Yukawas and discrete symmetries in F-theory compactifications without section

Published for SISSA by Springer Received: October 13, 2014 Accepted: November 10, 2014 Published: November 24, 2014 Iñaki Garcı́a-Etxebarria, Thomas W. Grimm and Jan Keitel Max Planck Institute for Physics, Föhringer Ring 6, 80805 Munich, Germany E-mail: , , Abstract: In the case of F-theory compactifications on genus-one fibrations without section there are naturally appearing discrete symmetries, which we argue to be associated to geometrically massive U(1) gauge symmetries. These discrete symmetries are shown to induce non-trivial selection rules for the allowed Yukawa couplings in SU(N ) gauge theories. The general discussion is exemplified using a concrete Calabi-Yau fourfold realizing an SU(5) GUT model. We observe that M2 instanton effects appear to play a key role in the generation of new superpotential terms and in the dynamics close to phase transition loci. Keywords: F-Theory, Superstring Vacua, String Duality ArXiv ePrint: 1408.6448 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP11(2014)125 JHEP11(2014)125 Yukawas and discrete symmetries in F-theory compactifications without section Contents 1 2 F-theory compactifications without section and Yukawa structures 2.1 Physics of F-theory compactifications without section 2.2 Discrete gauge symmetries 2.3 Four-dimensional Yukawa structures 2.4 String interpretation of the Higgsing 3 3 6 8 9 3 A class of elliptic fibrations with discrete symmetries 3.1 Hypersurface equation in P112 3.2 Non-Abelian matter curves and Yukawa points 3.3 Curve splitting and conifold transition 3.4 Discrete charges and forbidden Yukawa couplings 3.5 An explicit example without non-minimal singularities 12 12 13 16 17 17 4 Conclusions 18 1 Introduction F-theory [1] compactifications to four dimensions are typically defined by specifying a T 2 fibered Calabi-Yau fourfold. The traditional assumption is that the fibration has a section, i.e. there is an embedding of the basis divisor into the total space, almost everywhere intersecting the fiber at a point. All such models are birational to a Weierstrass model [2]. Restricting oneself to Calabi-Yau fourfolds defined by Weierstrass models (and thus, having at least one section) simplifies model building with non-Abelian gauge symmetries, since there are well understood techniques for reading off the low energy non-abelian gauge groups from the structure of a Weierstrass model.1 Considerable effort has been made to develop similar techniques for analyzing and engineering elliptically fibered Calabi-Yau manifolds that also give rise to Abelian gauge groups in the low energy effective theory. Initiated by the construction of the U(1)-restricted model in [4], the study of global F-theory compactifications with U(1) gauge factors can very roughly be divided into two approaches: (1) For a given U(1) gauge rank, one can determine the ambient space in which every elliptic fiber giving rise to such a low energy theory must be embeddable by using an old idea of Deligne [5]. Having obtained this space, one can then try to extract information about generic features of all such compactifications, such as all the matter representations can that possibly occur [6–11]. Non-generic elliptic fibers in Tate form 1 See for example table 4 of [3] for a comprehensive dictionary between vanishing degrees of the Weierstrass model and the associated gauge algebras. –1– JHEP11(2014)125 1 Introduction By now, not only the Abelian gauge groups themselves, but also purely Abelian matter states, often called singlet states, appear to be fairly well understood in four and six dimensions, both from a geometric [8, 29, 30] and a field theoretic perspective [15, 31, 32] using the Chern-Simons terms of the effective theory compactified on a circle. Recently, a proposal for counting the precise number of multiplets in F-theory compactifications to four dimensions has been made [33]. In contrast, Yukawa couplings in global compactifications have been much less studied so far, both those that involve singlets and those that do not. While their assumed geometrical counterparts, intersections of different matter curves in codimension 3 in the base manifold, have received attention [7–10, 12, 20, 30, 34], it appears crucial to point out that the relation to T-branes [35, 36], and in particular the low energy effective theory and local models [37–42] remain to be explored. Notably, beyond mathematical convenience there is no a priori physical reason to restrict oneself to T 2 fibrations with section. Calabi-Yau fourfolds with T 2 fiber but no section constitute perfectly respectable M-theory backgrounds, and they can admit a Ftheory limit. The physics of such backgrounds is rather interesting, and only recently it has been started to be systematically explored, mostly for the case of compactifications on threefolds [17–19].2 In this paper we extend the physical picture put forward in [19] to Calabi-Yau fourfold compactifications without section. We propose a closed string and an open string perspective on the massive U(1) arising in compactifications without section, and comment on the geometrical configurations realizing this duality. Furthermore, we explicitly describe how a non-Abelian gauge theory on seven-branes can be engineered in such geometries. This allows us to argue that models without section do have potentially fruitful model building properties, such as the natural appearance of certain discrete symmetries at low energies. These discrete symmetries can (and do) forbid certain Yukawa couplings from being generated, even though the Yukawa couplings are otherwise allowed by all continuous symmetries present at low energies. Let us remark that intersecting D6 brane scenarios with similar physical implications have recently been studied for example in [45–47]. As we were completing this paper, [20] appeared in which discrete symmetries in F-theory compactifications are also studied. 2 See also [43, 44] for earlier work on the topic. –2– JHEP11(2014)125 were studied in [12, 13]. (2) Conversely, one can take the stand and demand that given an arbitrary elliptically fibered Calabi-Yau manifold, one should be able to determine the low energy effective theory it gives rise to [14–16]. By breaking up the Calabi-Yau into its various building blocks and determining which of them can be treated separately, one can then systematically answer questions about entire classes of compactification manifolds [16] and find connections between them using Higgsings [20]. Alternatively, one could perform computer-aided scans over large numbers of compactifications as was done for example in [21, 22]. Naturally, these two approaches are not mutually exclusive and there exist many ways in which they overlap. Additionally, work has been done to understand the geometry associated to singularities in higher codimensions in the base manifold [23] and the relations between the different ways of (...truncated)