Root stacks and periodic decompositions

For an effective Cartier divisor D on a scheme X we may form an \({n}^{\text {th}}\) $n^{\text{th}}$  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 \(2n\) $2n$ -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.