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

We use ideas from the Strominger–Yau–Zaslow conjecture and microlocal sheaf theory to define graded modules associated with permissible $C^{\mathrm{\infty }}$-divisors on compact tropical manifolds. A $C^{\mathrm{\infty }}$-divisor is a generalization of a Lagrangian section on an integral affine manifold. The group of $C^{\mathrm{\infty }}$-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.