Hi, I really enjoyed Vivek’s talk and our discussions, which prompted me to come back here and discover a half-finished post from a while ago; let me finish it now.
Let $C$ be a Riemann surface. Let $\Sigma \subset T^* C$ be a complex curve (we think of it as a spectral curve for a Hitchin system on $C$.)
I will cheat by pretending that $T^* C$ is a hyperkahler space. In reality, what’s known is that a neighborhood of the zero section admits a canonical incomplete hyperkahler metric (depending on a choice of Kahler metric on $C$). Probably one should consider everything that I say below to be happening “very near the zero section”, in a sense I won’t try to make precise (though it would be very nice to do it!)
One of the complex structures in the hyperkahler structure on $T^* C$ is the obvious one coming from the complex structure on $C$. Let me call that one $J_3$, with corresponding Kahler form $\omega_3$. We also have the standard holomorphic symplectic form $\Omega_3$, whose real and imaginary parts are the Kahler forms for two of the other complex structures, $\Omega_3 = \omega_1 + i \omega_2$.
$\Sigma$ is a complex Lagrangian in $T^* C$ (for degree reasons), so in particular it is Lagrangian for $\omega_1$. (Or more generally for any real-linear combination of $\omega_1$ and $\omega_2$; but I’ll ignore that freedom for the rest of this post.) We have a natural complex structure compatible with $\omega_1$, namely $J_1$; so I’ll use that one.
$\Sigma$ is typically not an exact Lagrangian for $\omega_1$. Indeed, we have the standard complex Liouville $1$-form $\lambda$ with $d \lambda = \Omega_3$. Call its periods $Z_\gamma = \oint_\gamma \lambda$, for $\gamma \in H_1(\Sigma, Z)$. Generically $Z_\gamma$ are some arbitrary complex numbers — in particular they are neither purely real nor purely imaginary. Then $re\,Z_\gamma = \oint_\gamma (re\,\lambda)$ is an obstruction to $\Sigma$ being exact.
Despite the non-exactness of $\Sigma$, there are typically no $J_1$-holomorphic discs ending on $\Sigma$. The reason for this is: suppose that there is a $J_1$-holomorphic disc $D$ ending on $\Sigma$. Then, again for degree reasons, $D$ is complex Lagrangian for $\Omega_1 = \omega_2 + i \omega_3$. So in particular $\int_D \omega_2 = 0$, but letting $\gamma = [\partial D]$ this means $im\,Z_\gamma = 0$. Usually this isn’t true.
So in short, $\Sigma$ is not $\omega_1$-exact unless all $Z_\gamma$ are pure imaginary, but it doesn’t have any $J_1$-holomorphic discs ending on it unless some $Z_\gamma$ is purely real.
So, at least in my naive way of understanding things, it should be OK to use these non-exact spectral curves for microlocalization. When we deform $\Sigma$, the microlocalization map between local systems on $C$ and $\Sigma$ should be locally constant, as long as we don’t pass through a $\Sigma$ which has a holomorphic disc ending on it. When we do pass through such a $\Sigma$ we get a cluster transformation (or “cluster-like” if there is more than one disc appearing at once). I suppose that this should somehow be the non-exact analogue of the disc-surgery that you guys have developed for the exact Lagrangians.
In the next post I’ll try to describe how I think the microlocalization map for the non-exact spectral curves must look. But very briefly: I think it must be equal to what was called “nonabelianization” in the Gaiotto-Moore-Neitzke “Spectral Networks” paper. The “spectral network” in this context is just the locus of points $x$ on $C$ such that there exists a $J_1$-holomorphic disc in $T^* C$ whose boundary is a “bigon”, with one part on $\Sigma$ and one part on the cotangent fiber $T^*_x C$. Given the spectral network you can compute directly the nonabelianization map. As you deform $\Sigma$, the spectral network also deforms; then the nonabelianization map varies in a locally constant way, except that sometimes the spectral network it undergoes a jump in topology (associated with the appearance of a $J_1$-holomorphic disc with boundary entirely on $\Sigma$); then you can calculate a corresponding jump of the nonabelianization map and see that it gives a cluster or cluster-like transformation.
