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 parametrization
$$
p^\alpha=\frac1{2\pi\alpha'}\epsilon^{\alpha\beta}\partial_\beta\widetilde Z.
$$
Substitution into $S_2$ gives
$$
S_{\rm dual}[Y,\widetilde Z]
=\frac1{4\pi\alpha'}\int d^2\sigma
\left(G_{ij}(Y)\partial^\alpha Y^i\partial_\alpha Y^j
+V(Y)^{-1}\partial^\alpha\widetilde Z\partial_\alpha\widetilde Z\right).
$$
This is the <Buscher procedure> for the translation isometry in $z$, and its <T-duality> replaces $G_{zz}=V$ by $\widetilde G_{\widetilde z\widetilde z}=V^{-1}$.

Solution

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

{c}

The Buscher procedure derives T-dual backgrounds by replacing the derivative of an isometric target coordinate with an independent worldsheet one-form and imposing its flatness with a Lagrange multiplier. Integrating out either field yields the original or dual sigma model.


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 Incoming links: Buscher procedure