Group representations and the Euler characteristic of elliptically fibered Calabi–Yau threefolds

Abstract

Abstract. To every elliptic Calabi-Yau threefold with a section X there can be associated a Lie group G and a representation ρ of that group. The group is determined from the Weierstrass model, which has singularities that are generically rational double points; these double points lead to local factors of G which are either the corresponding A-D-E groups or some associated non-simply laced groups. The representation ρ is a sum of representations coming from the local factors of G, and of other representations which can be associated to the points at which the singularities are worse than generic.This construction first arose in physics, and the requirement of anomaly cancellation in the associated physical theory makes some surprising predictions about the connection between X and ρ. In particular, an explicit formula (in terms of ρ) for the Euler characteristic of X is predicted. We give a purely mathematical proof of that formula in this paper, introducing along the way a new invariant of elliptic Calabi-Yau threefolds. We also verify the other geometric predictions which are consequences of anomaly cancellation, under some (mild) hypotheses about the types of singularities which occur.As a byproduct we also discover a novel relation between the Coxeter number and the rank in the case of the simply laced groups in the "exceptional series" studied by Deligne.It was noted by Du Val [11] that certain surface singularities, now known as rational double points, are classified by the Dynkin diagrams of the simply laced Lie groups 1 of type A n , D n , E 6 , E 7 , E 8 . Du Val pointed out that the Dynkin diagram is the dual diagram to the intersection configuration of the exceptional divisors in the minimal resolution of the singularities. Further connections between these singularities and Lie groups were subsequently discovered by Brieskorn and Grothendieck [5].The resolutions of rational double points are crepant, that is, the pullback of the canonical divisor on the singular variety is the canonical divisor on the smoothResearch partially supported by the Harmon Duncombe foundation, by the Institute for Advanced Study, and by National Science Foundation grants DMS-9401447, DMS-9401495, DMS-9627351 and DMS-9706707. We thank the Institute for Advanced Study, the Mathematisches Forschunginstitut Oberwolfach, and the Institute for Theoretical Physics, Santa Barbara, for hospitality during various stages of this project.1 More precisely, Du Val recognized the combinatorial structure as occurring in the theory of finite reflection groups; the connection to Lie groups was made soon thereafter by Coxeter [7,8].2 minimal resolution. In particular, if the singular variety has trivial canonical class, so does its desingularization.One characterization of rational double points is as quotients of C 2 by finite subgroups of SL(2, C) [12]. Much recent work has been done by looking at the quotient of C 3 by a finite subgroup of SL(3, C) (see for example [20,19,30]). In this paper we consider another natural generalization of ...

Tate form and weak coupling limits in F-theory

“…This is also consistent with the result of [10]. The first hosts matter fields transforming in the 6, the antisymmetric representation of SU (4), which originates from the symmetry enhancement to the SO(7) group 8 .…”

“…The main purpose has been to provide a systematic way of solving the conifold problem afflicting the DW ansatz when applied to singular fibrations with unitary gauge groups: These singularities do not admit admissible crepant resolutions 10 . We also analyze the weak coupling limit of all gauge groups implemented by Tate forms, including the exceptional ones.…”

“…Introduction F-theory [1] is a powerful tool to engineer gauge theories appearing in type IIB string theory and in the hetorotic string theory using the geometry of elliptic fibrations [2,3]. When the structure of an elliptic fibration is seen through the eyes of string theory, the constraints coming from physics lead to surprising new mathematical results on the structure of elliptic fibrations [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Although F-theory is defined for elliptically fibered Calabi-Yau varieties, many of the mathematical results obtained in F-theory apply in a larger setup without restrictions on the dimension of the elliptic fibration and without assuming the Calabi-Yau condition [6,7,11].…”

“…These ansätze are now familiarly called Tate forms. The original list of Tate forms in [3] was corrected by Grassi and Morrison [10]. A Weierstrass model admitting a certain gauge group is not necessarily realized by one of the Tate forms.…”

“…We choose an SU (4) and an SU (5) model, since they display different properties of Yukawa couplings, as it will be clear in a moment. Also, they are the lowest rank representatives of the even and odd unitary series with more familiar enhancements: SU (2) enhances to the Kodaira singularity III and SU (3) to IV s on the O7-plane [10].…”