Root stacks and periodic decompositions

Abstract For an effective Cartier divisor D on a scheme X we may form an n^ {th} n th root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is 2n 2 n -periodic. For n=2 n = 2 this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For n > 2 n > 2 we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N -spherical functors and categorification of Euler’s continuants. arXiv: 2306. 13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.