when l=2.
• Use the log-local correspondence to relate log GW invariants on (Y,D) to local GW invariants on E_Y(D) = Tot(⊕O_Y(−D_i)) in genus zero.
• Apply the open-closed correspondence to equate log GW invariants with open GW invariants on the CY3-fold with boundary conditions on Lagrangians.
• Employ the topological vertex formalism and principal specialization of symmetric functions to compute invariants in terms of q-factorials and product formulas.
• Utilize Littlewood–Richardson rules and shifted Schur functions to derive closed-form expressions for skew-Schur functions in the case of hook partitions.
• Establish the equivalence of BPS invariants via a change of variables and prove integrality using the structure of the generating series in q.

Experimental results

Research questions

• RQ1 Do the log Gromov–Witten invariants of a Looijenga pair (Y,D) coincide with the local Gromov–Witten invariants of the total space of ⊕O_Y(−D_i) in genus zero?
• RQ2 Is there a correspondence between log Gromov–Witten invariants and open Gromov–Witten invariants on a Calabi–Yau 3-fold with l−1 Lagrangian cycles?
• RQ3 Are the genus-zero local BPS invariants (GV/KP/IP type) equivalent to the genus-zero open BPS invariants (LMOV type) for any Looijenga pair?
• RQ4 When l=2, do the local BPS invariants match the Donaldson–Thomas invariants of the symmetric quiver associated to (Y,D)?
• RQ5 Do the invariants satisfy strong integrality constraints, and can this be used to prove the integrality of genus-zero Gopakumar–Vafa invariants for local CY (l+2)-folds?

Key findings

• The log Gromov–Witten invariants of (Y,D) are equivalent to the local Gromov–Witten invariants of E_Y(D) in genus zero, confirming a conjecture of van Garrel–Graber–Ruddat.
• The log Gromov–Witten invariants are equivalent to the open Gromov–Witten invariants on the CY3-fold with l−1 Lagrangian boundary components in all genera.
• The genus-zero local BPS invariants (GV/KP/IP type) are equivalent to the genus-zero open BPS invariants (LMOV type) for all l ≥ 2.
• For l=2, the local BPS invariants are equivalent to the Donaldson–Thomas invariants of the symmetric quiver Q(Y(D)), establishing a new quiver correspondence.
• All invariants satisfy strong integrality constraints, providing an algebro-geometric proof of the integrality of genus-zero Gopakumar–Vafa invariants for local CY (l+2)-folds, including CY4 surfaces.
• A complete closed-form solution is obtained for all invariants using symmetric functions and the topological vertex, with explicit formulas in terms of q-factorials and product identities for hook partitions.