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Refined stable pair invariants for E-, M- and [p,q]-strings

Min-xin Huang, Albrecht Klemm, Maximilian Poretschkin

TL;DR

This work computes refined BPS invariants for local CY geometries built from del Pezzo bases and the half K3, using mirror symmetry, refined holomorphic anomaly equations, and modularity. It develops both a direct-integration approach and elliptic-curve mirror methods to extract $N^eta_{J_L,J_R}$, organizing results into Weyl orbits of $E_n$ and relating them to M-/F-/heterotic dualities. The authors provide detailed genus-zero and higher-genus computations, analyze toric and non-toric embeddings, and reveal a rich interplay between modular objects (Weierstrass data, Eisenstein and Jacobi forms) and physical spectra (E-strings, [p,q]-strings, M-strings). They also extend the framework to mass deformations and to the massive half K3 case, uncovering universal structures and suggesting deep connections to gauge-theory moduli and open-string dynamics. Overall, the paper furnishes a comprehensive, modular, and geometry-driven account of refined BPS spectra across a wide class of elliptic singularities with significant implications for F-theory and related dualities.

Abstract

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.

Refined stable pair invariants for E-, M- and [p,q]-strings

TL;DR

This work computes refined BPS invariants for local CY geometries built from del Pezzo bases and the half K3, using mirror symmetry, refined holomorphic anomaly equations, and modularity. It develops both a direct-integration approach and elliptic-curve mirror methods to extract , organizing results into Weyl orbits of and relating them to M-/F-/heterotic dualities. The authors provide detailed genus-zero and higher-genus computations, analyze toric and non-toric embeddings, and reveal a rich interplay between modular objects (Weierstrass data, Eisenstein and Jacobi forms) and physical spectra (E-strings, [p,q]-strings, M-strings). They also extend the framework to mass deformations and to the massive half K3 case, uncovering universal structures and suggesting deep connections to gauge-theory moduli and open-string dynamics. Overall, the paper furnishes a comprehensive, modular, and geometry-driven account of refined BPS spectra across a wide class of elliptic singularities with significant implications for F-theory and related dualities.

Abstract

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.

Paper Structure

This paper contains 62 sections, 323 equations, 21 figures, 16 tables.

Table of Contents

  1. Introduction
  2. The refined BPS invariants on local Calabi-Yau spaces
  3. Mathematical definition of the $N^\beta_{J_l,J_R}$
  4. The direct integration approach
  5. Elliptic curve mirrors and closed modular expressions
  6. The genus zero sector
  7. The genus zero sector
  8. Refining the higher genus sector in the rigid cases
  9. The gap condition
  10. Mirror symmetry for non-compact Calabi-Yau spaces with del Pezzo basis
  11. The Del Pezzo surfaces and the half K3 surface
  12. The general structure
  13. Algebraic realizations
  14. Del Pezzo surfaces and semi-stable $G$-bundles on elliptic curves ${\cal E}$
  15. Reduction of the structure group and almost del Pezzo surfaces
  16. ...and 47 more sections

Figures (21)

  • Figure 1:
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  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 16 more figures