On gravitational interactions for massive higher spins in $AdS_3$

In this paper we investigate gravitational interactions of massive higher spin fields in three dimensional $AdS$ space with arbitrary value of cosmological constant including flat Minkowski space. We use frame-like gauge description for such massive fields adopted to three-dimensional case. At first, we carefully analyze the procedure of switching on gravitational interactions in the linear approximation on the example of massive spin-3 field and then proceed with the generalization to the case of arbitrary integer spin field. As a result we construct a cubic interaction vertex linear in spin-2 field and quadratic in higher spin field on $AdS_3$ background. As in the massless case the vertex does not contain any higher derivative corrections to the Lagrangian and/or gauge transformations. Thus, even after switching on gravitational interactions, one can freely consider any massless or partially massless limits as well as the flat one.

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