\(D_5\) elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory
This research paper is a continuation of research done in [\textit{P. Aluffi} and \textit{M. Esole}, J. High Energy Phys. 2010, No. 2, Paper No. 020, 53 p. (2010; Zbl 1270.81145)], which studies some important aspects of elliptic fibrations by considering other models of elliptic curves than Weierstrass models. It is important to mention that \(F\)-theory has a decisive influence on providing a new and interesting perspective on the geometry of elliptic fibrations with new problems and challenges coming from physics. As an example, one can mention the duality between \(F\)-theory and the Heterotic string which has motivated the study of principle holomorphic \(G\)-bundles over elliptic fibrations by Friedman-Morgan-Witten (see [\textit{R. Friedman} et al., Commun. Math. Phys. 187, No. 3, 679--743 (1997; Zbl 0919.14010); J. Algebr. Geom. 8, No. 2, 279--401 (1999; Zbl 0937.14004)]). Indeed, since the advent of non-Weierstrass models, new ways of description of the strong coupling limit of certain non-trivial type IIB orientifold compactifications with brane-image-brane pairs by embedding them in \(F\)-theory have been suggested. As the authors pointed out, it is true that many of our insights gained on the structure of elliptic fibrations coming from \(F\)-theory are true without any assumptions on the dimension of the base and without assuming the Calabi-Yau condition (see [\textit{P. Aluffi} and \textit{M. Esole}, ``Chern class identities from tadpole matching in type IIB and F-theory'', J. High Energy Phys. 2009, No. 3, Article ID 0903, 32 p. (2009; \url{doi:10.1088/1126-6708/2009/03/032}); ibid. 2010, No. 2, Paper No. 020, 53 p. (2010; Zbl 1270.81145); \textit{M. Esole} and \textit{S.-T. Yau}, Adv. Theor. Math. Phys. 17, No. 6, 1195--1253 (2013; Zbl 1447.81171); \textit{R. Friedman} et al., Commun. Math. Phys. 187, No. 3, 679--743 (1997; Zbl 0919.14010); \textit{A. Grassi} and \textit{D. R. Morrison}, Commun. Number Theory Phys. 6, No. 1, 51--127 (2012; Zbl 1270.81174)]. In this paper, the authors focus on the elliptic fibrations whose generic fiber is an elliptic curve which is modeled by the complete intersection of two quadratic surfaces in \(P^3\) . In this respect, the authors classify all the singular fibers of a smooth elliptic fibration by utilizing the classification of pencils of quadratics by Sergre symbols which provide a better description of the singular fibers than the Kodaira symbols by detecting the degree of each of the components of a given singular fiber. For elliptic fibrations, the non-Kodaira singular fibers present themselves without involving singularities in the total spaces. The authors examine their physical relevance from the point of view of the weak coupling limit of \(F\)-theory which first introduced by \textit{A. Sen} [Nucl. Phys., B, Proc. Suppl. 68, 92--98 (1998; Zbl 0999.81520)]. They analyze some degenerations of these fibrations and obtain some non-trivial topological relations between the total space of elliptic fibration and certain divisors in its base. To be more specific, the obtained degeneration describes a theory of an orientifold with three brane-image-brane pairs which two of them are in the same homology class as the orientifold. This is done by considering a certain transition from type \(I_2\) to \(III\) described in Table 5 in the paper. In effect, the presented orientifold limit is related to the transition \(I_2 \rightarrow III\) when the brane-image-brane does not coincide with the orientifold. The authors also discuss the case when they coincide. For the first time, the authors offer Sen's orientifold limit for elliptic fibrations. Also they prove that the non-trivial relation between Euler characteristic of the elliptic fibration and the Euler characteristic of divisors corresponding to the orientifold and the brane-image-brane pairs holds at the level of the total Chern class of these loci.
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