what is the problem?

• Consider a flat symplectic fibration $\pi: E \to B$, $B$ is complex 1-dim, no singular fiber, no curvature, but may have non-trivial monodromy. Suppose $(B, \omega_B = -d\lambda_B$ is a Liouville domain. pick a reference point $b_0 \in B$, and reference fiber $F = \pi^{-1}(b_0)$. Assume $F_0$ is also a Liouville domain, with $\lambda_F$, and symplectic parallel transport preserves the fiberwise Liouville 1-form structure near the boundary. A fibered Lagrangian $L_\pi$ is given by a base Lagrangian $L_B \In B$, assuming $b_0 \in L_B$, and a fiber Lagrangian $L_F \In F$, and parallel transport $L_F$ along $L_B$.
• The question is, how do we make $L_\pi$ into a conical Lagrangian in the total space. First of all, it is not at all clear what does conical Lagrangian mean in the total space. We have to assume that, there is some relative 1-form $\lambda_\pi$, so that $\omega_\pi = d\lambda_\pi$ is a relative symplectic form for $\pi$. Then, we can do $\omega_{tot} = \omega_\pi + c_B \pi^* \omega_B$, where $c_B \gg 1$. This will not affect the symplectic connection, but helps make the horizontal distribution symplectic. So, we can consider the primitive $\lambda_{tot} = \lambda_\pi + c_B \lambda_B$. Suppose we already trivialized along the fiberwise boundary, we would have $\lambda_{tot}|_F = \lambda_\pi|_F$ and $\lambda_{tot}|_B = \lambda_\pi|_B + c_B \lambda_B$. The near the fiberwise boundary, the Liouville flow also factorize as a product $Z_F + Z_B$.
• Near the fiberwise boundary, the fibered Lagrangian looks like $L_B \times L_F$, and $L_F$ is preserved (i.e. tangent to) $Z_F$. However $L_B$ is not tangent to $Z_B$, so we don't have a conical Lagrangian here. How to conicalize?
• Maybe the question is: why do we want to conicalize? cannot we bound hol'c disk in some other ways? Like, fiberwise infinite wrapping of the Lagrangian? We hope so, but we don't know yet.
• One way to control the boundary of the holomorphic disk to not escaping to infinity is to use a $J$-psh function $\varphi$, and require $L$ to be conical outside a compact subset, in the sense that $d^c \varphi$ restricted to $TL$ to vanish. The proof is that, take a large enough sublevel set of $\varphi$, so that outside this set, all Lag are conical, and there are no intersection points. Then, all hol'c disk has to be contained in that sublevel set. Because for a disk to escape to here, one either has to have an interior maxium or a boundary maximum, and both are forbidden.
• To summarize, we can use J-psh function $\varphi$, conical Lagrangians (hence no intersections), to create a 'no-go' zone for holomorphic disks's $\varphi$-maximum. if we have a disk mapping to the defining domain of $\varphi$, then such a disk cannot touch the no-go zone. However, if a disk maps to a bigger space, only part of the disk is in the defining domain of $\varphi$, then the no-go result does not apply, since it can very well have no interior and boundary maximum of $\varphi$ on $D$.
• If we have not just a fibration, but a product setup, then we do not need to make Lagrangian eventually sum-conical, we can use two psh functions to bound the two directions separately. But it requires us to maintain the product structure of $J$, $L$, and the auxiliary psh functions, in order to prove the compactness of moduli space. It shouldn't have to, the moduli space should remain compact under compactly supported perturbation of the interior data, like $J$ and $L$, just the sneaky proof breaks down. That's why we want to avoid using product structure to prove moduli space compactness.
• why do we bother to attach the cylindrical end to the Liouville domain, and to extend the Lagrangian? if there is no intersections there, and no disks there, why not just forget them? because it was nice to have things complete to be safe, and then later you show you don't need those.

What's the setup?

