Geometry and Analysis Seminar

For any smooth projective surface $S$, we introduce BPS polynomials — Laurent polynomials in a formal variable $q$ — derived from the higher genus Gromov–Witten theory of the 3-fold $S \times {\mathbb P}^1$. When $S$ is a toric del Pezzo surface, we prove that these polynomials coincide with the Block–Göttsche polynomials defined in terms of tropical curve counts. Beyond the toric case, we conjecture that for surfaces $S_n$ obtained by blowing up ${\mathbb P}^2$ at $n$ general points, the evaluation of BPS polynomials at $q=-1$ yields Welschinger invariants, given by signed counts of real rational curves. We verify a relative version of this conjecture for all the surfaces $S_n$, and prove the main conjecture for n less than or equal to 6. This establishes a surprising link between real and complex curve enumerations, going via higher genus Gromov-Witten theory. Additionally, we propose a conjectural relationship between BPS polynomials and refined Donaldson–Thomas invariants. This is joint work with Hulya Arguz.