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Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ over to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}_{\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}_{\geq 0}$ component goes to $0$.

I would greatly appreciate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}_{\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}_{\geq 0}$ component goes to $0$.

I would greatly appreciate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ over to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}){\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}_{\geq 0}$ component goes to $0$.

I would greatly appreciate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ over to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}_{\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}_{\geq 0}$ component goes to $0$.

I would greatly appreciate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ over to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}^{\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}^{\geq 0}$ component goes to $0$.

I would greatly apprechiate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ this is given by $\omega=d\psi+\cos(\theta)d\phi$. Using $\omega$, we can form a covariant derivative on the associated bundle $\phi:\mathbb{S}^3\times_{U(1)}\mathfrak{u}(1)\rightarrow\mathbb{S}^2$. In the paper https://arxiv.org/abs/1705.02666, the author treats this as a covariant derivative on a bundle over a manifold with topology $\mathbb{C}^2$, such that the curvature is given by $d\omega$.

The only way I can think to extend $\phi$ over to a bundle over $\mathbb{C}^2$ is pulling it back via $h$ to a bundle over $\mathbb{S}^3$, then extending radially over $\mathbb{C}^2\cong\mathbb{S}^3\times\mathbb{R}){\geq 0}$. However, I am not sure how this would be well defined as the $\mathbb{R}_{\geq 0}$ component goes to $0$.

I would greatly appreciate if anyone could reconcile how to obtain such a bundle over $\mathbb{C}P^2$.