Root stacks and periodic decompositions

For an effective Cartier divisor D on a scheme X we may form an nᵗʰ 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.