2023-02-27

Discussion again with Vivek

Suppose we have fibration with singularity over torus on the A-side, and dual torus action on the B-side, can we say taking mirror is like taking fiber?

Step 0: no superpotential, no divisor. With group action. done.

Step 1: B-side add compactification to $\C^N$, and A-side add superpotential to get pair-of-pants.

Step 2: choose a GKZ chamber $C$, then on the B-side, we kill all the strata that we don't see. And on the A-side, we delete the stop that we don't use. These are modification of the stop.

So far, everything is trivial.

Step 3: Here is the fun part: we turn on the B-side superpotential, and turn on the A-side compactification.

Here we have two operations:

• localization: categoically kill some objects. geometrically, A-side, restriction to some Weinstein sector; on B-side, delete some unstable loci.
• deformation: categorically, we turn on $W_B$ add $D_A$, more disks.

Applying each of these operation should be fine.