= 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 $R'=\alpha'/R$. Fixed points, noncompact orbits and global identifications require additional analysis; a local spacelike <Killing vector> does not specify these data.
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