• We start from a product space, $B^o \times F$, each factor an exact symp space, and we have product Lagrangians $L_B \times L_F$, disks in $L_B$ and $L_F$ can be bounded separately.
• We consider another space $\pi: E \to B$, and $B^o \into B$, with trivialization $E^o :=\pi^{-1}(B^o) \cong B^o \times F$. We equip $E$ with a Kahler structure that when restricted to $E^o$ agrees with $B^o \times F$, namely $\varphi_E|_{E^o} = \varphi_{B^o} + \varphi_F$, also product type $J$.
• We consider Lagrangians in $E^o$, of the same old product type. The 'only' change is that, we allow for much more disks in $E$, not just in $E^o$.
• There exists psh $\varphi_B: B \to \R$ that extend $\varphi_{B^o}$.
• It make sense to ask, whether $\pi_F^*\varphi_{F}: E^o \to \R$ can be extended to $E$ as a psh function. Sadly, the answer is no.

What's the question?

• If we use product type Lagrangian, how to prevent disk from escaping to infinity in $E$? GPS says: attach a conical end to $E$, get $\hat E$. And extend $L_B \times L_F$ in some way, so that, after some bending, it has Legendrian boundary, and can be extended as a eventually conical Lagrangian.
• Now that we have compactness of moduli space, for disk with arbitrary input and output intersection points. We can ask for a more refined question, is it possible that
  • for some 'small' input intersections, the output is also small, hence is not an intersection that arises after conicalization
  • if input and output intersection are all in the product zone $E^o$, the disk cannot escape $E$.

something just clicked, felt very happy.

1. We all know the classical story, $[x, p] = \hbar $ (sorry, I will drop $i$ as math people don't care about constant...). We can realize this algebra (or faithful representation) of it in two ways (view $\hbar$ as some complex number)

• They act on $\C[x]$ by $$x \cdot f(x) = x f(x), \quad p \cdot f(x) = -\hbar \d_x f(x) $$
• The act on $\C[p]$ by $$ x \cdot g(p) = \hbar \d_p g(p), \quad p \cdot g(p) = p g(p). $$

actually, can we have more? can we just rotate the polarization as we want? what formula do we have? Fourier transformation $e^{-xp/\hbar}$ is an extreme version of rotation generated by Hamiltonian function $x^2+p^2$. metaplectic representation?

These are quantization of the symplectic space $T_p^*\C_x$, and with different choice of Lagrangian polarization (polarization just means foliating by Lagrangians).

2. Now comes the 'cylinder'. $T^*_w \C^*_u = C^*_u \times \C_w$. We have commutator relation $$ [w, u] = \hbar u $$ Again, we have two ways to quantize, let $u = e^x$, we get

• Representation is function on $u$, we get $$ u \mapsto u \cdot, \quad w \mapsto \hbar \d_x = \hbar u \d_u $$
• Representation is function on $w$, we get $$ u \mapsto e^{-\hbar \d_w}, \quad w \mapsto w \cdot $$

3. Now comes the '2 torus'. $C^*_u \times C^*_w$, say we set $u = e^x, w = e^y$, if $[x,y] = \hbar$, and $q= e^\hbar$, then we can do 'Baker-Hausdorff-Campbell'? $e^x e^y = e^{x+y + 1/2\hbar}, e^y e^x = e^{x+y-1/2 \hbar}$, so we get $$e^x e^y = e^\hbar e^y e^x, \quad uw = q wu $$ The so called quantum torus relation. It can be quantized symmetrically in two ways, either

• acting on function of $u$, $w f(u) = f(u/q)$
• acting on function of $w$, $u g(w) = g(qw)$.

hmm, it seems the rational version and elliptic version are most natural ones, where as the trig version is kind of half-baked...?

I am thinking about proving multiplicatve - multiplicative HMS, with Spencer, and with quantization.

Yuji proposed an interesting construction of category on a disk with stops. The bulk is decorated with some category $C$, and stops are decorated with something else, like $D_1,\cdots, D_n$. Then we have functors $D_i \to C$. We want to take some sort of global section on this.

