Question about the Aganagic-Vafa A-brane

Question

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?

Answer 1

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.