On Friday Sergey Galkin, Swarnava Mukhopadhyay and I uploaded a preprint to the arXiv, titled Graph potentials and moduli spaces of rank two bundles on a curve. It's been in the works since 2018, so it's great to have it finally out!
The paper discusses aspects of mirror symmetry for an interesting class of Fano varieties: moduli of rank 2 bundles on curves of genus $g\geq 2$. As you can tell from the page count, this takes some work: we introduce a class of Laurent polynomials (the graph potentials from the title), show that they have many interesting properties and that they are related to these Fano varieties via toric degenerations. This then allows us to conclude enumerative mirror symmetry in this case. Then we also use graph potentials to further study a conjecture on semiorthogonal decompositions for the derived categories of these Fano varieties, and give alternative geometric evidence for this conjecture.
A great introduction to this type of enumerative mirror symmetry for Fano varieties can be found in Tom Coates, Alessio Corti, Sergey Galkin, Vasily Golyshev, Alexander Kasprzyk: Mirror symmetry and Fano manifolds (arXiv version) introducing the Fanosearch program. Is there an equally accessible introduction to homological mirror symmetry of Fano varieties? Because this I mostly learnt from discussions with my coauthors, and less from reading it in the literature. Evidently there is a sizeable collection of papers discussing the homological mirror symmetry for Fano varieties program, but what is the go-to introduction?