Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

BPS invariants from $p$-adic integrals

Francesca Carocci, Giulio Orecchia, Dimitri Wyss

Abstract

We define $p$-adic BPS or $p$BPS-invariants for moduli spaces $M_{β,χ}$ 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 $μ_{can}$ on the $F$-analytic manifold associated to $M_{β,χ}$ and the $p$BPS-invariants are integrals of natural $\mathbb{G}_m$-gerbes with respect to $μ_{can}$. A similar construction can be done for meromorphic Higgs bundles on a curve. Our main theorem is a $χ$-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.

BPS invariants from $p$-adic integrals

Abstract

We define -adic BPS or BPS-invariants for moduli spaces of 1-dimensional sheaves on del Pezzo surfaces by means of integration over a non-archimedean local field . Our definition relies on a canonical measure on the -analytic manifold associated to and the BPS-invariants are integrals of natural -gerbes with respect to . A similar construction can be done for meromorphic Higgs bundles on a curve. Our main theorem is a -independence result for these BPS-invariants. For 1-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of BPS with usual BPS-invariants trough a result of Maulik-Shen.
Paper Structure (18 sections, 20 theorems, 67 equations)

This paper contains 18 sections, 20 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Cohomological $\chi$--independence and trace of Frobenius
  3. p-adic BPS function and its invariance
  4. Relation to previous works, consequences and final considerations
  5. Moduli spaces of sheaves and Higgs bundles
  6. Relative moduli spaces of sheaves on surfaces
  7. The open sub-stack of invertible sheaves
  8. Excluding the non-reduced locus
  9. Reduced curves
  10. Moduli of Higgs bundles
  11. $p$-adic integration
  12. The measure on the moduli spaces
  13. The canonical measure on moduli spaces of sheaves
  14. Gerbes and Tate duality
  15. ${\mathbb{G}}_m$-gerbes
  16. ...and 3 more sections

Key Result

Theorem 1.2.2

Let $S \to \mathop{\mathrm{\mathsf{Spec}}}\nolimits({\mathcal{O}})$ be either a smooth projective del Pezzo surface or a K3 surface satisfying Assumption K3as. The function $p\mathrm{BPS}_{\beta,\chi}: \mathrm{M}_{\beta,\chi}(k) \rightarrow {\mathbb{C}}$ satisfies the following two properties:

Theorems & Definitions (45)

  • Theorem 1.2.2: \ref{['mainthm']}
  • Definition 2.1.1
  • Theorem 2.1.3
  • proof
  • Lemma 2.1.4
  • proof
  • Theorem 2.1.5
  • Corollary 2.1.6
  • proof
  • Theorem 2.1.7
  • ...and 35 more