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General U(1)xU(1) F-theory Compactifications and Beyond: Geometry of unHiggsings and novel Matter Structure

Mirjam Cvetic, Denis Klevers, Hernan Piragua, Washington Taylor

Abstract

We construct the general form of an F-theory compactification with two U(1) factors based on a general elliptically fibered Calabi-Yau manifold with Mordell-Weil group of rank two. This construction produces broad classes of models with diverse matter spectra, including many that are not realized in earlier F-theory constructions with U(1)xU(1) gauge symmetry. Generic U(1)xU(1) models can be related to a Higgsed non-Abelian model with gauge group SU(2)xSU(2)xSU(3), SU(2)^3xSU(3), or a subgroup thereof. The nonlocal horizontal divisors of the Mordell-Weil group are replaced with local vertical divisors associated with the Cartan generators of non-Abelian gauge groups from Kodaira singularities. We give a global resolution of codimension two singularities of the Abelian model; we identify the full anomaly free matter content, and match it to the unHiggsed non-Abelian model. The non-Abelian Weierstrass model exhibits a new algebraic description of the singularities in the fibration that results in the first explicit construction of matter in the symmetric representation of SU(3). This matter is realized on double point singularities of the discriminant locus. The construction suggests a generalization to U(1)^k factors with k>2, which can be studied by Higgsing theories with larger non-Abelian gauge groups.

General U(1)xU(1) F-theory Compactifications and Beyond: Geometry of unHiggsings and novel Matter Structure

Abstract

We construct the general form of an F-theory compactification with two U(1) factors based on a general elliptically fibered Calabi-Yau manifold with Mordell-Weil group of rank two. This construction produces broad classes of models with diverse matter spectra, including many that are not realized in earlier F-theory constructions with U(1)xU(1) gauge symmetry. Generic U(1)xU(1) models can be related to a Higgsed non-Abelian model with gauge group SU(2)xSU(2)xSU(3), SU(2)^3xSU(3), or a subgroup thereof. The nonlocal horizontal divisors of the Mordell-Weil group are replaced with local vertical divisors associated with the Cartan generators of non-Abelian gauge groups from Kodaira singularities. We give a global resolution of codimension two singularities of the Abelian model; we identify the full anomaly free matter content, and match it to the unHiggsed non-Abelian model. The non-Abelian Weierstrass model exhibits a new algebraic description of the singularities in the fibration that results in the first explicit construction of matter in the symmetric representation of SU(3). This matter is realized on double point singularities of the discriminant locus. The construction suggests a generalization to U(1)^k factors with k>2, which can be studied by Higgsing theories with larger non-Abelian gauge groups.

Paper Structure

This paper contains 62 sections, 154 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. U(1)'s and unHiggsing
  3. Higgsing and a single U(1)
  4. Higgsing an SU(2)
  5. UnHiggsing a U(1)
  6. Higgsing and two U(1)'s: simple constructions
  7. Higgsing SU(2)$\times$SU(2)
  8. Higgsing SU(3)
  9. Hybrid models from SU(2)$\times$SU(2)$\times$SU(3)
  10. Singular divisors and higher charges
  11. Examples
  12. General elliptic fibrations with three sections
  13. A new elliptic curve
  14. Specialized models
  15. The Weierstrass form
  16. ...and 47 more sections

Figures (10)

  • Figure 1: Weight lattice of charges in a theory with a U$(1)\times$U(1) gauge group realized by Higgsing a theory with gauge group SU(2)$\times$SU(2). Dots in red indicate the adjoint, dots in blue the bifundamental and black dots are the fundamentals. The origin is indicated by an extra circle.
  • Figure 2: Weight lattice of charges in a theory with a U$(1)\times$U(1) gauge group realized by Higgsing a theory with gauge group SU(3). Dots in red indicate the adjoint and black dots are the fundamental and antifundamental. The origin is indicated by an extra circle.
  • Figure 3: Weight lattice of charges in a theory with a U$(1)\times$U(1) gauge group realized by Higgsing a theory with gauge group SU(2)$\times$SU(2)$\times$SU(3). The origin is indicated by an extra circle.
  • Figure 4: Weight lattice of charges in a theory with a U$(1)\times$U(1) gauge group realized by Higgsing a theory with gauge group SU(2)$\times$SU(2)$\times$SU(3), when the SU(3) carries matter in the symmetric representation, associated in F-theory with singular points on the curve $C$ carrying the SU(3) factor.
  • Figure 5: Cubic $\mathcal{E}$ with three rational points $P$, $Q$ and $R$ contained in the line $u=0$.
  • ...and 5 more figures