Chiral four-dimensional F-theory compactifications with SU(5) and multiple U(1)-factors

Abstract

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP 2 . We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)×U(1) and SU(5)×U(1)×U(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B = P 3 . We determine the general G 4 -flux that belongs to H (2,2) V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G 4 -flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New ChernSimons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.

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

“…Elliptic fibers with an increasing number of rational points and their corresponding elliptically fibered Calabi-Yau manifolds X with a Mordell-Weil group (MW-group) of rational sections of increasing rank have been systematically constructed and studied [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46]. 7 The free part of the MW-group leads to U(1)-gauge fields in F-theory 8 [2] and the torsion part yields non-simply connected gauge groups [53].…”

“…Further specializations of the X F i corresponding to toric tops [64,65] (see [66] for a systematic approach based on Tate's algorithm for elliptic fibers in Bl 1 P 2 (1, 1, 2)) permitted the engineering of toric Ftheory models with certain gauge groups, in particular with an SU(5) GUT-group. Some 4D examples of chiral SU(5) GUTs were constructed in this manner [42,43]. Since we determine here the intrinsic gauge groups and the non-toric MW-groups, as well as the full matter spectrum and the Yukawa couplings of X F i , our approach is complementary to these previous works.…”

“…We note that for evaluating this pairing in a concrete situation, the universal intersection relations π(S prove useful (cf. [35,39,42,77] for details), whereas π(S m · S n ) and π(S m · S 0 ) are modeldependent.…”

“…From a technical point of view, we are interested in all prime ideals, denoted throughout the paper by I (k) , of the loci (2.14) for every m. These are obtained by a primary decomposition of the complete intersection (2.14), cf. [42] for details. The codimension two variety in B associated to I (k) is denoted by V (I (k) ), which is the standard notation in algebraic geometry for an algebraic set, i.e.…”

Compactifications of deformed conifolds, branes and the geometry of qubits

“…The CP n+1 spaces also provide a natural base for constructions of elliptically fibered CalabiYau (n + 2)-folds, relevant to studies of F-theory compactifications to (8 − 2n) dimensions (cf. [18] and references therein). In the context of the AdS/CFT correspondence, CP n+1 or G 2 (R n+3 ) backgrounds, as opposed to compact Calabi-Yau (n + 1)-folds, have the possibility of avoiding the appearance of singular D-(p + 2) brane fluxes in the presence of anti-D-p branes (cf.…”