Let $\pi: P \rightarrow B$ be a principal circle bundle over $B$ and $\rho: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ an effective left action. Then, one can associate to the bundle $\pi$ a complex line bundle $\pi_{\rho}:P \times_{\rho} \mathbb{C} \rightarrow B$ by the canonical projection, where $$ P \times_{\rho} \mathbb{C} := \{[p,z]\in P \times \mathbb{C}\,|\, (p,z) \equiv (p\cdot\theta, \rho(\theta,z))\ \text{for some}\ \theta \in S^1 \} . $$
My question is the following: Define two left circle actions
$\rho_{1}, \rho_{2}: S^1 \times \mathbb{C} \rightarrow \mathbb{C}$ by
$$\rho_{1}(\theta, z)=e^{i\theta}z, \ \ \rho_{2}(\theta,z)=e^{-i\theta}z.$$
Then,
1) Are two associated bundle $\pi_{\rho_{j}}:P \times_{\rho_{j}}\mathbb{C} \rightarrow B$ ($j=1,2$) isomorphic as vector bundles?;
2) Are the two total spaces $P \times_{\rho_{j}}\mathbb{C}$ ($j=1,2$) mutually diffeomorphic?
I am happy to get to know the answer to each question. Thank you in advance.
Some time has passed, so I'm not sure if you still need the answer, but here it is:
The two line bundles will be duals of each other, so they will not be isomorphic, unless it's the trivial bundle.
Suppose $s_i$ are local sections of the circle bundle given in the following way. If $\phi_i : \pi^{-1}(U_i) \rightarrow U_i \times S^1$ is the local trivialisation of the bundle, then its inverse can be expressed as $\phi^{-1}_i(x,\theta) \rightarrow \theta + s_i(x)$ for some local section $s_i$ (I'm using the additive notation here so that exponentiation makes sense as a homomorphism later on). If $\psi_{ji}$ are transtion funtions for the bundle (going form $i$ to $j$ trivialisation), then the realtionship between $s$s is $s_i(x) = \psi_{ji}(x) + s_j(x)$.
Now we trivialize the corresponding complex lines in the following way $$\pi_\rho^{-1}(U_i) \ni [s_i(x),z] \rightarrow (x,z) \in U_i\times \mathbb{C}.$$ Then it is not hard to show that the transition functions for this line bundles are $e^{i\psi_{ji}}$ and $e^{-i\psi_{ji}}$ for the actions $\rho_1$ and $\rho_2$ respectively. That also shows that the trivialisation was well-defined since equivalent circle bundles will produce eqiuvalent line bundles. Finally, note that line bundles are duals of each other when their transition functions are inverses of each other.