Note on the spectral flow operator

The insertion of the spectral flow operator in a string scattering amplitude on ${\mathrm{AdS}}_3\times \mathcal{N}$ produces a change in the winding number of one of the incoming (or outgoing) states, making it possible to compute amplitudes of processes in which the winding number in ${\mathrm{AdS}}_3$ is not conserved. The insertion of such an operator, however, might seem artificial from the world sheet theory perspective, as it appears as an unintegrated vertex operator of conformal dimension zero that does not represent any normalizable state. Here, we show that the spectral flow operator naturally emerges in the Liouville field theory description of the Wess-Zumino-Witten (WZW) correlation functions once it is combined with a series of duality relations among conformal integrals. By considering multiple insertions of spectral flow operators, we study the dependence on the moduli for an arbitrary number of them, and we show explicitly that the amplitude does not depend on the specific locations of the accessory insertions in the world sheet, as required by consistency. This generalizes previous computations in which particular cases were considered. This can also be thought of as an alternative proof of the WZW-Liouville correspondence in the case of maximally winding-violating correlators.

Figure 1

Scheme of the $H_3^{+}$-Liouville correspondence. On the left, an $n$-point string scattering amplitude in ${\mathrm{AdS}}_3$ in which the total winding number is $\omega \equiv {\omega }_1+{\omega }_2+\cdots +{\omega }_n=1$; it is defined as an ($n+1$)-point correlation function in the $SL(2,\mathbb{R}{)}_k$ WZW model involving one (as $\omega =1$) spectral flow operator. On the right, the corresponding ($2n-3$)-point correlation function in LFT involving $n-2-\omega$ degenerate fields $V_{-1/(2b)}$ integrated on the sphere.