Let $X$ be a surface. We know for any loop $\gamma$ we can find a form $\alpha$ such that integral along $\gamma$ equals integral wedge $\alpha$ on the surface. My question is, are these forms ($\alpha$, get from duality) exactly those which integral along every loop is an integer?
I checked this is true for torus, so I want to know if is true in general.
Indeed, the Poincaré dual of a loop always gives an integer when integrated on any loop.
Let $C$ be a smooth loop on a compact oriented smooth surface $X$. The Poincaré dual $\eta_C$ of $C$ defines a cohomology class $[\eta_C] \in H^1(X)$ with the property that for any $\omega \in H^1(X)$,$$\int_C \omega = \int_X \omega \wedge \eta_C.$$Claim. For any loop $C'$, $\int_{C'} \eta_C$ is an integer.
Proof. We have$$\int_{C'} \eta_C = \int_X \eta_C \wedge \eta_{C'} = \#(C \cdot C')$$because the wedge product is dual to intersection of cycles. The intersection of two cycles is always an integer.$$\tag*{$\square$}$$
$\text{}$1. $\#(C \cdot C')$ is the intersection number of the two oriented curves $C$ and $C'$. They can be jiggled slightly so that they become transversal. In that case, each intersection point is counted with multiplicity $+1$ or $-1$ depending on the orientations of $C$ and $C'$. Thus, the intersection number is always an integer.
You can find a nice discussion of intersection theory in Guillemin and Pollack's "Differential Topology" or in Griffiths and Harris's "Principles of Algebraic Geometry".
$\text{}$2. In one version of Poincaré duality, closed forms are dual to cycles and under this duality, the wedge product of closed forms correspond to the intersection of cycles. I am not sure where this is discussed—maybe Griffiths and Harris's "Principles of Algebraic Geometry" or some text on algebraic topology.