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This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology.

Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex manifold, and $L$ a totally real submanifold of $M$. A map

$u:(\Sigma, \partial\Sigma)\to(M, L)$

gives rise to a bundle pair over $\Sigma$: a complex vector bundle $u^*TM$ over $\Sigma$, together with a totally real sub-bundle $u^*TL$ over $\partial\Sigma$.

Question: Is there a nice description for the Maslov index of this bundle pair, in terms of a topological invariant of $u$? For instance, in terms of the homology class $u_*[\Sigma]\in H_2(M, L)$, or in terms of the homotopy equivalence class of $u$?

Motivating special case: if $\partial\Sigma=\emptyset$, then the Maslov index of the bundle pair $(u^*TM, \emptyset)$ is $2\langle c_1(TM), u_*[\Sigma]\rangle$.

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1 Answer 1

up vote 8 down vote accepted

When you have a vector bundle on a manifold $X$ with boundary, trivialised over $\partial X$, there are characteristic classes valued in $H^\ast (X,\partial X)$. Here, when $L$ is orientable, the Maslov index is twice the first Chern class of $u^\ast TM$ relative to the trivialisation on the boundary induced by $L$, evaluated on $[\Sigma,\partial \Sigma]$.

When $\Sigma$ is closed, the Chern number of $u^\ast TM$ is the signed count of zeroes of a transversely-vanishing section $s$ of $u^\ast\Lambda^{max}_{\mathbb{C}}TM$.

When there is an orientable boundary condition, the relative Chern number is the same thing, but you choose $s$ non-vanishing along the boundary and tangent to the real line sub-bundle $\Lambda^{max}_{\mathbb{R}} TL$.

This doesn't make sense when $u^*|_{\partial \Sigma} TL \to \partial \Sigma$ is not orientable: its top exterior power then has no non-vanishing section. Besides, the Maslov index is odd in this case.

ADDED: Here's a proof using the method of Robbin's appendix to McDuff-Salamon ("$J$-holomorphic curves and symplectic topology"). Robbin characterises the boundary Maslov index as an invariant of bundle pairs (complex vector $E$ bundle over a surface, totally real sub-bundle $F$ over the boundary) which is additive under direct sum and under sewing boundaries and is suitably normalised for line bundles over the disc. The uniqueness proof, by "pair-of pants induction", still applies when $F$ is assumed orientable. The invariant "twice the relative Chern number" evidently satisfies the direct sum and sewing properties, and the section $z\mapsto z$ of the trivial line bundle over the disc satisfies the standard Maslov-index 2 boundary condition. Done!

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Thanks! Can you suggest a good reference for this material? –  macbeth May 1 '10 at 14:08
    
Well, I would have suggested McDuff-Salamon... You could try the "Intro to symp. topology" if the "J-hol. curves" book doesn't have what you want. When I'm in the office I'll try to find a precise reference. –  Tim Perutz May 2 '10 at 13:42
    
Cheers. McD-S as a rule seem uninterested in the with-boundary case; I couldn't find this treated there. –  macbeth May 2 '10 at 15:58
5  
It follows from what they do say - see added paragraph. [Hm. Macbeth bamboozled by McDuff. It sounds strangely familiar... I feel I should warn you that McDuff is not a "man of woman born"!] –  Tim Perutz May 3 '10 at 17:51
2  
The holomorphic bubbles have caused me much toil and trouble. Thanks for the extra details! –  macbeth May 3 '10 at 20:04

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