Today I gave a talk in the MSRI irregular singularities seminar, and afterward was talking with Valerio Toledano Laredo about isomonodromy. In particular now I actually know what isomonodromy is:
Recall that the formal type of an ODE is some $\bigoplus R_i \otimes S_i$ where the $R_i$ are regular connections and the $S_i$ are rank one irregular connections. For simplicity let me assume that the $R_i$ are all rank one and the $S_i$ all have the same valuation $\nu$ but different leading terms $\lambda_i$. That is, the formal type has “just one level”.
Isomonodromy (or anyway the so to speak part of isomonodromy specific to the wild case) just means you vary around the quantities $\lambda_i$, without letting them collide.
Note that in the knot picture, what is going on is that the link of the singularity is wandering around in a family of Legendrians. So, recall from [STZ]:
Theorem. Hamiltonian isotopy $\Lambda \to \Lambda’$ induces an equivalence of categories $Sh_\Lambda(X) \to Sh_{\Lambda’}(X)$.
I think what is actually proven is the slightly stronger looking:
Reformulation. Let $\mathfrak{B}$ be the space parameterizing Hamiltonian isotopies of $\Lambda$. (It is a quotient of some open subset in the space of Hamiltonians.) Then $\mathfrak{B}$ carries a family of categories with an flat connection.
Corollary. Let $M(\Lambda)$ be some moduli space formed from this category, e.g. the moduli of microlocal rank one sheaves. Then there is an algebraic action of $\pi_1(\mathfrak{B}, \Lambda)$ on $M(\Lambda)$.
Remark. The statement that there is a local system of categories is much stronger than the statement that there is a local system of moduli spaces. From the second statement, it would not follow that isomonodromy gives an algebraic map on the Betti moduli space, since it is coming from integrating a connection; from the first, it does. Note also how the reformulation is stronger than the theorem: from the theorem taken literally it follows only that the space of paths in $\mathfrak{B}$ acts; from the reformulation we learn this action factors through $\pi_1$. Probably really there is some more correct reformulation in the appropriately higher categorical language.
Remark. The discussion above shows that the usual notion of isomonodromy action is included in the above Corollary. Note also that it will follow formally that it respects any structures on the moduli space which have a categorical origin, e.g. in dimension 2, the Poisson structure coming from the self exts of the sheaves.
Project. Let’s build a bunch of interesting algebra actions this way!
(I’ll discuss it more with Valerio soon.)
In particular, consider the construction of Alexei and Zhiwei of a DAHA action on certain equivalued affine Springer fibres. Taking equivalued affine Springer fibre <–> wild Hitchin fibration over P^1 <— nonabelian Hodge —> Betti moduli space of an irregular connection, you should believe that the DAHA action is built half out of isomonodromy and half out of Chern classes of tautological bundles. But constructing this and checking the relations should be much easier on the Betti side, because isomonodromy actually acts on the space rather than just on its cohomology and has a description in terms of knot moves. Perhaps more importantly, A & Z were limited to the finite dimensional representations because that’s when the Hitchin system is controllable. But we need suffer no such limitation: we should be able to produce a rep for every decoration of the knot, i.e. one for each partition…
The $\mathbf{Z}/(p+q)$-action on $\mathcal{M}_1$ of a $(p,q)$-torus knot is an example. A while ago I was looking for something of infinite order. I tried the link of the plane curve singularity uniformized by $(t^4,t^6+t^7)$. It can be parametrized in $S^1 \times \mathbf{C}$ as
$$
(e^{i\theta},e^{6i\theta}+e^{7i\theta})
$$
There are a circles worth of front projections to $S^1 \times \mathbf{R}$, i.e.
$$
\Phi_t = \text{image of }(e^{i\theta},\mathrm{Re}(e^{it}(e^{6i\theta}+e^{7i\theta})))
$$
which are animated here . (At the time I thought this was in some sense the hyperKahler rotation of the moduli space. I no longer believe that this is the most natural way to extract a Legendrian knot from a plane curve singularity, but it works.)
Each frame in the animation is the closure of a positive braid of length 19, from frame to frame they can differ by Reidemeister 3 moves. Last year I spent several hours cataloging them, from my notes:
(I couldn’t get the latex array to work, I uploaded a screenshot here)
But I think this too might have finite order…
My interpretation of this discussion:
In Nadler’s Springer theory paper he finds that the braid action of the Weyl group on the Springer sheaf comes from varying the regular parameter, and that this is a toy model of the Hitchin system. Vivek wants to study the category version of this action (Seidel, Thomas) in the irregular singularity case. We understand that a loop in the parameter space has a finite number of discrete walls (where the homeomorphism type of the front diagram changes) and these are made up of Reidemeister moves. David responds with a class of examples in which he has already formed a loop of knots and catalogued the Reidemeister moves which relate the corresponding front diagrams (hence categories).
Cool stuff, guys!
One question: when you have a 13 31 because these transpositions commute, you may change the notion of normality of rulings. Do we know how to account for the effect of this on the category (or need to)?
Is $\pi_1$ of the space $\mathfrak{B}(\La)$ of Hamiltonian isotopies related to $\pi_1$ of the nonsingular locus $\mathcal{B}\smallsetminus\mathcal{B}_{sing}$ of the Hitchin base? As in the group formed by branch points of the spectral curve braiding around each other? This is my (quite possibly wrong!) impression of what the “extended” mapping class group action on these spaces should be, i.e. there is an action of an extension of the MCG by $\pi_1(\mathcal{B}\smallsetminus\mathcal{B}_{sing})$.
Another thought: Every cluster variety has a canonical automorphism, and this group should be where it lives in this case (the “spectrum generator” or “half-monodromy operator”). For irregular singularities for $SL_2$, where there are a set of Stokes rays all on equal footing, it acts by partially rotating the singularity, carrying each Stokes ray to its successor. I think it is an open problem to describe in general, so it would be interesting if a recipe for an obvious sequence of cluster transformations (read: knot moves) presents itself here.
there are of course a lot of other irregular singularities which have such a move, e.g. any of the torus knots…