BPS invariants from p-adic integrals

We define p -adic BPS or p BPS invariants for moduli spaces M, of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field F. Our definition relies on a canonical measure ₂₀₍ on the F -analytic manifold associated to M, and the p BPS invariants are integrals of natural {G}ₘ gerbes with respect to ₂₀₍. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a -independence result for these p BPS invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of p BPS with usual BPS invariants through a result of Maulik and Shen Cohomological -independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586.