Almost toric mutations

Question

I'm trying to understand the details of the almost toric mutation process as explained in Section 8.4 in https://arxiv.org/pdf/2110.08643.pdf. More specifically, given an almost toric fibration $f: (M,\omega) \rightarrow B$ from a symplectic manifold $(M,\omega)$ to a base $B$, the process of mutation briefly involves cutting the base diagram $B$ into 2 along the eigen direction of the monodromy matrix, applying the monodromy to one of the halves and glueing the two new halves back.

It is unclear to me why this process gives an almost toric fibration on a manifold which is symplectomorphic to $(M,\omega)$ and I've not been able to find a full proof anywhere.

It is of course possible that I'm missing something simple, so any help would be appreciated.

Answer 1

Mutation doesn't even change the integral affine base, which is why it doesn't change the symplectic manifold. All you're doing is changing the way the integral affine base is drawn. If you're given an integral affine manifold, the way you get a picture is by making some branch cuts to pick a fundamental domain in the universal cover of its regular locus, then the developing map gives you an immersion into Euclidean space and hence a picture. Changing branch cuts (of which mutation is a special case) just amounts to picking a different fundamental domain in the same integral affine manifold.