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Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G = \mu^{-1}(0)/G$. This construction clearly depends on the choice of moment map. However, I wonder if it is still unique up to some sort of (very?) weak equivalence in the symplectic category?

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Given a Hamiltonian group action, moment maps may only differ by constant addition. So you seem to be comparing the reduced spaces at different levels. Let me state the two extreme cases.

When $G$ is a torus, any constant addition to a moment map is also a moment map. In a paper "Birational equivalence in the symplectic category (1989)" by Guillemin and Sternberg, authors showed that reduced spaces at regular levels are related by blowing up and down. I do not know the recent progress though. It might be helpful to read papers citing their paper.

The other extreme case is when $G$ is semisimple. In this case, the moment map, which is $G$-equivariant, is unique by the semisimplicity. Then the reduced space is unique and there is nothing to do.

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