Exact path integral for 3D higher spin gravity

Extending the works of [Phys. Rev. Lett. 115, 161304 (2015)] and [Phys. Rev. D 93, 064014 ( 2016)], we study three dimensional Euclidean higher spin gravity with negative cosmological constant. This theory can be formulated in terms of SL(N, C) Chern-Simons theory. By introducing auxiliary fields, we rewrite it in a supersymmetric way and compute its partition function exactly by using the localization method. We obtain a good expression for the partition function in terms of characters for the vacuum and primaries in 2D unitary conformal field theory with W-N symmetry. We also check that the coefficients of the character expansion are positive integers and exhibit Cardy formula in the large central charge limit.