Twisted points of quotient stacks, integration and BPS-invariants

TL;DR

This paper develops a formalism connecting $p$-adic integrals on quotient stacks with BPS-invariants in moduli problems, generalizing orbifold formulas and recovering known invariance results for sheaves on del Pezzo surfaces.

Contribution

It introduces a new formula relating $p$-adic integrals to the cyclotomic inertia stack and applies it to compute refined BPS-invariants, extending orbifold formulas to broader contexts.

Findings

Derived a formula expressing $p$-adic integrals via the cyclotomic inertia stack.

Showed that refined BPS-invariants are computed locally by $p$-adic integrals.

Recovered $ ext{chi}$-independence of invariants for sheaves on del Pezzo surfaces.

Abstract

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 problems associated to hereditary abelian categories with symmetric Euler pairing, 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 for $1$-dimensional sheaves on del Pezzo surfaces 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.