 Solution

The classical elimination in parts b and c discarded the field-dependent Gaussian functional integral determinant. At one loop that determinant is a local curvature coupling and supplies the Buscher rules dilaton shiftThe metric alone consequently need not obey ${\tilde{R}}_{\mu \nu }=0$. The relevant leading sigma-model beta function in the dual background instead containsThe new dilaton term cancels the failure of the dual metric to be Ricci-flat, so the complete metric-dilaton background remains conformal and physically T-dual to the original one.