# Polchinski String Theory (8.4.38) T-duality of two compactified dimensions

I am confused about a sentence in Polchinski's String theory chapter 8 p 255 when he works out the example of the full $$T$$-duality with two compact dimensions. He writes

"A simultaneous $$T$$-duality on $$X^{24,25}$$ acts as $$\rho\rightarrow -1/\rho$$ with $$\tau$$ invariant."

Here $$\rho$$ and $$\tau$$ are two complex fields that contain the four moduli $$G_{24,24}, G_{24,25}, G_{25,25}$$ and $$B_{24,25}$$.

I am not able to derive this. What am I missing?

Details

The start is the non-linear model $$S= \frac{1}{4\pi\alpha'}\int d^2\sigma \, \sqrt{g} \left[ \left(g^{ab} G_{\mu\nu}(X) + i \epsilon^{ab} B_{\mu\nu} \right)\partial_a X^\mu \partial_b X^\nu + \alpha' R \Phi(X) \right]$$ Introduce the moduli $$\rho$$ and $$\tau$$ in (8.4.36) and (8.4.37), i.e. $$G_{24,24} = \frac{\alpha' \rho_2}{R^2\tau_2}; \quad G_{25,25} = \frac{\alpha' \rho_2 |\tau|^2 }{R^2\tau_2} ;\quad G_{24,25} = \frac{ \alpha' \rho_2\tau_1}{R^2\tau_2};\quad B_{24,25} = \frac{\alpha'\rho_1}{R^2}$$ with inverse $$\rho_1 = \frac{R^2}{\alpha'} B_{24,25} ;\quad \rho_2 = \frac{R^2}{\alpha'} \sqrt{ G} ;\quad \tau_1 = \frac{G_{24,25}}{G_{24,24}};\quad \tau_2 = \frac{\sqrt{ G}}{ G_{24,24}}$$ where $$G= \det G_{mn}$$.

A simultaneous $$T$$-duality on $$X^{24,25}$$ changes $$R\longrightarrow \alpha' /R$$ and leaves the $$X$$ invariant. This implies that also the $$G_{mn}$$ and $$B_{mn}$$ are invariant (I think?) and thus $$\rho_1 \longrightarrow \rho_1'= \frac{\alpha'^2}{R^4} \rho_1;\quad \rho_2 \longrightarrow \rho_2'= \frac{\alpha'^2}{R^4} \rho_2$$ and leaves $$\tau$$ unchanged. But that does not correspond to $$\rho\longrightarrow -1/\rho$$. What have I missed?