We first investigate Abelian duality at the classical and quntum level . In the quantum considerations we suppose our manifold is closed . We briefly introduce T-duality and the importance of Abelian duality in the string context is declared . At first , it was assumed that the manifold is without boundary , but then we consider our manifold to have a boundary and therefore , boundary conditions came into play . After calculations in which general boundary conditions take part , we show that under the action of Abelian duality Neumann boundary condition is replaced by Dirichlet boundary condition and vice versa . Finally we calculate the partition function and the two-point function , which is considered on the upper half-plane , then by using the Cayley transform we show what it will be when we have a disk instead .