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I have some trouble in understanding the notion of a moment map for the Lie group $S^1$: \ In the book "Moment maps, cobordism and Hamiltonian group actions" it is said on page 15, which you can find online as http://www.ma.huji.ac.il/~karshon/monograph/chap-hamiltonian.pdf that a moment map for the Lie group $S^1$ is given by the equation $d\Phi= \iota(\zeta_m) \omega$. Can you tell me what the $\iota$ in this equation is and also why we can consider $d\Phi$ as a two-form? I would really appreciate, if you could help me.

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$\iota$ denotes contraction and $d\Phi$ is actually a 1-form, not a 2-form. That is, if $X$ is a vector field, $\iota(X)\omega$ is the 1-form $\omega(X,\cdot)$.

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