GV-Spectroscopy for F-theory on genus-one fibrations

Abstract

We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau d-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of sixdimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with G = Z 4 that is engineered by fibrations of bi-quadrics in P 3 . We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

On genus one fibered Calabi-Yau threefolds with 5-sections

“…In general it is very difficult to match these invariants with geometric quantities. However, if X is elliptically or genus one fibered some of these multiplicities are directly related to the multiplicities of reducible fibers [13,90]. We will use this relation in Section 5 to obtain general expressions for the multiplicities and structures of the reducible fibers of generic genus one fibered Calabi-Yau threefolds with 5-sections.…”

“…Let us choose a basis Π = (ω 1 , ω 0 ) of periods at the large volume point with Numerical analytic continuation 13 produces the corresponding monodromy matrices…”

“…To determine the field content of the effective theories, we apply GV-spectroscopy [13] to the elliptic fibrations. This provides us with base independent expressions for the multiplicities of reducible fibers over isolated points in the base 14 , which in turn correspond to hypermultiplets in the six-dimensional effective supergravity.…”

“…These can be used as fibers to construct a large class of elliptically fibered Calabi-Yau d-folds that via F-theory lead to effective supergravities with gauge group U (1) × U (1). We apply a recently developed technique based on the so-called fiber Gopakumar-Vafa invariants [13] to determine the spectrum of matter representations and show that the resulting theories allow a Higgs transition U (1) × U (1) → U (1) → Z 5 .…”

Orders of Vanishing and U(1) Charges in F-theory

“…As an example, consider F-theory models with just a U(1) gauge group. Explicit constructions of this type have realized charges up to q = ±6 [18,19,57,24,65,67], even though indirect arguments suggest F-theory U(1) models should be able to admit charges at least as large as q = ±21 [21]. Since only a limited number of charges have actually been observed in F-theory models, one might imagine it would be difficult to make statements about arbitrary charges.…”