Let $(M , \omega)$ be a symplectic manifold, $J$ a compatible almost complex structure. We call pseudo holomorphic strip a solution $u : \mathbb R \times I \to M$ of the equation $\partial_s u + J \partial_t u = 0.$ Given a pair of Lagrangian submanifolds $L_0, L_1$, such a strip is said to be bounded by the pair if $u(s, i) \in L_i, i = 0, 1.$ Under mild conditions, such a strip has limits as $s \to \infty$ that are intersection points between the Lagrangian submanifolds.
Robbin & Salamon proved that If the Lagrangian intersections are transverse, the Fredholm index between suitable Sobolev spaces of this linearized Cauchy-Gromov-Riemann operator coincides with the Maslov-Viterbo of the strip. Their proof involves general considerations for linear operators of the form $\partial_s + A_s$ defined on the space of paths $\mathbb R$ to some Hilbert space and rely on reducing the problem to a finite dimensional Hilbert space.
However the eventual result admits a purely intrinsic formulation : it states that the Fredholm index of a Dirac operator is given in terms of a characteristic class. This seems like a particular instance of the Atiyah-Singer index theorem, only on a manifold with boundary (the strip) and with totally real boundary conditions.
Can this particular result (index = Maslov class) be obtained through a less coordinate-bound and maybe more striking way ?
Best Answer
In some sense this really goes back to pre-index theory days to Vekua and was one of the motivations for the index theorem (for a reference to Vekua see Gromov's psuedo-holomorphic curves paper and I think there is long discussion in Booss and Bleecker). Vekua proved the following. Take a map from $f:S^1 \to \mathbb{C}^*$. Consider the operator $u \mapsto \bar \partial u + a u + b \bar u$ on the domain $ u\in C^\infty(D,\mathbb{C})$ so that $ Re(\bar f u|{\partial D})=0 $ mapping to $C^\infty(D,\mathbb{C})$. This operator has index equal to the degree of $f$. You can translate this to a Maslov index. It forces the boundary values of $f$ to lie in this line which spins around according to the degree of $f$. The index is really rather easy to compute once you prove that this is a Fredholm boundary value problem. To compute the index, use homotopy invariance of the index to reduce to a model computation so that $f$ is homotopic to $z^k$ for some $k \in \mathbb{Z}$ and $a=b=0$. Then the kernel consist of holomorphic functions $u$ on the disk which satisfy the boundary condition, if $k\ge 0$ these are the real span of the polynomials $iz^k, z^{k-1}-z^{k+1},z^{k-2}-z^{k+2},\ldots$ so that its dimension is $k+1$. For $k \ge 0$ the kernel is zero. The cokernel consists of anti-holomorphic $v$ functions which satisfy the adjoint boundary condition $Im(f v|_{\partial D^2})=0$. For $k\le 0$ a real basis is given by $\bar z^k, \bar z^{k-1}+\bar z^{k+1},\bar z^{k-2}+\bar z^{k+2},\ldots$. Thus the index is $k$ no mater the sign of $k$.