BPS invariants from \(p\)-adic integrals

We define \(p\)-adic BPS or \(p\)BPS-invariants for moduli spaces \(M_{\beta,\chi}\) of 1-dimensional sheaves on del Pezzo surfaces by means of integration over a non-archimedean local field \(F\) . Our definition relies on a canonical measure \(\mu_{can}\) on the \(F\)-analytic manifold associated to \(M_{\beta,\chi}\) and the \(p\)BPS-invariants are integrals of natural \(\mathbb{G}_m\)-gerbes with respect to \(\mu_{can}\). A similar construction can be done for meromorphic Higgs bundles on a curve. Our main theorem is a \(\chi\)-independence result for these \(p\)BPS-invariants. For 1-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of \(p\)BPS with usual BPS-invariants trough a result of Maulik-Shen.