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BPS invariants from p-adic integrals

Part of: Algebraic number theory: local and $p$-adic fields Projective and enumerative geometry

Published online by Cambridge University Press:  30 May 2024

Francesca Carocci Open the ORCID record for Francesca Carocci [Opens in a new window] ,
Giulio Orecchia  and
Dimitri Wyss
Francesca Carocci
Affiliation:
Mathematics Department, University of Geneva, Office 6.05, Rue du Conseil-Général 7-9, CH-1204 Geneva, Switzerland francesca.carocci@unige.ch
Giulio Orecchia
Affiliation:
White Oak Asset Management, Rue du Rhône 17, CH-1204 Geneva, Switzerland giulioorecchia@gmail.com
Dimitri Wyss
Affiliation:
Ecole Polytechnique Fédérale de Lausanne (EPFL), Chair of Arithmetic Geometry, CH-1015 Lausanne, Switzerland dimitri.wyss@epfl.ch
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Abstract

We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ 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 $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

Keywords

moduli of sheavesDonaldson–Thomas invariantsp-adic integration

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.)

Information

Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 7 , July 2024 , pp. 1525 - 1550
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© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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