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 ?