Spin-1 Bulk Local States in AdS/CFT and Carrollian Holography

Abstract: We construct the bulk local states of a massive spin-1 field in AdS$_3$/CFT$_2$ and three-dimensional Carrollian holography. In AdS, covariance under the isometries that leave a bulk point invariant determines the three components of the vector there and splits the solution into two branches of definite helicity, whose inner products reproduce the massive vector two-point function of AdS$_3$. By expanding the massive vector HKLL representation in boundary descendants, one can reproduce the three component states term by term. The two branches together describe the parity-invariant Proca field, and one alone the single propagating mode of Maxwell--Chern--Simons theory. The same conditions are solved in the induced representation of the Carrollian theory, where the solution again splits into two branches. Under the flat limit of the generators the highest weight module becomes the induced module, and the AdS states and Green's functions become their Carrollian counterparts, which supports the induced representation as the home of massive bulk excitations in Carrollian holography. This is the first bulk local state built with spin, complete from the state itself to its bulk-to-bulk propagator and to the limit relating the two descriptions.