Recent developments in higher genus curve counting: January 6 – February 28, 2025

Organized by:

• Qile Chen (Boston College)
• Felix Janda (University of Illinois Urbana-Champaign)
• Sheldon Katz (University of Illinois Urbana-Champaign)
• Melissa Liu (Columbia University)
• John Pardon (SCGP)
• Rachel Webb (Cornell University)

Modern curve-counting theories were in part inspired by the work of physicists yet have active lives of their own as interesting and rich mathematical notions with connections to many areas of mathematics. A plethora of enumerative invariants have been developed, including Gromov–Witten (GW) invariants, GLSM invariants, Fan–Jarvis–Ruan–Witten (FJRW) invariants, as well as variants of these curve-counting theories.  Many conjectures about enumerative invariants have arisen from physics, providing both deep insight as well as strategies for effective computation.  Several of these conjectures have been proven in recent years, sometimes in their original form, and other times after the conjecture has been translated into a mathematically more natural framework.  This program will focus on the higher genus curve counts from multiple angles, including geometric, computational and categorical perspectives.

On the geometric side, the program will investigate various moduli spaces recently introduced as tools to understand higher genus invariants, including (but not limited to) desingularizations of moduli of stable maps, moduli of Mixed-Spin-P fields, and logarithmic gauged linear sigma models, as well as recent progress on skein-valued invariant counts of higher genus holomorphic curves with Lagrangian boundary conditions in Calabi-Yau threefolds and higher genus open BPS invariants.

On the computational side, higher genus GW invariants of Calabi–Yau threefolds are expected to satisfy universal properties such as Yamaguchi–Yau finite generation,  the BCOV holomorphic anomaly equations, and the Castelnuovo bound, which have been established for quintic Calabi–Yau threefolds.  Progress and difficulties in generalizing these results to other Calabi–Yau threefolds, and to more general targets, will be investigated during the program, with the participation of physicists for synergistic effect.  Building on these results and other developments, the g=2 Gromov–Witten invariants of the quintic have been rigorously determined.  There is much room and hope for further exciting progress, as physicists have a prediction for the Gromov–Witten invariants of the quintic up to genus 64.

Most of the above enumerative invariants are expected to agree with the corresponding enumerative categorical invariants constructed by Caldararu–Tu: the Fukaya category of a symplectic manifold for GW theory; the wrapped Fukaya category of a symplectic Landau-Ginzburg model for FJRW theory; the derived category of a Calabi–Yau manifold for BCOV theory; the category of matrix factorizations for B-model FJRW theory. From this point of view, categorical/homological mirror symmetry (which can be viewed as a version of genus-zero open mirror symmetry) implies enumerative mirror symmetry for all genera. The main challenge lies in identifying categorical invariants with geometric ones, which is one of the main themes of this program.