According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the equations

$\sum_{i=1}^{k+3}l_i^1|X_i|^2=c_1$, $\sum_{i=1}^{k+3}l_i^2|X_i|^2=c_2$, $\sum_{i=1}^{k+3}\phi_i=c_3$

In the above, $X_i=\rho_ie^{i\phi_i}$ denotes the coordinates on $\mathbb{C}^{k+3}$, and $X$ is defined as the quotient $X=\mu^{-1}(r_1,\cdot\cdot\cdot,r_k)/G_\mathbb{R}$. Here $G\cong(\mathbb{C}^\ast)^k$, and $\mu$ is induced by the Hamiltonian $G_\mathbb{R}$-action. Also $l_i^1,l_i^2\in\mathbb{Z}$, and we require that $\sum_{i=1}^{k+3}l_i^\alpha=0$ for $\alpha=1,2$. $c_i$ are fiexed constants.

In the special case when $X=\mathbb{C}^3$ and $G$ is trivial, clearly $k=0$ and we get the equations which characterize a Harvey-Lawson fiber.

I think generically ($c_3\neq0$ or $c_3=0$ but $c_1\cdot c_2\neq0$ and $c_1\neq c_2$), an Aganagic-Vafa A-brane is just a special Lagrangian fiber of the Harvey-Lawson fibration, so it should be diffeomorphic to $T^2\times\mathbb{R}$. But in both of the two papers mentioned above, the topology of $L_{AV}$ is taken to be $\mathbb{R}^2\times S^1$. Why?