2026-02-27

Talked with Kifung and Conan about how 3d MS will transport brane to brane. There are some partial success, but still more to be understood.

One interesting thing is that they somehow don't need to specify which side is 3dA-side and which is 3d B-side. Their prescription is somewhat simple, just take fiber product of Lagrangians in (possibly shifted) (real or complex) symplectic manifold, then apply Fuk or Coh to the fiber product.

One thing we need to be worried about is that: mirror symmetry depends on a choice of torus, otherwise it won't be functorial. So, it is wrong to say, I give you a complex manifold, and then I find 'the' mirror of it. What might be better said is that, I give you a Kahler manifold with roughly, a Lagrangian torus fibration, then the mirror is another Kahler manifold with torus manifold. Then we can check if the two sides are mirror to each other.

What do I know about toric mirror symmetry?

• If the B-side is Fano, or weakly fano (including CY case), then the other side is some LG model on torus. Fano has no complex moduli, but lots of Kahler moduli; similarly, LG side has no Kahler moduli, but lots of complex moduli (in the form of varying the superpotential). To be fair, I should say if we only do exact symp deformation on the A-side, then there is no A-side deformation.
• I also know about the toric GIT story, how the asymptotic limit of the Fuk of the fiber on one side, matches with the different phase on the B-side.

In this story, no skeleton, since that already favors one side to be the A-side.

Now, how to think about 3d MS? It couldn't possibly be a functor, sending any holomorphic Lagrangian brane on one side to something on the other side. Since $T^*pt$ is mirror to $T^*pt$.

It is more like, there is a theory called physics. That theory is something like a 2-category, which admits a functor to Cat. So that a theory $T$, under this functor is sent to a category $C(T)$. Now, this category can be realized either using A-model, namely, have some space $X_A(T)$, then apply A-Cat to A-space, we get $C_A(X_A(T)) = C(T) = C_B(X_B(T))$.

That means, we don't have a functor from A-Space to B-space, or vice versa. Rather we have two different realization of the category.

The slogan is that, for $G$ compact Lie, like $S^1, SU(2)$, we have $T^*[1](pt/G) = \mathfrak{g}^\vee / G$ is 3d dual to $BFM(G)$.

Why we need a shifted cotangent $T^*[1]M$? Here are two possible origins:

• We also have the curious thing that $LM = T[-1]M$ for $M$ nice (does $M$ have to be a scheme? can it be a stack? a smooth manifold? a singular one?). Then $T[-1]M$ is dual bundle to $T^*[1]M$.
• If we call $T^*[1]M$ as the 1-shifted symp mfd, then any $Y \to M \to T^*[1]M$ for $Y \to M$ a symp fibration (in real sense), gives a 1-shifted Lagrangian in $T^*[1]M$.
• In particular, if $G$ acts on $Y$ Hamiltonianly, namely, we have $Y/G \to \mathfrak{g}^\vee /G$, then $Y/G$ is a 1-Lagrangian in the 1-shifted cotangent bundle $T^*[1](BG)$.

Suppose we have two Lagrangian brane living over the zero-section $M \in T^*M$, $Y_i \to M$.

• Let $Z_0$ denote the naive fiber product $Y_1 \times_{T^*M} Y_2$ inside $$.
• Let $Z_1$ denote the fiber product inside $T^*[1]M$, then

$$ Z_1 = Z_0 \times_M T^*M $$ Namely, the space acquires a new cotangent fiber direction. And that will make the whole things 0-symplectic.

Sanity check, $Z_1$ should be 0-shifted symplectic manifold. Yes, by PTVV, Calaque. intersection of n-Lag will give you (n-1)-symp.

OK, to play it safe and interesting, I will give you a 'real' one $$ X = T^*[1](B U(1)) , \quad X^! = T^* \C^*.$$ The is the pure gauge, real group case.

A Lag $L$ in $X$ is a symplectic space $Y \to pt$, but with a Hamiltonian $S^1$-action, so we have $$ Y/U(1) \to \R / U(1). $$

What's the 3d dual Lag? wait wait, here you are asking as if 3d mirror symmetry is a functor, no it is more like correspondence, you cannot ask for too much.

And be aware, here I somehow broke the symmetry. I didn't ask for $Y$ to be Kahler, and I cannot go back and do B-model on stuff here. Should I insist on Kahler? Or should I stick with hol'c symplectic stuff?

I will work with the more restrictive holomorphic symp setting.

What is the space they consider?

• Symp space: $X = [\mathcal{X} / / G_\C]$, where hol'c symp $\mathcal{X}$ is a $G_\C$-Ham space, and we do holc Ham reduction.
• Lag subspace $[Y / G_\C] \to [\mathcal{X} / / G_\C]$, here we assume $Y \to \mathcal X$ is a $G_\C$-invariant Lag morphism (with correct complex moment map value as $\mathcal X$).

What do you do then? You need to define the A-hom and B-hom between these Branes.

It is very confusing at this moment:

• What is the role of Gammage-Hilburn-Mazel-Gee skeleton in $X$?
• What is the role of equivariant, or Kahler parameter in $X$, in defining the hol'c symp reduction?
• Is A-hom simply taking Fukaya category of whatever fiber product?
• Is B-hom simply taking Coh of the fiber product?
• What about MF?
• How to reconcile with Doan-Rezchikov's Fukaya-Seidel interpretation for intersection of hol'c Lag, Fueter equation?
• Are we doing shifted space or not?
• What does physics give me? Just category O Koszul duality between symplectic resolution with skeleton support condition? Or this Cat O is only a nice piece of a more general story?
• I like GHMG's 2-categorical story, 2-muPerv is mirror to 2-muCoh. Then, they take some sort of trace, to descend to 1-muSh mirror to 1-muKsh.
• I want to extend the above story, so that one side skeleton become more 'stacky', and the other side skeleton become more 'spiky'. The equivalence should still holds. It is just, one need to be careful in defining the hom.