BPS invariants from $p$-adic integrals

Francesca Carocci; Giulio Orecchia; Dimitri Wyss

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arXiv · 2112.12103

BPS invariants from $p$-adic integrals

Francesca Carocci·Giulio Orecchia·Dimitri Wyss

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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.

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Francesca Carocci, Giulio Orecchia, Dimitri Wyss. 2021-12-22. BPS invariants from $p$-adic integrals. https://arxiv.org/abs/2112.12103

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