Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 306 / 4

The target metric is a coupling of the string nonlinear sigma model. Quantum consistency requires the gauge-fixed worldsheet theory to preserve Weyl invariance, so its sigma-model beta functions must vanish. With no B-field or varying dilaton, the metric beta function begins asTherefore vanishing of the worldsheet Weyl anomaly requiresto leading order in ${\alpha }^{\prime }$: the target-space metric must be Ricci-flat at this order.

The $p$-dependent part of the first-order action can be completed to a square:The Gaussian functional integral over $p$ therefore leavesafter the stipulated omission of its functional determinant. Identifying $X^{\mu }=(Y^i,Z)$ gives exactly $S_1[X]$, establishing the classical equivalence.

Integrating by parts makes the $Z$ dependence $-i\int Z{\partial }_{\alpha }{p}^{\alpha }$. Its functional integral imposes the constraint ${\partial }_{\alpha }{p}^{\alpha }=0$. On the simply connected plane, the Poincare lemma therefore permits the local and global parametrizationSubstitution into $S_2$ givesThis is the Buscher procedure for the translation isometry in $z$, and its T-duality replaces $G_{zz}=V$ by ${\tilde{G}}_{\tilde{z}\tilde{z}}=V^{-1}$.

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.