What are the compact Lagrangian submanifolds of a twisted cotangent bundle? In Hamiltonian dynamics and symplectic geometry a twisted cotangent bundle is the cotangent space $T^*N$ of a closed (compact without boundary) $n$-manifold $N$ equipped with a twisted symplectic structure: $T^*N$ carries the canonical symplectic structure $\omega=d\lambda$, where $\lambda$ is the Liouville 1-form. One can "twist" $\omega$ by adding a closed two-form $\sigma$ on $N$ as follows:
$$
\omega_{\sigma}:=\omega + \pi^*\sigma.
$$
Here $\pi:T^*N \to N$ denotes the footpoint map. It is easy to check that $\omega_{\sigma}$ is symplectic. Twisted cotangent bundles play an important role in Hamiltonian dynamics, but I am here interested in their symplectic topology. Many classical questions in symplectic topology concern the closed Lagrangian submanifolds of $(T^*N,\omega)$. But what about closed Lagrangian submanifolds in $(T^*N,\omega_{\sigma})$? Does anyone know a non-trivial example (meaning $\sigma$ is not exact) where $(T^*N,\omega_{\sigma})$ contains closed Lagrangian submanifolds with "good properties" (say weakly exact, monotone etc.)? Are any general statements known? Any examples, ideas, references or proofs will be highly appreciated!
It is easy to find non-compact Lagrangians in $(T^*N, \omega_{\sigma})$: If $X\subset N$ is a submanifold such that $\sigma|_{X}=0$ then its conormal space 
$$
\nu^*(X):=\{ p\in T^*N\ |\ p|_{TX}\equiv 0 \}\subset T^*N
$$ 
is a non-compact Lagrangian submanifold. My questions therefore concerns closed Lagrangian submanifolds! It is easy to find closed Lagrangians when $\sigma$ is exact. Hence, my interest is really in the case when $\sigma$ is not exact.
Thanks in advance!
 A: Let's begin by pointing out the following: you will not find monotone examples for the simple reason that a nontrivial such deformations creates a class of nonzero symplectic area, while the Chern class is always vanishing. The best you could hope for is Calabi-Yau, and such examples indeed exist. However, I know of no examples where a closed Lagrangian has a nonvanishing Floer homology.
Now a general observation: In the case when $T^*N \setminus N$ has no second cohomology with $\mathbb{R}$-coefficients, e.g. if $N=S^2,$ then for small closed forms $\sigma$ one can use Moser's trick to show that any compact Lagrangian submanifold of $T^*N \setminus N$ is preserved (up to smooth isotopy) after turning on a sufficiently small magnetic potential.
A more concrete example: taking $\sigma$ to be the area form on $S^2,$ we obtain the total space of the line bundle $\mathcal{O}(-2)$ on $\mathbb{C}P^1$ with its standard Kähler form. (The first reference coming to my mind is 2.4A in [Y. Eliashberg and L. Polterovich; Unknottedness of Lagrangian surfaces in symplectic 4-manifolds] but maybe there is something more to the point). Unlike $T^*S^2$, the latter symplectic manifold is an open toric Calabi-Yau manifold. Unfortunately, according to Theorem 5 in [Ritter; Floer theory for negative line bundles via Gromov-Witten invariants], its symplectic homology vanishes: this twisted cotangent bundle therefore contains no Lagrangians with interesting Floer homology. See [Ritter-Smith; The monotone wrapped Fukaya category and the open-closed string map] where a closed-open map is constructed in this setting.
A side note: if you compactify a subset of the total space of $\mathcal{O}(-2)$ to the Hirzebruch surface $F_2(\alpha)$ as studied in [Fukaya-Ohta-Ono-Oh; Toric degeneration and non-displaceable Lagrangian tori in $S^2\times S^2$] then Fukaya category actually becomes nontrivial.
