A projective toric variety \(X_P\) carries an action of the compact torus \(T_c \cong (S^1)^n \subseteq T\), and the moment map \[ \mu : X_P \to M_\RR \cong \RR^n \] has image exactly \(P\), with \(\inverseof{\mu}(\text{relative interior of } F)\) the orbit attached to the face \(F\). So \(P\) is the quotient \(X_P / T_c\), and the torus orbits of \(X_P\) correspond to the faces of \(P\), the orbit of a face \(F\) having dimension \(\dim F\).
To construct \(\mu\), realise \(X_\Sigma\) as a quotient of an open subset of \(\CC^{\Sigma(1)}\) by a torus of rank \(\size\Sigma(1) - n\); the moment map of that torus is \[ \mu_\Sigma : \CC^{\size \Sigma(1)} \to \RR^{\size\Sigma(1) - n} , \] and, for \(X_\Sigma\) smooth and projective, \(X_\Sigma\) is the symplectic quotient \(\mu_\Sigma^{-1}(c)/T_c'\) for a regular value \(c\) determined by \(P\), where \(T_c'\) is the compact part of that torus. For \(\PP^n\), \(\mu([z]) = \abs{z}^{-2}(\abs{z_0}^2, \ldots, \abs{z_n}^2)\) has image the standard simplex, whose vertices are the images of the coordinate points.