Scattering of low lying states in the black hole atmosphere

We investigate finite ${\alpha }^{\prime }$ effects in string theory on a black hole background. By explicitly computing tree-level scattering amplitudes, we confirm a duality between seemingly different states recently conjectured by Giveon, Itzhaki, and Kutasov. We verify that the relevant 3-point functions factorize in such a way that the duality between oscillator and winding states becomes manifest. This leads us to determine the precise normalization of the dual vertex operators, and confirms at the level of the interacting theory the identification of states suggested by the analysis of the spectrum. This result implies a duality between two seemingly distinct mechanisms driving the violation of the string winding number in the black hole atmosphere.

Figure 1

The Euclidean black hole geometry resembles a semi-infinite cigar whose compact direction represents the periodic Euclidean time. The black hole horizon is located at the tip of the cigar ($r=0$). The black hole atmosphere corresponds to the cap, a region where the curvature in string units is of order $1/k$. Asymptotically, the geometry tends to a cylinder of radius $\sqrt{k}$ in string units. Strings can have nonzero winding number around the time direction $\theta$. Such winding numbers, however, can be violated in a scattering process due to the topology of the space.