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?


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?


1 Answer 1


By lack of response from anyone, let me refer to Giveon et al. Target Space Duality in String theory, hep-th/9401139v1, specifically section 2 which works out the relevant symmetries and how the moduli transform under these symmetries in detail.


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