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What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:

$$ 1+\Sigma_i z_i = 0 $$

By exact, I mean with respect to the standard Liouville structure coming from being an affine variety. For example, can we provide restrictions on their homotopy type? In his thesis, Sheridan has constructed a very interesting immersed Lagrangian sphere $L$ using tropical geometry and computed the $A_\infty$ algebra $HF^*(L,L)$, where $HF^*$ is the Floer cohomology. But I cannot see any obvious embedded exact Lagrangians in there (EDIT: As Zack points out below there are some very natural exact Lagrangian tori in the pair of pants coming from the fact that there are natural Liouville subdomains isomorphic to $(\mathbb{C}^*)^n$). Because the classification of real surfaces is so explicit, the case $n=2$ may be the most interesting to consider. Of course if one assumes a sufficiently strong version of homological mirror symmetry, it should be possible to say something. But I am more interested in what can be deduced using "standard techniques."

Somewhat more speculatively, are there any compact embedded exact Lagrangians which are not tori in the generalized pair of pants?

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  • $\begingroup$ If nothing else, it has $n+2$ Liouville subdomains isomorphic to $(\mathbb{C}^*)^n$, each of which contains an exact Lagrangian torus. $\endgroup$
    – Zack
    Commented May 23, 2014 at 7:22
  • $\begingroup$ Maybe it's helpful to notice that it is expected that there is an $A_\infty$-functor from the Fukaya category of the pair of pants to the Fukaya category of $(\mathbb{C}^\ast)^{n+1}\times\mathbb{C}$ given by Lagrangian correspondence. $\endgroup$
    – YHBKJ
    Commented May 24, 2014 at 4:29
  • $\begingroup$ That doesn't seem quite right ... $Fuk((\mathbb{C}^*)^{n+1} \times \mathbb{C})$ should be empty (maybe equip it with some sort of partial-wrapping ?). There's some similar construction in work of Smith and Abouzaid, Auroux, Katzarkov which involves a blowup of $(\mathbb{C}^*)^{n+1} \times \mathbb{C}$ (but does not seem simpler than working with the hypersurface itself). $\endgroup$
    – user36931
    Commented May 24, 2014 at 10:53

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