A Looijenga pair is a pair (X,D) with X a smooth complex projective surface and D a singular anticanonical divisor in X. I will describe a series of correspondences relating five different classes of string-theory motivated invariants specified by the geometry of (X,D):
– the log Gromov–Witten theory of (X,D),
– the Gromov–Witten theory of X twisted by the sum of the dual line bundles to the irreducible components of D,
– the open Gromov–Witten theory of special Lagrangians in a toric Calabi–Yau 3-fold determined by (X,D)
– the Donaldson–Thomas theory of a symmetric quiver specified by (X,D), and
– a class of BPS invariants considered in different contexts by Klemm–Pandharipande, Ionel–Parker, and Labastida–Marino–Ooguri–Vafa.
I will also show how the problem of computing all these invariants is closed-form solvable. This is joint work with P. Bousseau (Saclay/ETH) and M. van Garrel (Warwick/Birmingham).