BPS invariants from $p$-adic integrals

Abstract

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.

24 pages, comments welcome!