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On Flux Quantization in F-Theory

Andres Collinucci, Raffaele Savelli

TL;DR

This work establishes that in F-theory, smooth elliptic Calabi–Yau fourfolds with a Weierstrass presentation always yield integrally quantized $G_4$ flux, so half-integral quantization must originate from singularities on 7-branes. By deploying toric resolutions and explicit SU(2) and Sp(N) singularities, the authors relate $G_4$ quantization to Freed-Witten constraints in perturbative IIB via Sen's limit and a D9–anti-D9 tachyon condensation picture, finding half-integer fluxes precisely when D7-branes wrap non-spin divisors. They derive general formulas for curvature- and flux-induced D3 tadpoles, and show that brane recombination/bound-state processes preserve total tadpoles while distributing flux between geometrical and gauge sectors. The results unify M-/F-theory and IIB perspectives, illuminating how half-integral fluxes pin Freed-Witten-type anomalies to Cartan fluxes on D7 stacks and clarifying the role of base geometry, blow-ups, and B-field choices in flux quantization and tadpole cancellation.

Abstract

We study the problem of four-form flux quantization in F-theory compactifications. We prove that for smooth, elliptically fibered Calabi-Yau fourfolds with a Weierstrass representation, the flux is always integrally quantized. This implies that any possible half-integral quantization effects must come from 7-branes, i.e. from singularities of the fourfold. We subsequently analyze the quantization rule on explicit fourfolds with Sp(N) singularities, and connect our findings via Sen's limit to IIB string theory. Via direct computations we find that the four-form is half-integrally quantized whenever the corresponding 7-brane stacks wrap non-spin complex surfaces, in accordance with the perturbative Freed-Witten anomaly. Our calculations on the fourfolds are done via toric techniques, whereas in IIB we rely on Sen's tachyon condensation picture to treat bound states of branes. Finally, we give general formulae for the curvature- and flux-induced D3 tadpoles for general fourfolds with Sp(N) singularities.

On Flux Quantization in F-Theory

TL;DR

This work establishes that in F-theory, smooth elliptic Calabi–Yau fourfolds with a Weierstrass presentation always yield integrally quantized flux, so half-integral quantization must originate from singularities on 7-branes. By deploying toric resolutions and explicit SU(2) and Sp(N) singularities, the authors relate quantization to Freed-Witten constraints in perturbative IIB via Sen's limit and a D9–anti-D9 tachyon condensation picture, finding half-integer fluxes precisely when D7-branes wrap non-spin divisors. They derive general formulas for curvature- and flux-induced D3 tadpoles, and show that brane recombination/bound-state processes preserve total tadpoles while distributing flux between geometrical and gauge sectors. The results unify M-/F-theory and IIB perspectives, illuminating how half-integral fluxes pin Freed-Witten-type anomalies to Cartan fluxes on D7 stacks and clarifying the role of base geometry, blow-ups, and B-field choices in flux quantization and tadpole cancellation.

Abstract

We study the problem of four-form flux quantization in F-theory compactifications. We prove that for smooth, elliptically fibered Calabi-Yau fourfolds with a Weierstrass representation, the flux is always integrally quantized. This implies that any possible half-integral quantization effects must come from 7-branes, i.e. from singularities of the fourfold. We subsequently analyze the quantization rule on explicit fourfolds with Sp(N) singularities, and connect our findings via Sen's limit to IIB string theory. Via direct computations we find that the four-form is half-integrally quantized whenever the corresponding 7-brane stacks wrap non-spin complex surfaces, in accordance with the perturbative Freed-Witten anomaly. Our calculations on the fourfolds are done via toric techniques, whereas in IIB we rely on Sen's tachyon condensation picture to treat bound states of branes. Finally, we give general formulae for the curvature- and flux-induced D3 tadpoles for general fourfolds with Sp(N) singularities.

Paper Structure

This paper contains 23 sections, 144 equations, 1 table.

Table of Contents

  1. Introduction
  2. Setting up the problem
  3. Smooth Calabi-Yau fourfolds
  4. Proposition:
  5. Proof:
  6. The perturbative type IIB perspective
  7. Review of Sen's limit, and tachyon condensation
  8. Non-spin O7-plane
  9. O3-planes
  10. Remark
  11. Singular Calabi-Yau fourfolds
  12. SU(2) singularities
  13. Sen's limit of the SU(2) configuration
  14. Bound states via tachyon condensation
  15. Unorientifolded case
  16. ...and 8 more sections