Chern-Simons Gauge Theory and the AdS_3/CFT_2 Correspondence

The bulk partition function of pure Chern-Simons theory on a three-manifold is a state in the space of conformal blocks of the dual boundary RCFT, and therefore transforms non-trivially under the boundary modular group. In contrast the bulk partition function of AdS_3 string theory is the modular-invariant partition function of the dual CFT on the boundary. This is a puzzle because AdS_3 string theory formally reduces to pure Chern-Simons theory at long distances. We study this puzzle in the context of massive Chern-Simons theory. We show that the puzzle is resolved in this context by the appearance of a chiral “spectator boson” in the boundary CFT which restores modular invariance. It couples to the conformal metric but not to the gauge field on the boundary. Consequently, we find a generalization of the standard Chern-Simons/RCFT correspondence involving “nonholomorphic conformal blocks” and nonrational boundary CFTs. These generalizations appear in the long-distance limit of AdS_3 string theory, where the role of the spectator boson is played by other degrees of freedom in the theory