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?



Question about the Aganagic-Vafa A-brane
I believe that the special Lagrangians that Aganagic-Vafa want to consider are contained in special fibres of the Harvey-Lawson fibration. First, we had better
take $k=0$ in your equations, since otherwise the three equations will give a subspace which is of too high dimension to be a Lagrangian. (But these equations do generalize to higher dimension if you take more equations of the form $\sum l\_i^2|X\_i|^2=c$.) Taking specifically
the Harvey-Lawson fibration $(X\_1,X\_2,X\_3)\mapsto (|X\_1|^2-|X\_2|^2, |X\_1|^2-|X\_3|^2,
Im(X\_1X\_2X\_3))$, it is not difficult to see that set of critical points of the fibration are given by the points where at least two of the coordinates $X\_1,X\_2,$
and $X\_3$ are zero. A fibre over a critical value (except for the fibre over zero) is actually a union of two manifolds, each homeomorphic to ${\mathbb R}^2\times S^1$.


To see this topologically, it is best to think of the fact that there is a $T^2$-action which acts fibrewise: this is given by $(\theta\_1,\theta\_2)$
acts by $(X\_1,X\_2,X\_3)\mapsto (\exp(i\theta\_1)X\_1,\exp(i\theta\_2)X\_2,
\exp(-i(\theta\_1+\theta\_2))X\_3)$. A general fibre of the Harvey-Lawson fibration then has this $T^2$ acting freely, and the quotient is ${\mathbb R}$, with
coordinate given by $Re(X\_1X\_2X\_3)$. However, the $T^2$-orbits whose dimension is
$<2$ are precisely the orbits of points where at least two of the coordinates
are $0$. From this one sees that the general singular fibre is obtained by taking
${\mathbb R}\times T^2$ and contracting $\{0\}\times T^2$ to $\{0\}\times S^1$.
This decomposes as a union of two copies of ${\mathbb R}^2\times S^1$. Aganagic-Vafa are using one of these two copies. There are a lot more details of this kind of construction in my paper "Examples of special Lagrangian fibrations," <http://arxiv.org/abs/math/0012002>.


