Twisted points of quotient stacks, integration and BPS-invariants

We study \(p\)-adic manifolds associated with twisted points of quotient stacks \(\mathcal{X} = [U/G]\) and their quotient spaces \(\pi:\mathcal{X} \to X\). We prove several structural results about the fibres of \(\pi\) and derive in particular a formula expressing \(p\)-adic integrals on \(X\) in terms of the cyclotomic inertia stack of \(\mathcal{X}\), generalizing the orbifold formula for Deligne-Mumford stacks. We then apply our formalism to moduli stacks of \(1\)-dimensional sheaves on del Pezzo surfaces and show that their refined BPS-invariants are computed locally on the coarse moduli space by a \(p\)-adic integral. As a consequence we recover the \(\chi\)-independence of these invariants previously proven by Maulik-Shen. Along the way we derive a new formula for the plethystic logarithm on the \(\lambda\)-ring of functions on \(k\)-linear stacks, which might be of independent interest.