Tate’s algorithm and F-theory

By Sheldon Katz, D. Morrison, Sakura Schafer-Nameki, J. Sully. 2011

185 citations recorded by the source corpus.

The “Tate forms” for elliptically fibered Calabi-Yau manifolds are reconsidered in order to determine their general validity. We point out that there were some implicit assumptions made in the original derivation of these “Tate forms” from the Tate algorithm. By a careful analysis of the Tate algorithm itself, we deduce that the “Tateforms” (without any futher divisiblity assumptions) do not hold in some instances and have to be replaced by a new type of ansatz. Furthermore, we give examples in which the existence of a “Tate form” can be globally obstructed, i.e., the change of coordinates does not extend globally to sections of the entire base of the elliptic fibration. These results have implications both for model-building and for the exploration of the landscape of F-theory vacua.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/12317f406e76a8daea3a05d7cb8435049036e380

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Citations in the corpus

U(1) symmetries in F-theory GUTs with multiple sections (2012, background): “The form that the elliptic fibration must take in order to induce a non-Abelian singularity, or equivalently gauge group, over a divisor in the base is very well understood and given a fibration the non-Abelian structure can be discerned in an algorithmic way [20, 21].”
F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections (2013, unknown): “(5)See [53] for a reconsideration of the validity of the Tate model in global F-theory compactifications.”
Yukawas, G-flux, and spectral covers from resolved Calabi-Yau’s (2011, methodology): “ns so that suitable bm’s actually exist. We will always assume that this condition is satisﬁed and, moreover, that the bm’s are suitably generic9. 8Because we are dealing with SU(5) the subtleties of [43] do not apply and we can use a Tate model without loss of generality. 9Nongeneric choices of b m”
Classifying bases for 6D F-theory models (2012, methodology): “heory, and codimension two singularities carry matter ﬁelds. The gauge algebra along a given component Cof the discriminant locus can be determined by the Kodaira classiﬁcation and the Tate algorithm [31, 16, 32, 33, 9] in terms of the degrees of vanishing of f,g,∆ along the curve C(see Table 1). In”
G(4)-flux, chiral matter and singularity resolution in F-theory compactifications (2011, background): “This is because the non-abelian gauge groups, the matter spectrum and the Yukawa interactions of a model are in oneto-one correspondence with the singularities in the fibre of the Calabi-Yau four-fold over loci of, respectively, complex co-dimension one, two and three on the base (see [10] for a des”
MODELS OF PARTICLE PHYSICS FROM TYPE IIB STRING THEORY AND F-THEORY: A REVIEW (2012, background): “acGenerally this is possible but there are some exceptions [173].”
Atomic classification of 6D SCFTs (2015, methodology): “If 2 ≤ m ≤ 4, then (using [75]), the equation can be put into the form”
Geometric engineering in toric F-theory and GUTs with U(1) gauge factors (2013, result): “(5) tops and xed base manifold B= P3. The minimal number r min = rankMW T(X) is the number of toric U(1) factors. All values r min rr max are realized by a fourfold. of the compactication manifold [26, 27, 20, 28, 29]. In this section, we contrast this with the Abelian gauge group factors arising fr”
Matter and singularities (2011, methodology): “This “Tate form” of the description of the elliptic fibration is often used in the mathematical analysis of singular elliptic fibrations [26, 9, 15], but is less convenient for a global description of the elliptic fibration in the context of F-theory, where the Weierstrass form allows a systematic u”
Anomalies and the Euler characteristic of elliptic Calabi-Yau threefolds (2011, background): “The generalized Weierstrass forms proposed in [4,21] can all be written in the general form (see [21, Appendix A])”
F-theory on all toric hypersurface fibrations and its Higgs branches (2014, methodology): “structure of XFi by means of Tate’s algorithm [59–61] (see [62,63] for recent refinements) or upon addition of tops.”