When are Maslov $0$ disks non-trivial in $\pi_2(M,L)$?

Question

My goal is to better understand the Maslov-index of pseudoholomorphic disks.

For a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L\subset M$, the Maslov-index of a pseudoholomorphic disk with boundary on $L$ is given by a homomorphism $$ \mu:\pi_2(M,L)\to \mathbb{Z}. $$

If $[u]=0\in\pi_2(M,L)$, then the Maslov-index $\mu([u])$ is zero as well.

What can be said about the converse? Are there any special situations in which $[u]=0\in\pi_2(M,L)$ is equivalent to $\mu([u])=0$?

If, for example, $\pi_2(M,L)$ is torsion, then $\mu([u])=0$ for all $[u]\in\pi_2(M,L)$. However, I don't know any explicit example where $\pi_2(M,L)$ is torsion, so I would really like to see such an example.

Answer 1

A holomorphic disc class is quite different from a holomorphic disc, there may not be any holomorphic disc in a disc class. So what you asked in your title is different from what you described below.

I'd like to point out a simple example of a Maslov index 0 holomorphic disc whose class in $\pi_2(X,L)$ is non-trivial.

Consider $X=\mathbb{C}^2\setminus\{xy=1\}$ and the Lagrangian submanifold $L\subset X$ defined by

$L:=\big\{\log|xy-1|=C,|x|=|y|\big\}$

for some fixed constant $C\neq0$. It's clear that $L$ is topologically a torus. Then $L$ bounds a Maslov index 0 holomorphic disc whose class $\beta$ is non-trivial in $\pi_2(X,L)$.

One way to see this is to use the Hamiltonian isotopy between

$L'=\big\{\log|xy-1|=C,|x|>|y|\big\}$

treated as a Lagrangian submanifold in $\mathbb{C}^2$ and a torus fiber $K$ of the standard moment map on $\mathbb{C}^2$. Then we are led to the computation of some Maslov index 2 open Gromov-Witten invariants in some class $\alpha\in\pi_2(\mathbb{C}^2,K)$, which is known to be 1.

Answer 2

On the topological level, since $\mu : \pi_2(M,L) \to \mathbb Z$ is a group homomorphism, if it is nontrivial, then you get an exact sequence $$1 \to \ker \mu \to \pi_2(M,L) \to \mathbb Z \to 0$$ and of course there are situations in which $\ker \mu \neq 0$. In fact, if $\mu \neq 0$, then $\ker \mu = 0$ if and only if $\pi_2(M,L) \simeq \mathbb Z$. There are numerous examples with $\pi_2(M,L) \not\simeq \mathbb Z$, for example Clifford tori in $\mathbb{CP}^n$, where $\pi_2(M,L) \simeq \mathbb Z^{n+1}$.

Of course, as YHBKJ pointed out, not every class in $\pi_2(M,L)$ is going to have a holomorphic representative. This depends on the almost complex structure, but beyond that nonzero classes with zero area or classes with negative area will never have such a representative.

To complete the thought, I should add that a Maslov $0$ disk will be trivial in $\pi_2(M,L)$ if and only if it is constant. One direction is trivial. For the other one notice that a nontrivial holomorphic disk must have positive area, therefore it can't represent the trivial class in $\pi_2(M,L)$.