Where does this come from? Consider a family of LG model over a base $\C$. With total space $X$, function $W$ on it, and in addition, a function $\pi: X \to \C$. OK, you can say that we can combine $W$ and $\pi$ together to have a 2d base, $\C^2_{x,y}$, with some singularity curve $S \In \C^2$. We decree that $Re(y) > R$ is the stop. For example, say $F$ is given by $y^2 = x^3$. And the stop is given by $Re(y) > 10$, and when a singularity falls into the stop. the thing is, instead of integrating out $x$ first, then do $y$, Yuji integrated out $y$ first. That's new and brave! (well maybe we did this as well without realizing it, when we have the $\pi, W$ stuff).

consider a simpler case, $y = x^2$ as singularity, and $Re(y) > 1$ as stop. If we integrate $y$ first, then on the $x$ space, we are left with a cool coefficient system, it would be zero cat when $Re(x^2) > 1$. If we do 'infinitesimal Fukaya category', then we do opposite thimble ending on some singularity. Why the wrapping stops? because upstairs, the seed of the Lagrangian in the singularity $S$ get stopped when wrapping.

Now suppose we have something that is like $\{y=x^2\} \cup \{y=0\}$, so we have something that never escapes. what do we say about the 'nonescaping' one? I want to say, first this one is degenerate.

Let's try another one $\{y=x^2\} \cup \{y=-x^2\}$, right the one that Yuji was considering. The two branches was escaping at different places.

an object is a Lagrangian (or just totally real submanifold), so we have a (constructible) sheaf of category on it, and we want to a global section of object over it.

new space and new function.

I want to study $[1]-(1)$ quiver. In the sense of how to see it as framed zastava space.

In the nicest setting, max of smooth psh function is still psh, but with kink when the dominant term switch over. To solve this problem, people developed softmax, which is a smearing of max function. When we softmax a bunch of psh function, the outcome is smooth and psh.

Another application is the following: suppose we have a bunch of locally defined psh function $u_\alpha$, living on some locally finite open cover $\Omega_\alpha$ (say extending continuously to the closure of $\Omega_\alpha$). If we take max of these whole collection of functions, that certainly does not make sense. If we take max at each point $z$, then the problem is that if $z$ moves out of the boundary of certain $\Omega_\beta$, $u_\beta$ will suddenly not avaiable for doing max, it would be a disaster when the 'weight bearer' of the group suddenly leave, we would have a cliff fall over. The only case where everything is safe, is when $u_\beta$ is already relatively 'retired' near the boundary $\Omega_\beta$, as the real work is taken up by some other $u_\alpha$, then there is no problem. Then, you can take pointwise max, the thing will still be continuous psh.

Now, we don't want to do convolution to regularize continuous psh. We want to do softmax. The problem with softmax is that, each term needs a room of epsilon to smooth over. This is usually no problem, the fuzzy uncertainty for $u_\beta$ by $\eta_\beta$ is tolerable, if the bump-up of $u_\beta$ still won't catch the low-day of the best $u_\alpha$, then it is safe to retire, byebye safe trip.

Next, we consider Richberg's theorem. Input a strictly psh function on a manifold $X$

• Do a locally finite covering of $\Omega$, so that each $\Omega_\alpha$ is in a coordinate patch
• Do some convolution smoothing for each patch $u_\alpha$.

I want a wiki / blog tool, that is online and easy to use and share.

overleaf is good, but not good at sharing or updating.

notability is good and smooth, but not good at sharing.

In order to prove some Liouville pair is a Weinstein pair, we need to know that the stop is good.

If our space is like $\R \times \R_-$, one factor of space is cutting off some factor, but leaving some other factors intact. How does the fiber look like? It would be just the stop-fiber from the relevant factor, times the entire space from the non-partipating factor. So to show the Weinstein-ness, one just need to show that the two factors are. Now, where is the participating factor? It is about some smoothable function, that only involves center of mass variables. So, it is as if in the cotangent bundle case.

Now, how about the other factors? What do we need to show? What do we have already? Those other factor is locally a product, which we assume is Weinstein already.