352 research outputs found
Higher Dimensional Elliptic Fibrations and Zariski Decompositions
We study the existence and properties of birationally equivalent models for
elliptically fibered varieties. In particular these have either the structure
of Mori fiber spaces or, assuming some standard conjectures, minimal models
with a Zariski decomposition compatible with the elliptic fibration. We prove
relations between the birational invariants of the elliptically fibered
variety, the base of the fibration and of its Jacobian.Comment: 16 pages; Accepted for publication in Communications in Contemporary
Mathematic
Weierstrass models of elliptic toric K3 hypersurfaces and symplectic cuts
We study elliptically fibered K3 surfaces, with sections, in toric Fano
threefolds which satisfy certain combinatorial properties relevant to
F-theory/Heterotic duality. We show that some of these conditions are
equivalent to the existence of an appropriate notion of a Weierstrass model
adapted to the toric context. Moreover, we show that if in addition other
conditions are satisfied, there exists a toric semistable degeneration of the
elliptic K3 surface which is compatible with the elliptic fibration and
F-theory/Heterotic duality.Comment: References adde
Discrete Symmetries in Heterotic/F-theory Duality and Mirror Symmetry
We study aspects of Heterotic/F-theory duality for compactifications with
Abelian discrete gauge symmetries. We consider F-theory compactifications on
genus-one fibered Calabi-Yau manifolds with n-sections, associated with the
Tate-Shafarevich group Z_n. Such models are obtained by studying first a
specific toric set-up whose associated Heterotic vector bundle has structure
group Z_n. By employing a conjectured Heterotic/F-theory mirror symmetry we
construct dual geometries of these original toric models, where in the stable
degeneration limit we obtain a discrete gauge symmetry of order two and three,
for compactifications to six dimensions. We provide explicit constructions of
mirror-pairs for symmetric examples with Z_2 and Z_3, in six dimensions. The
Heterotic models with symmetric discrete symmetries are related in field theory
to a Higgsing of Heterotic models with two symmetric abelian U(1) gauge
factors, where due to the Stuckelberg mechanism only a diagonal U(1) factor
remains massless, and thus after Higgsing only a diagonal discrete symmetry of
order n is present in the Heterotic models and detected via Heterotic/F-theory
duality. These constructions also provide further evidence for the conjectured
mirror symmetry in Heterotic/F-theory at the level of fibrations with torsional
sections and those with multi-sections.Comment: 25 pages, 4 figure
Matter From Geometry Without Resolution
We utilize the deformation theory of algebraic singularities to study charged
matter in compactifications of M-theory, F-theory, and type IIa string theory
on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is
more physical than that of resolution. We describe how two-cycles can be
identified and systematically studied after deformation. For ADE singularities,
we realize non-trivial ADE representations as sublattices of Z^N, where N is
the multiplicity of the codimension one singularity before deformation. We give
a method for the determination of Picard-Lefschetz vanishing cycles in this
context and utilize this method for one-parameter smooth deformations of ADE
singularities. We give a general map from junctions to weights and demonstrate
that Freudenthal's recursion formula applied to junctions correctly reproduces
the structure of high-dimensional ADE representations, including the 126 of
SO(10) and the 43,758 of E_6. We identify the Weyl group action in some
examples, and verify its order in others. We describe the codimension two
localization of matter in F-theory in the case of heterotic duality or simple
normal crossing and demonstrate the branching of adjoint representations.
Finally, we demonstrate geometrically that deformations correctly reproduce the
appearance of non-simply-laced algebras induced by monodromy around codimension
two singularities, showing the reduction of D_4 to G_2 in an example. A
companion mathematical paper will follow.Comment: 30 pages + references, appendices. v2: references and two figures
added, typos correcte
Non-Abelian Gauge Symmetry and the Higgs Mechanism in F-theory
Singular fiber resolution does not describe the spontaneous breaking of gauge
symmetry in F-theory, as the corresponding branch of the moduli space does not
exist in the theory. Accordingly, even non-abelian gauge theories have not been
fully understood in global F-theory compactifications. We present a systematic
discussion of using singularity deformation, which does describe the
spontaneous breaking of gauge symmetry in F-theory, to study non-abelian gauge
symmetry. Since this branch of the moduli space also exists in the defining
M-theory compactification, it provides the only known description of gauge
theory states which exists in both pictures; they are string junctions in
F-theory. We discuss how global deformations give rise to local deformations,
and also give examples where local deformation can be utilized even in models
where a global deformation does not exist. Utilizing deformations, we study a
number of new examples, including non-perturbative descriptions of and
gauge theories on seven-branes which do not admit a weakly coupled type
IIb description. It may be of phenomenological interest that these
non-perturbative descriptions do not exist for higher rank theories.Comment: 30 pages. v2: Updated codes, added references, and discussed how
local deformation can be utilized even when a global deformation does not
exist (the case of non-Higgsable clusters). v3: final version, published in
Communications in Mathematical Physic
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