Yesterday I went to an extremely interesting talk of Rafe Mazzeo about the ends of the moduli of Higgs bundles.
Roughly speaking, what was happening is that he described a new compactification — a sort of blowup of the usual compactification at the boundary — in which the limiting behavior corresponds to the Higgs field decoupling.
Specifically, he consider rescaling limits of the Higgs field, i.e. the limit as $t \to \infty$ of $t \Phi$. You wind up with a Higgs field equation $[\Phi, \Phi] = 0$, which cannot literally be satisfied on a compact surface, so it blows up somewhere. As best as I could understand, the geometric picture was that as you take the rescaling limit of the spectral curve, it turns into something number of copies of the zero section together with a bunch of vertical fibres; the vertical fibres are where this field is blowing up.
A particularly interesting thing was the mentions he made of the asymptotics he was observing near these poles: the expression $r^{-3/2}$ appeared. So — the Airy equation? Anyway I asked him after whether he knew what his discussion looked like in the Betti picture; he didn’t, but it reminded me of an idea I discussed with Sean Keel over the summer and then forgot about until now: maybe wild / knotty character varieties can be interepreted as boundary components of the moduli of usual character varieties.
maybe wild / knotty character varieties can be interepreted as boundary components of the moduli of usual character varieties.
The easier thing to see is reversed: degenerate a point in a more wild character variety to a point in a less wild character variety. E.g. M_1(Hopf link) are 5 boundary components of M_1(trefoil). I wonder whether this works in the de Rham model. Can you degenerate $y” = (x^3 + ax + b) y$ to $y” = (x^2 + c)y$ in a sensible way?
This compactification will play a role in the proof that knotty character varieties are cluster varieties. According to Gross-Hacking-Keel, it is enough to prove they are log CY with maximally degenerate boundary.