Boundary Maslov index of holomorphic disks in Calabi-Yau manifolds

Question

Let $u \colon \Sigma^2 \to M^{2n}$ be a holomorphic disk (so $\Sigma = \{z \in \mathbb{C} \colon |z| \leq 1\}$) in a compact Calabi-Yau manifold $M$ of real dimension $2n$ with boundary on a Lagrangian submanifold $L^n \subset M^{2n}$. Note that the normal bundle $N\Sigma \to \Sigma$ is a complex vector bundle of complex rank $n-1$. At points of $\partial \Sigma$, let's orthogonally decompose $TL|_{\partial \Sigma} = T(\partial \Sigma)|_{\partial \Sigma} \oplus F$, so that $F \subset N\Sigma|_{\partial \Sigma}$ is a totally real subbundle of real rank $n-1$.

I am interested in the boundary Maslov index $\mu(N\Sigma, F) \in \mathbb{Z}$. My question is: Among all possible pairs $(u, L)$ of holomorphic curves with Lagrangian boundary, can $\mu(N\Sigma, F)$ attain every possible integer value, or are there restrictions (coming from, say, the Calabi-Yau assumption on $M$)?

I am not a symplectic geometer, so I don't know what to expect. In particular, I don't know whether there are standard examples of holomorphic curves with Lagrangian boundary for which $\mu(N\Sigma, F)$ is easily computable.

Answer 1

I assume you're asking about embedded or at least immersed discs, in order to make sense of the normal bundle. If so, let's fix an immersion $\iota\colon\Sigma\to M$ and observe that the pullback bundle-pair $(\iota^*TM,\iota^*TL)$ splits as $(T\Sigma\oplus N\Sigma,TS^1\oplus F)$ for some totally real subbundle $F\subset\iota^*TL$. The total Maslov number $\mu$ of the disc splits as $2+n$ where $n$ is the Maslov index of the normal component (which is what you're interested in). So we might as well ask about the ordinary Maslov number and then subtract 2.

First thing to notice is that the Maslov number needs to be even if $L$ is orientable (this is because the orientation allows you to lift the classifying map for the bundle pair to the double cover which classifies orientable bundle pairs).

Now it's easy to get any positive even number as the Maslov index of a disc. To see this, note that any symplectic manifold (in particular a compact Calabi-Yau) contains a Darboux ball (symplectomorphic to an open ball in $\mathbb{C}^n$, and inside a Darboux ball you can find a Lagrangian product torus which bounds discs of Maslov index 2, 4, 6, etc. This gives you normal Maslov indices 0, 2, 4, etc. I imagine it's possible to cook up examples with any negative Maslov index.

If you're interested in things like this, you might be interested in the following papers by Globevnik and Oh:

Josip Globevnik. Perturbation by analytic discs along maximal real submanifolds of C N . Math. Z., 217(2):287–316, 1994.

Yong-Geun Oh. Riemann-Hilbert problem and application to the perturbation theory of analytic discs. Kyungpook Math. J., 35(1):39–75, 1995

They study holomorphic discs in complex manifolds with boundary on a totally real submanifold, and show that there is a splitting theorem like the Birkhoff-Grothendieck splitting for holomorphic vector bundles over $\mathbb{CP}^1$. Namely, every holomorphic bundle pair (holomorphic bundle over the disc together with a totally real subbundle around the boundary) splits as a direct sum of line bundles.