Now how is this related to what I have been learning from you guys? A nice dream is that there is some kind of flow which takes the non-exact spectral curves into exact Lagrangians, deforming the periods in such a way that they become purely imaginary, without any period becoming purely real along the way, so that no discs can appear — for example, a flow which leaves the imaginary parts constant and just dilates the real parts to zero. Ideally the microlocalization operation would deform in a locally constant way along this flow, so this would match up the two stories directly. If we then imagine applying this flow to a Lagrangian which has some period already purely real, then the flow would take that period to zero, i.e. it would collapse the corresponding 1-cycle; that matches with my very vague understanding of what happens in the middle of your disc surgery.
I think Harold actually told me a candidate for such a flow a few months ago, but now I forget all details! Harold, do you remember?
Hi Andy, thanks for writing this!
At certain points I have believed, or anyway hoped, the following thing: after the “hyperkahler rotation” you describe, each connected component of the complement of the walls in the Hitchin base contains one $\omega_1$-exact Lagrangian curve. Is the idea that the flow you describe contracts each such component to this point, and that if you instead try and apply it on the wall, then what happens is that you flow to a singular spectral curve where that disk has collapsed?
Another question: in the next post, are you going to describe what disks appear when they are allowed to have one boundary on the fibre and one on the spectral curve?
I’m not quite sure how to think about the endpoints of this putative flow. My initial guess would be that they are not spectral curves at all — i.e. not $\Omega_3$-Lagrangians anymore, just $\omega_1$-Lagrangians. Certainly, if the flow goes the way I described above — where the imaginary parts of the periods stay fixed and only the real parts move — I think it can’t remain within the space of spectral curves for a fixed underlying curve $C$ (and unless $G = SL(2)$, I would be surprised if it could remain within the space of spectral curves even if we allow $C$ to deform).
Yes, I am going to try to describe what discs are supposed to appear.
Actually, their becoming not spectral curves is maybe better — in the NZ setting, there’s no reason to care that your Lagrangians are holomorphic…
Unfortunately I think you covered most of what I’d be able to say, in particular the picture of flowing the central charges to the Imaginary axis! There was some attempt at formulating a kind of geodesic flow on the space of embedded curves, but nothing very encouraging ever came of it…
I’ll ask Knut Smoczyk about this. He’s an expert on Lagrangian flows. In the flow that Smoszyk studies, the symplectic form changes by the exterior derivative of the mean curvature form defining the flow (recall the normal bundle and cotangent bundle of a Lagrangian are identified), so one might hope that the symplectic primitive changes by that one-form itself.
There is an obvious analogous, though less Riemannian-geometric, guess in this setting, which would be to flow (negatively) by the one-form whose periods you want to kill. But in this context that is equivalent to Liouville contraction for the associated vector field.
Knut points out that the mean curvature form only preserves Lagrangianicity (i.e. is closed) in the Kahler-Einstein case (e.g., Calami-Yau), in which case it represents the Maslov class. So in order for this flow to kill the symplectic primitive the Maslov class must be a multiple of it, i.e. the Lagrangian should be “monotone.” (A version of the flow exists in the non-KE case, too.)
Even if we don’t have monotonicity, we can still do Liouville dilation, only it won’t necessarily head toward something minimal.
The 1-form I want to kill is the real part of the Liouville form on T*C. But what do you mean by “flowing by” this 1-form?
Restrict that one-form to your Lagrangian, then use the symplectic form to make it a vector field. (I mean, if $\omega$ is a symplectic form on $M$ and $L\subset M$ is Lagrangian, then we have an isomorphism $\omega: T^*L \rightarrow N_L,$ where $N_L = TM\vert_L/TL$ is the normal bundle to $L$ in $M$.) Flow by that vector field.