Stable maps to Looijenga pairs: orbifold examples

Abstract

In [15] we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs (Y, D) with Y a smooth projective complex surface and D = D1 + • • • + D l an anticanonical divisor on Y with each Di smooth and nef. In this paper we explore the generalisation to Y being a smooth Deligne-Mumford stack with projective coarse moduli space of dimension 2, and Di nef Q-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi-Yau surfaces, and for each of them we provide closed form solutions of the maximal contact log Gromov-Witten theory of the pair (Y, D), the local Gromov-Witten theory of the total space of i OY (−Di), and the open Gromov-Witten of toric orbi-branes in a Calabi-Yau 3-orbifold associated to (Y, D). We also consider new examples of BPS integral structures underlying these invariants, and relate them to the Donaldson-Thomas theory of a symmetric quiver specified by (Y, D), and to a class of open/closed BPS invariants.

On the log–local principle for the toric boundary

“…It remains open whether the conjecture holds in the more restrictive setting of a log Calabi-Yau variety with only point insertions. The present paper as well as [7,8] provide evidence for it.…”

“…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.…”

Proof of two multivariate $q$-binomial sums arising in Gromov-Witten theory

“…The purpose of this note is to prove two conjectured multivariate q-binomial summation identities from [1]. There, Bousseau, Brini and van Garrel deal with the computation of Gromov-Witten invariants of log Calabi-Yau surfaces (Looijenga pairs).…”