210 research outputs found
The Tate Form on Steroids: Resolution and Higher Codimension Fibers
F-theory on singular elliptically fibered Calabi-Yau four-folds provides a
setting to geometrically study four-dimensional N=1 supersymmetric gauge
theories, including matter and Yukawa couplings. The gauge degrees of freedom
arise from the codimension 1 singular loci, the matter and Yukawa couplings are
generated at enhanced singularities in higher codimension. We construct the
resolution of the singular Tate form for an elliptic Calabi-Yau four-fold with
an ADE type singularity in codimension 1 and study the structure of the fibers
in codimension 2 and 3. We determine the fibers in higher codimension which in
general are of Kodaira type along minimal singular loci, and are thus
consistent with the low energy gauge-theoretic intuition. Furthermore, we
provide a complementary description of the fibers in higher codimension, which
will also be applicable to non-minimal singularities. The irreducible
components in the fiber in codimension 2 correspond to weights of
representations of the ADE gauge group. These can split further in codimension
3 in a way that is consistent with the generation of Yukawa couplings. Applying
this reasoning, we then venture out to study non-minimal singularities, which
occur for A type along codimension 3, and for D and E also in codimension 2.
The fibers in this case are non-Kodaira, however some insight into these
singularities can be gained by considering the splitting of fiber components
along higher codimension, which are shown to be consistent with matter and
Yukawa couplings for the corresponding gauge groups.Comment: 75 pages, v2: clarifications and references added, JHEP versio
The Gravitational Sector of 2d (0,2) F-theory Vacua
F-theory compactifications on Calabi-Yau fivefolds give rise to
two-dimensional N=(0,2) supersymmetric field theories coupled to gravity. We
explore the dilaton supergravity defined by the moduli sector of such
compactifications. The massless moduli spectrum is found by uplifting Type IIB
compactifications on Calabi-Yau fourfolds. This spectrum matches expectations
from duality with M-theory on the same elliptic fibration. The latter defines
an N=2 Supersymmetric Quantum Mechanics related to the 2d (0,2) F-theory
supergravity via circle reduction. Using our recent results on the
gravitational anomalies of duality twisted D3-branes wrapping curves in
Calabi-Yau fivefolds we show that the F-theory spectrum is anomaly free. We
match the classical Chern-Simons terms of the M-theory Super Quantum Mechanics
to one-loop contributions to the effective action by S^1 reduction of the dual
F-theory.Comment: 19 pages, v2: JHEP versio
Chiral 2d Theories from N=4 SYM with Varying Coupling
We study 2d chiral theories arising from 4d N=4 Super-Yang Mills (SYM) with
varying coupling tau. The 2d theory is obtained by dimensional reduction of N=4
SYM on a complex curve with a partial topological twist that accounts for the
non-constant tau. The resulting 2d theories can preserve (0,n) with n = 2, 4,
6, 8 chiral supersymmetry, and have a natural realization in terms of strings
from wrapped D3-branes in F-theory. We determine the twisted dimensional
reduction, as well as the spectrum and anomaly polynomials of the resulting
strings in various dimensions. We complement this by considering the dual
M-theory configurations, which can either be realized in terms of M5-branes
wrapped on complex surfaces, or M2-branes on curves that result in 1d
supersymmetric quantum mechanics.Comment: 78 pages, 2 figures, v2: references adde
When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion
We explore novel gauge enhancements from abelian to non-simply-connected
gauge groups in F-theory. To this end we consider complex structure
deformations of elliptic fibrations with a Mordell--Weil group of rank one and
identify the conditions under which the generating section becomes torsional.
For the specific case of Z2 torsion we construct the generic solution to these
conditions and show that the associated F-theory compactification exhibits the
global gauge group [SU(2) x SU(4)]/Z2 x SU(2). The subsolution with gauge group
SU(2)/Z2 x SU(2), for which we provide a global resolution, is related by a
further complex structure deformation to a genus-one fibration with a bisection
whose Jacobian has a Z2 torsional section. While an analysis of the spectrum on
the Jacobian fibration reveals an SU(2)/Z2 x Z2 gauge theory, reproducing this
result from the bisection geometry raises some conceptual puzzles about
F-theory on genus-one fibrations.Comment: 51 page
Tate's Algorithm for F-theory GUTs with two U(1)s
We present a systematic study of elliptic fibrations for F-theory
realizations of gauge theories with two U(1) factors. In particular, we
determine a new class of SU(5) x U(1)^2 fibrations, which can be used to
engineer Grand Unified Theories, with multiple, differently charged, 10 matter
representations. To determine these models we apply Tate's algorithm to
elliptic fibrations with two U(1) symmetries, which are realized in terms of a
cubic in P^2. In the process, we find fibers which are not characterized solely
in terms of vanishing orders, but with some additional specialization, which
plays a key role in the construction of these novel SU(5) models with multiple
10 matter. We also determine a table of Tate-like forms for Kodaira fibers with
two U(1)s.Comment: 59 pages, 7 figure
Box Graphs and Singular Fibers
We determine the higher codimension fibers of elliptically fibered Calabi-Yau
fourfolds with section by studying the three-dimensional N=2 supersymmetric
gauge theory with matter which describes the low energy effective theory of
M-theory compactified on the associated Weierstrass model, a singular model of
the fourfold. Each phase of the Coulomb branch of this theory corresponds to a
particular resolution of the Weierstrass model, and we show that these have a
concise description in terms of decorated box graphs based on the
representation graph of the matter multiplets, or alternatively by a class of
convex paths on said graph. Transitions between phases have a simple
interpretation as `flopping' of the path, and in the geometry correspond to
actual flop transitions. This description of the phases enables us to enumerate
and determine the entire network between them, with various matter
representations for all reductive Lie groups. Furthermore, we observe that each
network of phases carries the structure of a (quasi-)minuscule representation
of a specific Lie algebra. Interpreted from a geometric point of view, this
analysis determines the generators of the cone of effective curves as well as
the network of flop transitions between crepant resolutions of singular
elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types
in codimensions two and three, and we find new, non-Kodaira, fiber types for
E_6, E_7 and E_8.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber
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