Moduli space for $CP^N$ and $T^{*} CP^N$ in $\mathcal{N}=2$ SUSY

Question

For complex $\phi$ in $U(1)$ gauge theory, $$\begin{align} |\phi_1|^2 + |\phi_2|^2 +\cdots |\phi_N|^2 =r \end{align}$$ This equation $|\phi|^2=r$, describes sphere $S^{2N-1}$. Dividing the space of this solution by the gauge group $U(1)$ we obtain that the moduli space for $\phi$ which is $\mathbf{CP}^{N-1}$

This procedure is based on the explanation in Witten's paper of "Phase of $\mathcal{N}=2$ theories in two dimensions". (Above situation corresponds to $\mathcal{N}=2$ supersymmetric $U(1)$ gauge theory with $N$ chiral superfields. Here i solve the equation for minimizng potential energy.)

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Here what i want to extend this idea to following equations,
(This situation corresponds to $\mathcal{N}=2$ supersymmetric $U(1)$ gauge theory with $N$ chiral superfields and $N$ anti-chiral superfields. )

For same complex $\phi$ in $U(1)$ gauge theory, we have $$\begin{align} |\phi_1|^2 + |\phi_2|^2 \cdots +|\phi_N|^2 -|\phi_{N+1}|^2 - |\phi_{N+2}|^2 \cdots - |\phi_{2N}|^2 =r \end{align}$$

The results for this moduli space is known as $T^* \mathbf{CP}^{N-1}$ where $T^*$ represents cotangent bundle.

Here i want to know why this space is $T^* \mathbf{CP}^{N-1}$.

Can anyone give some explanation about this?

This post imported from StackExchange Physics at 2014-12-23 15:10 (UTC), posted by SE-user phy_math

Comments

Comment to the question (v3): Consider adding references in order to receive useful and focused answers.

This post imported from StackExchange Physics at 2014-12-23 15:10 (UTC), posted by SE-user Qmechanic

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I cannot imagine your sources simply state that this is $T^*\mathbb{C}P^{N-1}$ without at least some explanation.

This post imported from StackExchange Physics at 2014-12-23 15:10 (UTC), posted by SE-user ACuriousMind