Universal Maslov class of a Bohr–Sommerfeld Lagrangian embedding into a pseudo-Einstein manifold

We show that in the case of a Bohr–Sommerfeld Lagrangian embedding into a pseudo-Einstein symplectic manifold, a certain universal 1-cohomology class, analogous to the Maslov class, can be defined. In contrast to the Maslov index, the presented class is directly related to the minimality problem for Lagrangian submanifolds if the ambient pseudo-Einstein manifold admits a Kähler–Einstein metric. We interpret the presented class geometrically as a certain obstruction to the continuation of one-dimensional supercycles from the Lagrangian submanifold to the ambient symplectic manifold.