Stable maps to Looijenga pairs

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 = D1 + • • • + D l ∈ | − KY | 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 i OY (−Di), (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-Mariño-Ooguri-Vafa. We furthermore provide a complete closed-form solution to the calculation of all these invariants.

On the log–local principle for the toric boundary

“…At the level of BPS invariants [12-14, 19, 29], this is proven for the pair of ℙ 2 and smooth cubic in [5,6] and in higher genus in [9]. In [7,8], we extend the correspondences to the non-toric and higher genus/refined setting and include open Gromov-Witten invariants, their underlying open BPS counts, as well as quiver Donaldson-Thomas invariants to the set of correspondences. Another direction is the relationship between local and orbifold invariants [3,30].…”

“…Relation to [26] and [7,8] After this paper was finished, we received the manuscript [26] where the log-local principle is considered for simple normal crossings divisors. The respective strategies have different flavours in the proof and complementary virtues in the outcome: [26] consider the log/local correspondence for 𝑋 smooth and 𝐷 𝑗 a hyperplane section, with a beautiful geometric argument reducing the simple normal crossings case to the case of smooth pairs, and with no restrictions on 𝑋.…”

“…The combinatorial pathway we pursued in the toric setting allows on the other hand to relax the hypotheses on the smoothness of 𝑋, the normal crossings nature of 𝐷, and the very ampleness of 𝐷 𝑗 , and it lends itself to a wider application to the case when 𝐷 is not the toric boundary and the refinement to include all-genus invariants. We consider this specifically in the follow-up papers [7,8], where we prove the log-local principle for log Calabi-Yau surfaces with the components of the anticanonical divisor smooth and numerically effective and suitably reformulate it to, and verify it for, the higher genus theory in these cases. In addition, we extend the correspondences to include open Gromov-Witten invariants, the various underlying BPS counts, and quiver Donaldson-Thomas invariants.…”

Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

“…According to [14], the invariant N β (Y, D) should be thought of as enumerating suitable holomorphic discs in the log Calabi-Yau U (see [6] for recent related results).…”

“…It should be possible to extend Theorem 1.2 to a larger class of quivers and log Calabi-Yau surfaces, by relying on the recent results of [6].…”