Abstract
The Mordell-Weil lattices (MW lattices) associated to rational elliptic surfaces are classified into 74 types. Among them, there are cases in which the MW lattice is none of the weight lattices of simple Lie algebras or direct sums thereof. We study how such “non-Cartan MW lattices” are realized in the six-dimensional heterotic/F-theory compactifications. In this paper, we focus on non-Cartan MW lattices that are torsion free and whose associated singularity lattices are sublattices of A7. For the heterotic string compactification, a non-Cartan MW lattice yields an instanton gauge group H with one or more U(1) group(s). We give a method for computing massless spectra via the index theorem and show that the U(1) instanton number is limited to be a multiple of some particular non-one integer. On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over ℙ1. Except for some cases, we obtain such geometries for specific distributions of instantons. All the spectrum derived from those geometries completely match with the heterotic results.
Highlights
F-theory [1,2,3] has a unique feature in modern particle physics model building based on string theory
On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over P1
For every case of the non-Cartan type, where we have succeeded in finding an equation for the RES-fibered space, we show the complete match of the six-dimensional massless spectra read off from the Weierstrass equations on the F-theory side and those obtained by the index computations on the heterotic side for a special choice of instanton distribution
Summary
F-theory [1,2,3] has a unique feature in modern particle physics model building based on string theory. For every case of the non-Cartan type, where we have succeeded in finding an equation for the RES-fibered space, we show the complete match of the six-dimensional massless spectra read off from the Weierstrass equations on the F-theory side and those obtained by the index computations on the heterotic side for a special choice of instanton distribution.. They are constructed from the geometry of the No. case with a particular choice of instanton distributions. Appendix C shows the explicit forms of the functions f , g of the Weierstrass equations and the discriminant ∆ for various cases considered in the text
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