24 research outputs found
On Abelian and Discrete Symmetries in F-Theory
In this dissertation, we systematically construct and study global F-theory compactifications with abelian and discrete gauge groups. These constructions are of fundamental relevance for both conceptual and phenomenological reasons.
In the case of abelian symmetries, we systematically engineer compactifications that support U(1)U(1) and U(1)U(1)U(1) gauge groups. The engineered geometries are elliptic fibrations with Mordell-Weil group rank two and three respectively. The bases of the fibrations are arbitrary, but as proofs of concept, we explicit create examples with bases and . We study the low energy physics of these compactifications, we calculate the matter spectrum and confirm that it is anomaly free. In 4D compactifications, the flux is designed and the existence of Yukawa couplings is verified.
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
explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond
to extremal transitions induced by blow-ups in the 2D toric varieties. The discrete gauge groups and are naturally found when and are used as fiber ambient spaces. We also find the first realization of matter with U(1) charge three.
Finally, we study the discrete gauge group in detail. We find the three elements of the Tate-Shafarevich (TS) group. We make use of the Higgs mechanism with the charge three hypermultiplets and the Kaluza-Klein reduction from 6D to 5D. The results are interpreted from the F- M- theory duality perspective. In F-theory, compactifications over any of the three elements of the TS groups yield the same low energy physics, however, M-theory compactifications over the same elements give rise to different gauge groups
Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors
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)xU(1) and SU(5)xU(1)xU(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 Chern-Simons 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.Comment: 100 pages, 11 figures, 7 appendices; v3: minor changes requested by
the referee, typos corrected, references added to the introductio
Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) x U(1) x U(1) Gauge Symmetry
We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abelian
gauge symmetry by constructing the general elliptically fibered Calabi-Yau
manifolds with a rank three Mordell-Weil group of rational sections. The
general elliptic fiber is shown to be a complete intersection of two
non-generic quadrics in P^3 and resolved elliptic fibrations are obtained by
embedding the fiber as the generic Calabi-Yau complete intersection into Bl_3
P^3, the blow-up of P^3 at three points. For a fixed base B, there are finitely
many Calabi-Yau elliptic fibrations. Thus, F-theory compactifications on these
Calabi-Yau manifolds are shown to be labeled by integral points in reflexive
polytopes constructed from the nef-partition of Bl_3 P^3. We determine all 14
massless matter representations to six and four dimensions by an explicit study
of the codimension two singularities of the elliptic fibration. We obtain three
matter representations charged under all three U(1)-factors, most notably a
tri-fundamental representation. The existence of these representations, which
are not present in generic perturbative Type II compactifications, signifies an
intriguing universal structure of codimension two singularities of the elliptic
fibrations with higher rank Mordell-Weil groups. We also compute explicitly the
corresponding 14 multiplicities of massless hypermultiplets of a
six-dimensional F-theory compactification for a general base B.Comment: 48 pages, 1 figur
General U(1)xU(1) F-theory Compactifications and Beyond: Geometry of unHiggsings and novel Matter Structure
We construct the general form of an F-theory compactification with two U(1)
factors based on a general elliptically fibered Calabi-Yau manifold with
Mordell-Weil group of rank two. This construction produces broad classes of
models with diverse matter spectra, including many that are not realized in
earlier F-theory constructions with U(1)xU(1) gauge symmetry. Generic U(1)xU(1)
models can be related to a Higgsed non-Abelian model with gauge group
SU(2)xSU(2)xSU(3), SU(2)^3xSU(3), or a subgroup thereof. The nonlocal
horizontal divisors of the Mordell-Weil group are replaced with local vertical
divisors associated with the Cartan generators of non-Abelian gauge groups from
Kodaira singularities. We give a global resolution of codimension two
singularities of the Abelian model; we identify the full anomaly free matter
content, and match it to the unHiggsed non-Abelian model. The non-Abelian
Weierstrass model exhibits a new algebraic description of the singularities in
the fibration that results in the first explicit construction of matter in the
symmetric representation of SU(3). This matter is realized on double point
singularities of the discriminant locus. The construction suggests a
generalization to U(1)^k factors with k>2, which can be studied by Higgsing
theories with larger non-Abelian gauge groups.Comment: 83 pages, 10 figures; v2: minor correction
F-Theory Vacua with Z_3 Gauge Symmetry
Discrete gauge groups naturally arise in F-theory compactifications on
genus-one fibered Calabi-Yau manifolds. Such geometries appear in families that
are parameterized by the Tate-Shafarevich group of the genus-one fibration.
While the F-theory compactification on any element of this family gives rise to
the same physics, the corresponding M-theory compactifications on these
geometries differ and are obtained by a fluxed circle reduction of the former.
In this note, we focus on an element of order three in the Tate-Shafarevich
group of the general cubic. We discuss how the different M-theory vacua and the
associated discrete gauge groups can be obtained by Higgsing of a pair of
five-dimensional U(1) symmetries. The Higgs fields arise from vanishing cycles
in -fibers that appear at certain codimension two loci in the base. We
explicitly identify all three curves that give rise to the corresponding Higgs
fields. In this analysis the investigation of different resolved phases of the
underlying geometry plays a crucial r\^ole.Comment: 13 page
F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
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
P^2, P^1x P^1 and the recently studied P^2(1,1,2), yield F-theory realizations
of SUGRA theories with discrete gauge groups Z_3, Z_2 and Z_4. 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 I_2-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 dP_1 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.Comment: 129 pages, 32 figures, 44 tables v2: minor changes, references adde