Instituto de Ciencias Matemáticas, Madrid
Stability data, irregular connections and tropical curves
I will give an overview of recent joint work with S. Filippini and J. Stoppa, in which we construct isomonodromic families of irregular meromorphic connections on P1, with values in the derivations of a class of infinite-dimensional Poisson algebras. Our main results concern the limits of the families as we vary a scaling parameter R. In the R → 0 “conformal limit” we recover a semi-classical version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for DT invariants). In the R → ∞ “large complex structure limit” the families relate to tropical curves in the plane and tropical/GW invariants. The connections we construct are a rough but rigorous approximation to the (mostly conjectural) four-dimensional tt*-connections introduced by Gaiotto–Moore–Neitzke.