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

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.)

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 $T^* \mathbf{CP}^{N-1}$.

In Kentaro hori et al's Mirror symmetry textbook, i found some interesting remarks seems to related with this problem. For four matter fields with charge $1,1,-1,-1$, the vacua equation can be reduced to \begin{align} |\phi_1|^2 + |\phi_2|^2 - |\phi_3|^2 - |\phi_4|^2 =r \end{align} For $r>>0$ this vacua makes $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$ over $\mathbf{CP}^1$ where the base is $(\phi_1, \phi_2)$ and for $r<<0$ case vacua makes $\mathcal{O}(-1) \oplus \mathcal{O}(-1)$ over $\mathbf{CP}^1$ where the base is $(\phi_3, \phi_4)$.
Can one say that the $T^{*}\mathbf{CP}^1 = \mathcal{O}(-1) \oplus \mathcal{O}(-1)$ over $\mathbf{CP}^1$ ?
• I cannot imagine your sources simply state that this is $T^*\mathbb{C}P^{N-1}$ without at least some explanation. Dec 22, 2014 at 15:06
• My reference is witten's paper of Phase of N=2 theories in two dimensions. For $T^{*} CP^{N-1}$, I refer Gadde's \$2d index and surface operator" Dec 24, 2014 at 8:03