arXiv:1109.1923v2 [hep-th] 5 Oct 2011 The AdS 3 central charge in string theory Jan Troost a a Laboratoire de Physique Th´ eorique Unit´ e Mixte du CNRS et de l’ ´ Ecole Normale Sup´ erieure associ´ ee ` a l’Universit´ e Pierre et Marie Curie 6 UMR 8549 ´ Ecole Normale Sup´ erieure 24 Rue Lhomond Paris 75005, France Abstract We evaluate the vacuum expectation value of the central charge operator in string theory in an AdS 3 vacuum. Our calculation provides a rare non-zero one-point function on a spherical worldsheet. The evaluation involves the regularization both of a worldsheet ultraviolet divergence (associated to the infinite volume of the conformal Killing group), and a space-time infrared divergence (corresponding to the infinite volume of space-time). The two divergences conspire to give a finite result, which is the classical general relativity value for the central charge, corrected in bosonic string theory by an infinite series of tree level higher derivative terms. 1