Physics of F-theory compactifications without section

Lara B. Anderson 0 1 3 Inaki Garca-Etxebarria 0 1 2 Thomas W. Grimm 0 1 2 Jan Keitel 0 1 2 Open Access 0 1 c The Authors. 0 1 0 F ohringer Ring 6, 80805 Munich , Germany 1 850 West Campus Drive , Blacksburg, VA 24061 , U.S.A 2 Max Planck Institute for Physics 3 Department of Physics , Robeson Hall, 0435, Virginia Tech We study the physics of F-theory compactifications on genus-one fibrations without section by using an M-theory dual description. obtained by considering M-theory on a Calabi-Yau threefold is compared with a sixdimensional F-theory effective action reduced on an additional circle. We propose that the six-dimensional effective action of these setups admits geometrically massive U(1) vectors with a charged hypermultiplet spectrum. The absence of a section induces NS-NS and R-R three-form fluxes in F-theory that are non-trivially supported along the circle and induce a shift-gauging of certain axions with respect to the Kaluza-Klein vector. In the five-dimensional effective theory the Kaluza-Klein vector and the massive U(1)s combine into a linear combination that is massless. This U(1) is identified with the massless U(1) corresponding to the multi-section of the Calabi-Yau threefold in M-theory. We confirm this interpretation by computing the one-loop Chern-Simons terms for the massless vectors of the five-dimensional setup by integrating out all massive states. A closed formula is found that accounts for the hypermultiplets charged under the massive U(1)s. 1 Introduction 2 Six-dimensional action of F-theory on multi-section threefolds Review of massless U(1) in F-theory Massive U(1) and the Stuckelberg mechanism Massless U(1) on a circle and its M-theory dual Background flux and the M-theory to F-theory limit for multi-sections Fluxed circle reduction and M-theory comparison 3 Fluxed S1 reduction of the six-dimensional theory 4 Examples: transitions removing the section Constructing (X, X ) pairs with general base manifold Physics of the conifold transition Explicit examples with base P2 Chern-Simons terms A close look at the model with (a, b) = (0, 3) A close look at the model with (a, b) = (0, 2) Explicit formulas for the Chern-Simons terms 5 Conclusions 5.1 Open questions and future directions of study A Geometric description of the matter multiplets in X B Non-existence of a section for X Introduction F-theory, as introduced in [1], provides a beautiful geometric reformulation of Type IIB string theory with varying string coupling. Not only has it been explored from a formal perspective, but, more recently, it has also found exciting applications to realistic model building, starting with [25]. The underlying idea of F-theory is to identify the complexified two-torus. Such an interpretation is motivated by the existence of the non-perturbative SL(2, Z) symmetry of Type IIB. Remarkably, this construction extends to situations in One can thus consider backgrounds in which the T 2 is fibered over some compact base manifold. If the effective theory is to be supersymmetric the entire T 2 fibration X must be a Calabi-Yau manifold. So far, most of the literature has focused on a subclass of T 2 fibrations X that are simpler to analyze. Namely, it has largely been assumed that X has a section, that is, a global meromorphic embedding of the base into the total space of the fibration; or equivalently, a canonical choice of point in the fiber well defined everywhere (except possibly at some lower-dimensional loci in the base where the fiber degenerates). All such fibrations can be birationally transformed [6] into a Weierstrass model of the form y2 = x3 + f xz4 + gz6 with (x, y, z) coordinates of a P2,3,1, and f, g functions on the base of the fibration. A models is physically simpler to treat, because the existence of a section implies the absence of certain fluxes, as we will explain in more detail later on. Geometrically, the restriction to Weierstrass models facilitated model building with non-Abelian gauge symmetries, as the widely used algorithm of [8] (see also [9, 10] for later extensions) could be applied directly to models with Weierstrass form. We emphasize, however, that while the assumption of having a section simplifies the analysis, it is in no way necessary for the consistency of the physics, or the existence of an F-theory limit. In fact, it is very easy to construct T 2 fibrations with no section that serve as natural backgrounds for F-theory and we analyze explicitly various examples below. For completeness, let us also note that the approach taken by [11, 12] provides a convenient and more general way of generating non-Abelian gauge symmetries also for models without section. Based on this observation, in this paper we want to explore the physics of F-theory backgrounds X in which the T 2 does not have a section, and thus no Weierstrass model. This case remains basically unexplored, with the exception of the recent works [13, 14] (which appeared while this work was in progress), and some remarks in [7] that will play a role in our analysis below. We will focus on the formal aspects of this class of Ftheory backgrounds, uncovering some interesting characteristics of the resulting effective We will argue that a massive U(1) symmetry in the resulting six-dimensional theory coming from F-theory on X plays an essential role in a proper understanding of the theory. In fact, one of the important results in this paper is a proposal for a method of computing the massless and part of the massive spectrum of F-theory on a fibration X without section. We will test this proposal in a particular class of examples where the origin and properties of this massive U(1) are particularly transparent namely, examples where X is obtained from a conifold transition from a Calabi-Yau threefold X with two sections. Note that massive U(1)s in F-theory have recently been investigated in [1517]. In fact, for the cases studied in detail in this paper there exist both geometrical and physical reasons for why the Calabi-Yau manifolds X with bi-section are naturally related to fibrations X with two independent sections. Geometrically, by transitioning to a different manifold X the bi-section can be split into two independent sections. Physically, the massive U(1) becomes massless in that limit. Recently, the study of massless U(1) gauge symmetries in global F-theory compactifications has been a heavily investigated topic. Geometrically, the number of the Abelian gauge fields corresponds to the rank of the Mordell-Weil group of the fibration. As the Mordell-Weil group is generated by the sections, there is a direct correspondence between the number of independent sections and the number of U(1) generators. Let us note here that starting with the U(1)-restricted models of [15], continued by a systematic six-dimensional analysis of single U(1) models [18] and extended to more general treatments of multiple U(1) factors [1927] both with holomorphic and n (...truncated)