There has been 2 earlier posts about new(ish) papers recently, a first one and a second one, where the new(ish) refers to the fact that they are split off and expanded from a preprint from 2 years ago by Sergey Galkin, Swarnava Mukhopadhyay and myself. Now the third installment, called Graph potentials and symplectic geometry of moduli spaces of vector bundles.

In the first paper we introduced graph potentials, a family of Laurent polynomials with interesting properties. In the current paper we explain why we were motivated in the first place to introduce these graph potentials: to understand mirror symmetry for moduli spaces of rank 2 bundles on curves.

This forms part of the mirror symmetry picture for Fano varieties, which is maybe slightly less familiar territory than mirror symmetry for Calabi–Yau varieties for most. The mirror to a Fano variety $X$ is a Landau–Ginzburg model $f\colon Y\to\mathbb{A}^1$ (where $Y$ is a smooth quasiprojective variety and $f$ is a regular function), so mirror symmetry is "less symmetric" in the sense that it involves objects of a different nature. But there are still 2 comparisons:

  • the symplectic geometry of $X$ "equals" the complex geometry of $(Y,f)$
  • the complex geometry of $X$ "equals" the symplectic geometry of $(Y,f)$

We will be interested in the first comparison, and we are concerned with enumerative mirror symmetry. The enumerative invariant on the symplectic side is Gromov–Witten theory, and the quantum period in particular. This is a generating series for counting rational curves on $X$. On the complex side we are looking for a Hodge-theoretic invariant (this is similar to the initial success story of mirror symmetry for the quintic 3-fold!) which will be the classical period. We only need $Y$ to be a torus for this, so that $f$ is in fact a Laurent polynomial.

This type of enumerative mirror symmetry is only looking at a certain "fingerprint" of the objects involved, and there is plenty of things which are not being compared. But it is an excellent litmus test for checking whether your candidate mirror construction is indeed valid. And this is exactly what we do!

Theorem The classical period of the graph potential equals the quantum period of the moduli space of rank 2 bundles on a curve.

Last week I gave a talk about this work in Oberwolfach, so I'll be writing an extended abstract for it soon. If you are interested in a slightly more technical overview, without having to read the paper (or just its introduction) I hope that will satisfy your needs.