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Tate Form and Weak Coupling Limits in F-theory

Mboyo Esole, Raffaele Savelli

TL;DR

This work systematically analyzes weak coupling limits of F-theory with Tate-form gauge groups, highlighting conifold singularities that obstruct crepant resolutions in the Donagi-Wijnholt limit for unitary groups. It introduces two refined weak-coupling regimes—the quadric-cone and suspended pinch point (spp) limits—that replace conifold points with singularities amenable to crepant resolutions compatible with orientifold symmetry, preserving unitary or yielding orthogonal/gauge structures as appropriate. The authors provide detailed geometric resolutions, brane spectra, and matter-curves analyses for SU(n) and certain exceptional groups, and study the resulting IIB Calabi–Yau threefolds, uncovering rich phenomenological implications for GUTs and brane dynamics. While these constructions illuminate viable weakly coupled uplifts and their matter content, they also reveal subtleties such as D7-tadpole cancellation constraints and the limitations of perturbative open-string pictures for exceptional groups, guiding future explorations of fully consistent F-theory–IIB dual descriptions.

Abstract

We consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several non-equivalent directions to flow from strong to weak coupling leading to different gauge groups.

Tate Form and Weak Coupling Limits in F-theory

TL;DR

This work systematically analyzes weak coupling limits of F-theory with Tate-form gauge groups, highlighting conifold singularities that obstruct crepant resolutions in the Donagi-Wijnholt limit for unitary groups. It introduces two refined weak-coupling regimes—the quadric-cone and suspended pinch point (spp) limits—that replace conifold points with singularities amenable to crepant resolutions compatible with orientifold symmetry, preserving unitary or yielding orthogonal/gauge structures as appropriate. The authors provide detailed geometric resolutions, brane spectra, and matter-curves analyses for SU(n) and certain exceptional groups, and study the resulting IIB Calabi–Yau threefolds, uncovering rich phenomenological implications for GUTs and brane dynamics. While these constructions illuminate viable weakly coupled uplifts and their matter content, they also reveal subtleties such as D7-tadpole cancellation constraints and the limitations of perturbative open-string pictures for exceptional groups, guiding future explorations of fully consistent F-theory–IIB dual descriptions.

Abstract

We consider the weak coupling limit of F-theory in the presence of non-Abelian gauge groups implemented using the traditional ansatz coming from Tate's algorithm. We classify the types of singularities that could appear in the weak coupling limit and explain their resolution. In particular, the weak coupling limit of SU(n) gauge groups leads to an orientifold theory which suffers from conifold singulaties that do not admit a crepant resolution compatible with the orientifold involution. We present a simple resolution to this problem by introducing a new weak coupling regime that admits singularities compatible with both a crepant resolution and an orientifold symmetry. We also comment on possible applications of the new limit to model building. We finally discuss other unexpected phenomena as for example the existence of several non-equivalent directions to flow from strong to weak coupling leading to different gauge groups.

Paper Structure

This paper contains 32 sections, 6 theorems, 122 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Elliptic curves and Weierstrass models: a quick review
  3. Elliptic fibration
  4. Formulaire
  5. Weak coupling limit of elliptic fibrations
  6. Sen's limit of Weierstrass models
  7. The orientifold interpretation of Sen's limit
  8. Sen's limit for a Weierstrass model in Tate form
  9. The Donagi-Wijnholt ansatz
  10. Gauge groups and singularities in the weak coupling limit
  11. Singular double covers from the Donagi-Wijnholt limit
  12. $SO(k)$ , $G_2$, $\mathop{\rm {}Sp}(1)$ and $\mathop{\rm SU}(3)$ and quadric cone singularities
  13. Exceptional groups $F_4,E_6, E_7, E_8$ and Whitney umbrella
  14. Unitary groups and conifold singularities.
  15. Solving the conifold problem for the Tate form of unitary gauge groups
  16. ...and 17 more sections

Key Result

Proposition A.3

Let $B$ be any variety over a field of characteristic different than $2$. There is a one-to-one correspondence between double covers $\rho:X\rightarrow B$ and pairs $(\mathscr{L}, \underline{O})$ consisting of a line bundle $\mathscr{L}$ and the divisor $\underline{O}\in \mathscr{L}^{2}$.

Theorems & Definitions (11)

  • Definition 2.1: Elliptic curve and Weierstrass normal equation
  • Remark 3.1
  • Definition A.1: Finite maps
  • Definition A.2: double cover
  • Proposition A.3: Characterization of the space of double covers over a fixed base (CSMiranda)
  • Proposition A.4: Algebraic properties of double covers
  • Proposition A.5: Characterization of smooth double covers
  • Proposition A.6: Resolution of double covers (see for example CS
  • Proposition A.7
  • proof
  • ...and 1 more