string theory

derive the supersymmetric Yang-Mills theory:
Attaching Chan-Paton charges (internal, current algebra?) to the endpoints of open strings, they live in the fundamental rep. Orientability influences the number of dim of rep and thus the symmetry group (eg. orientable~ $U (N)$ , unorientable~sym- $U S p (N)$ , asym- $S O (N)$ ). Using basis and CP matrices, strings become the adjoint rep ( $?$ ).
Applying T-duality to a open string on the compactified dimensions, the discrete momentum will be mapped to a fractional winding number (when Wilson line has $θ_{i} \neq θ_{j}$ , namely D-branes are separate) or an integer one (when overlapping). Here, $θ_{i}$ are the labels of Wilson line, related to the momentum and thus being the angular coordinates of D-branes in T-dual theory, with connection (excited states) flat since the compactified dimension is of tiny scale. (similar to standing waves in a box) When N D-branes overlap, the symmetry $U (1)^{N}$ is enhanced to $U (N)$ (symmetry enhancement).
M theory