We discuss a simple example of an F-theory compactification on a Calabi-Yau fourfold where background fluxes, nonperturbative effects from Euclidean D3 instantons and gauge dynamics on D7 branes allow us to fix all closed and open string moduli. We explicitly check that the known higher order corrections to the potential, which we neglect in our leading approximation, only shift the results by a small amount. In our exploration of the model, we encounter interesting new phenomena, including examples of transitions where D7 branes absorb O3 planes, while changing topology to preserve the net D3 charge.

We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" which efficiently describes this relationship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a number of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.

We discuss a simple example of an F-theory compactification on a Calabi-Yau fourfold where background fluxes, nonperturbative effects from Euclidean D3 instantons and gauge dynamics on D7 branes allow us to fix all closed and open string moduli. We explicitly check that the known higher order corrections to the potential, which we neglect in our leading approximation, only shift the results by a small amount. In our exploration of the model, we encounter interesting new phenomena, including examples of transitions where D7 branes absorb O3 planes, while changing topology to preserve the net D3 charge.

We develop geometric techniques to determine the spectrum and the chiral indices of matter multiplets for four-dimensional F-theory compactifications on elliptic Calabi-Yau fourfolds with rank two Mordell-Weil group. The general elliptic fiber is the Calabi-Yau onefold in dP 2 . We classify its resolved elliptic fibrations over a general base B. The study of singularities of these fibrations leads to explicit matter representations, that we determine both for U(1)×U(1) and SU(5)×U(1)×U(1) constructions. We determine for the first time certain matter curves and surfaces using techniques involving prime ideals. The vertical cohomology ring of these fourfolds is calculated for both cases and general formulas for the Euler numbers are derived. Explicit calculations are presented for a specific base B = P 3 . We determine the general G 4 -flux that belongs to H (2,2) V of the resolved Calabi-Yau fourfolds. As a by-product, we derive for the first time all conditions on G 4 -flux in general F-theory compactifications with a non-holomorphic zero section. These conditions have to be formulated after a circle reduction in terms of Chern-Simons terms on the 3D Coulomb branch and invoke M-theory/F-theory duality. New ChernSimons terms are generated by Kaluza-Klein states of the circle compactification. We explicitly perform the relevant field theory computations, that yield non-vanishing results precisely for fourfolds with a non-holomorphic zero section. Taking into account the new Chern-Simons terms, all 4D matter chiralities are determined via 3D M-theory/F-theory duality. We independently check these chiralities using the subset of matter surfaces we determined. The presented techniques are general and do not rely on toric data.

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 ...