Global qualifications of Abelian T-duality

A local Buscher procedure sets a flat auxiliary connection equal to a coordinate derivative only on a simply connected patch. A full compact-circle Polyakov path integral must also sum flat-connection holonomies, match the period of the dual multiplier, and include momentum and winding modes. A free circle isometry then gives the radius relation $R^{\prime }={\alpha }^{\prime }/R$. Fixed points, noncompact orbits and global identifications require additional analysis; a local spacelike Killing vector does not specify these data.