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?


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.

  • $\begingroup$ Thank you very much for your answer. I'm sorry that I didn't write the question clearly, I have refined it now. Your answer is very helpful for me. I think the problem is that in the paper of Fang and Liu, they didn't require the Aganagic-Vafa A-brane to be contained in a generic singular fiber, but they still claim that the topology of the fiber should be $\mathbb{C}\times S^1$, this is not correct. $\endgroup$ – Acky Nov 29 '13 at 12:04
  • $\begingroup$ I have actually read your paper mentioned above several month ago, this is inspiring and of great importance for SYZ mirror symmetry, and can also be viewed as an example of the compactification of positive Lagrangian fibrations, which is introduced in your famous Topological Mirror Symmetry. On the other hand, it should also be viewed as an example of the almost toric fibration in the sense of Leung-Symington in higher dimensions. This seems very interesting. $\endgroup$ – Acky Nov 29 '13 at 12:08
  • $\begingroup$ I think what you described above is just an Aganagic-Vafa brane associated to an unknot. In general, we can associate every knot $K\subset S^3$ a Lagrangian submanifold $L_K$ with topology $\mathbb{R}^2\times S^1$, then such a Lagrangian submanifold may not come from a Lagrangian fiber. $\endgroup$ – YHBKJ Nov 30 '14 at 13:14

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