  In this thesis, we study a class of special Lagrangian submanifolds of toric
Calabi-Yau manifolds and construct their mirrors using some techniques
developed in the SYZ programme. We present a justification on the conjecture on
the mirror construction of D- branes in Aganagic-Vafa [2]. We apply the
techniques employed in Chan-Lau-Leung [8] and Chan [6], which give the SYZ
mirror construction for D-branes. Recently, Chan-Lau-Leung [8] has given an
explicit mirror construction for toric Calabi-Yau manifolds as complex
algebraic varieties. Inspired by the SYZ transformation of D-branes in
Leung-Yau-Zaslow[29], Chan[6] has defined a generalised mirror construction for
a larger class of Lagrangian subspaces, which is important to our situation.
The results point towards a need for further quantum correction of the SYZ
transform for Lagrangian subspaces in order to recover physicists mirror
prediction of counting holomorphic discs in Ooguri-Vafa [34]. When compare to
physicists prediction of the Aganagic-Vafa mirror branes, such naive SYZ
transform fails to coincide with the prediction. The main reasons lies in the
fact that the SYZ transformation does not capture open Gromov-Witten invariants
of the A-brane. Through the special case of Aganagic-Vafa A-branes, we are able
to modify the SYZ transform that involves further quantum correction using
discs counting of the A-brane. However it is not known how such a modified SYZ
transform can be constructed and works in general. We believe that the explicit
formulas and computations in the special case we consider here can give hints
to the correct modification in general in future development of the SYZ
programme.
