F-theory on all toric hypersurface fibrations and its Higgs branches

Denis Klevers 0 1 2 3 5 Damian Kaloni Mayorga Pena 0 1 2 4 Paul-Konstantin Oehlmann 0 1 2 4 Philadelphia 0 1 2 PA 0 1 2 U.S.A. 0 1 2 0 Open Access , c The Authors 1 Nussallee 12 , 53115 Bonn , Germany 2 CH-1211 , Geneva 23 , Switzerland 3 Theory Group, Physics Department , CERN 4 Bethe Center for Theoretical Physics, Physikalisches Institut der Universit at Bonn 5 Department of Physics and Astronomy, University of Pennsylvania We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in P2, P1 P1 and the recently studied P2(1, 1, 2), yield F-theory realizations of SUGRA theories with discrete gauge groups Z3, Z2 and Z4. This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I2-fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. We also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP1 has a U(1)-gauge field - induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3. tries, Supergravity Models ArXiv ePrint: 1408.4808 1 Introduction & summary of results Summary of results Structure of the paper 3.2 Fibration with discrete gauge symmetry Geometry & physics of F-theory backgrounds Genus-one, Jacobian and elliptic fibrations with Mordell-Weil groups Divisors on genus-one fibrations and their intersections The spectrum of F-theory on genus-one fibrations Explicit examples: Calabi-Yau hypersurfaces in 2D toric varieties Analysis of F-theory on toric hypersurface fibrations Three basic ingredients: the cubic, biquadric and quartic Constructing toric hypersurface fibrations Fibration by cubic curves: XF1 and its specializations Fibration with gauge groups of rank 1, 2 and no discrete gauge symmetry Fibrations with gauge groups of rank 3: selfdual polyhedra Fibrations with gauge groups of rank 4, 5 and no MW-torsion Fibrations with gauge groups of rank 5 and 6 and MW-torsion Fibration by the biquadric: XF2 Fibration by the quartic: XF4 Polyhedron F1: GF1 = Z3 Polyhedron F2: GF2 = U(1) Z2 Polyhedron F4: GF4 = SU(2) Z4 Polyhedron F3: GF3 = U(1) Polyhedron F5: GF5 = U(1)2 Polyhedron F6: GF6 = SU(2) U(1) Polyhedron F7: GF7 = U(1)3 Polyhedron F8: GF8 = SU(2)2 U(1) Polyhedron F9: GF9 = SU(2) U(1)2 Polyhedron F12: GF12 = SU(2)2 U(1)2 Polyhedron F11: GF11 = SU(3) SU(2) U(1) Polyhedron F14: GF14 = SU(3) SU(2)2 U(1) Polyhedron F16: GF16 = SU(3)3/Z3 Polyhedron F13: GF13 = (SU(4) SU(2)2)/Z2 Polyhedron F15: GF15 = SU(2)4/Z2 U(1) The toric Higgs branch of F-theory Toric Higgsing: an example Matching the charged spectrum Matching of the neutral spectrum: Higgsing & Euler numbers Allowed regions for base P Higgsings to theories with discrete gauge symmetries The complete Higgsing chain A Anomaly cancellation conditions in 6D Additional data on toric hypersurface fibrations C Euler numbers of the Calabi-Yau threefolds XFi D The full Higgs chain of toric hypersurface fibrations Group theoretical decomposition of representations Introduction & summary of results F-theory [13] is a non-perturbative formulation of Type IIB string theory with backreacted 7-branes, that is manifestly invariant under the SL(2, Z)-duality symmetry of the theory. String backgrounds constructed via F-theory are not only located in the heart of the web of string dualities, but also allow for the construction of phenomenologically appealing local GUT-models [47], which has recently rekindled a lot of interest into the subject. defined up to SL(2, Z)-transformations, by a quantity, that only depends on the SL(2, Z)torus-fibrations over B. In particular, for a supersymmetric and tadpole-canceling setup Most of the torus-fibered Calabi-Yau manifolds X that have been studied are algebraic, that is they are realized as complete intersections in some ambient space.1 In these constructions, the torus fiber over B is realized as an algebraic curve C of genus one. In addition, many examples considered in the literature are elliptically fibered, meaning that 1For recent advances on Calabi-Yau manifolds constructed as determinantal and Pfaffian varieties, X has a section B X, which was traditionally assumed to be holomorphic. These elliptically fibered Calabi-Yau manifolds have fruitful applications, e.g. for the construction of semi-realistic GUTs in global F-theory compactifications starting with [10, 11] or the classification and study of 6D SCFTs [1214]. Despite these successes, addressing open conceptual questions e.g. regarding the finiteness of the F-theory landscape2 or which consistent 6D and 4D supergravity (SUGRA) theories can be realized in F-theory,3 as well as the understanding of the geometric origins of discrete symmetries or analogous field theoretic mechanisms, crucial for the phenomenology of F-theory models, requires to broaden the class of Calabi-Yau manifolds X used for F-theory compactifications. In fact, using the well-developed map between the geometry of F-theory and SUGRA theories, see [20, 21] for the complete map in 6D and [22] for results about certain topological terms in 4D,4 one finds that the Calabi-Yau manifolds realizing many known consistent SUGRA theories, in particular those with U(1) symmetries [24] or discrete gauge groups, are still unknown [25].5 For the search of an F-theory realization of these theories it is crucial to construct new classes of Calabi-Yau manifolds X admitting new geometric features and to deduce the general SUGRA theories that arise in F-theory compactifications on these X.6 There has been a lot of recent progress in systematically extending the set of CalabiYau manifolds X that can be used for F-theory compactifications. The different approaches can be roughly sorted into two groups. The (...truncated)