Small resolutions of SU(5)-models in F-theory

By Mboyo Esole, Shing-Tung Yau. 2011

164 citations recorded by the source corpus.

We provide an explicit desingularization and study the resulting fiber geometry of elliptically fibered four-folds defined by Weierstrass models admitting a split $\tilde{A}_4$ singularity over a divisor of the discriminant locus. Such varieties are used to geometrically engineer SU(5) grand unified theories in F-theory. The desingularization is given by a small resolution of singularities. The $\tilde{A}_4$ fiber naturally appears after resolving the singularities in codimension-one in the base. The remaining higher codimension singularities are then beautifully described by a four-dimensional affine binomial variety which leads to six different small resolutions of the elliptically fibered four-fold. These six small resolutions define distinct four-folds connected to each other by a network of flop transitions forming a dihedral group. The location of these exotic fibers in the base is mapped to conifold points of the three-folds that defines the type IIB orientifold limit of the F-theory. The full resolution has interesting properties, specially for fibers in codimension-three: the rank of the singular fiber does not necessary increase and the fibers are not necessary in the list of Kodaira and some are not even (extended) Dynkin diagrams.

Subfields: Geometry and Topology, Mathematical Physics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/ca8cab4749515e138b6ba3d1bd827100c2870de8

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Citations in the corpus

6d Conformal matter (2014, background): “This is rather analogous to the fact that in F-theory on Calabi-Yau fourfolds, the structure of a Yukawa point [29] which is transparently realized in gauge theory terms can sometimes be obscure just from the resolution of singular fibers [65].”
Anomaly cancellation and abelian gauge symmetries in F-theory (2012, methodology): “These intersection properties can be checked by performing an explicit resolution, employing compact toric methods in [32, 33, 34, 35, 36, 37, 38, 39] and the local methods and their extensions in [40, 12].”
G(4)-flux, chiral matter and singularity resolution in F-theory compactifications (2011, methodology): “There, the singularity corresponds to the non-extended E6 Dynkin diagram or T − 3,3,3 [25].”
F-theory fluxes, chirality and Chern-Simons theories (2011, methodology): “The toric resolutions are equivalent, at least at co-dimension one and two relevant here, to the small resolutions performed for an SU(5) gauge group in [20, 13].”
F-theory on all toric hypersurface fibrations and its Higgs branches (2014, methodology): “Finally, it would be interesting to use the toric hypersurface fibrations studied here also for compactifications of M-theory to engineer 3D N = 2 gauge theories and to study their Coulomb-branches and phase structures, see [106–109] (and also the seminal works [110, 111]) for recent detailed analys”
Classifying bases for 6D F-theory models (2012, background): “Recent progress in understanding codimension two singularities and associated matter content appears in [8, 35, 9, 36].”
F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections (2013, background): “A recent reconsideration and careful analysis has closed remaining gaps in the understanding of singularities at higher codimension [19, 20, 21].”
Matter and singularities (2011, background): “For compactification on an elliptically fibered Calabi-Yau threefold, s, t are the only two local coordinates needed on the base.”
Anomaly equations and intersection theory (2011, background): “These contain information on the local matter we obtain [11, 12, 35, 36, 37, 38, 39, 40, 41].”
U(1) symmetries in F-theory GUTs with multiple sections (2012, methodology): “This resolution process has been studied with different techniques in the recent F-theory SU(5) GUT literature in a number of papers [22, 23, 24, 6, 7, 8] (see [25] for other gauge groups) and we use the process described in [6, 8].”
G-flux in F-theory and algebraic cycles (2011, background): “In [82], the necessity for small resolutions in F-theory was stressed.”