Graded modules associated with permissible $C^{\infty}$-divisors on tropical manifolds

We use ideas from the Strominger--Yau--Zaslow conjecture and microlocal sheaf theory to define graded modules associated with permissible $C^{\infty}$-divisors on compact tropical manifolds. A $C^{\infty}$-divisor is a generalization of a Lagrangian section on an integral affine manifold. The group of $C^{\infty}$-divisors on a tropical manifold surjects onto the Picard group. We also prove a Riemann--Roch formula for compact tropical curves and integral affine manifolds admitting Hessian forms. Our approach differs from the tropical Riemann--Roch theorem established by Gathmann and Kerber.