I gave a talk yesterday on mirror symmetry, in particular the toric ansatz. There are a few different versions of the actual ansatz but the common theme is that the mirror to a Fano variety should be a cluster variety with cluster charts indexed by the toric degenerations of the Fano. Ilten showed that polytope (or, at least polygon) mutations induce qG deformations between toric Fanos and so part of the mirror symmetry conjectures in this vein are that are enough mutations to connect all toric degenerations of a given Fano variety.
Of course there’s a problem if your Fano doesn’t admit any degenerations to a toric variety! This is the surjectivity component of said conjectures. I’ve discussed class TG – the class of Fano varieties that do degenerate to toric varieties – a few times in the past, and this talk was intended to be an exposition of the very limited current knowledge on the matter.
To complement the talk notes, I wrote out an example of moving between polytope mutations, Laurent polynomial (or ‘algebraic’) mutations, and cluster mutations (transition functions between the cluster charts on the mirror cluster variety). The cluster algebraic objects in play are really ‘cluster algebras with potential’. This is capturing the notion dating back to Przyjalkowski that a ‘weak mirror’ to a Fano is a Landau-Ginzburg model: a regular function on a torus, or a Laurent polynomial, with certain compatibilities regarding the enumerative geometry of the Fano. After all, a regular function on a cluster variety is locally just a Laurent polynomial on each cluster chart. The cluster framework incorporates all the (infinitely many) weak mirrors into a single place to create a canonical mirror object.
If there isn’t enough symplectic geometry for you here, note that there are emerging beautiful syntheses between toric degenerations and Lagrangian tori. Vianna recently constructed infinite collections of monotone Lagrangian tori inside smooth del Pezzo surfaces indexed by generalised Markov triples, exactly those that classify the degenerations of the same del Pezzo surfaces to (fake) weighted projective spaces from the work of Hacking-Prokhorov. Moreover, these share the same mutation properties as these degenerations from the work of Akhtar-Kasprzyk, who classified how fake weighted projective spaces (or Fano simplices) mutate. I’ll post again in more detail on this correspondence soon.