Buscher dilaton shift (${\phi }^{\prime }=\phi -\frac{1}{2}\log g_{yy}$)

The quantum measure in abelian T-duality changes the conventional dilaton by ${\phi }^{\prime }=\phi -\frac{1}{2}\log g_{yy}$ in alpha-prime-one units for a spacelike isometry. The regulated Gaussian integration over the auxiliary gauge field produces this curvature coupling. It is consistent with invariance of the string-frame density $e^{-2\phi }\sqrt{\det g}$ under the local Buscher rules.