F-theory fluxes, Chirality and Chern-Simons theories

Abstract

We study the charged chiral matter spectrum of four-dimensional F-theory compactifications on elliptically fibered Calabi-Yau fourfolds by using the dual M-theory description. A chiral spectrum can be induced by M-theory four-form flux on the fully resolved Calabi-Yau fourfold. In M-theory this flux yields three-dimensional Chern-Simons couplings in the Coulomb branch of the gauge theory. In the F-theory compactification on an additional circle these couplings are only generated by one-loop corrections with charged fermions running in the loop. This identification allows us to infer the net number of chiral matter fields of the four-dimensional effective theory. The chirality formulas can be evaluated by using the intersection numbers and the cones of effective curves of the resolved fourfolds. We argue that a study of the effective curves also allows to follow the resolution process at each co-dimension. To write simple chirality formulas we suggest to use the effective curves involved in the resolution process to determine the matter surfaces and to connect with the group theory at co-dimension two in the base. We exemplify our methods on examples with SU(5) and SU(5)xU(1) gauge group.

Figures (5)

• Figure 1: The effective curves corresponding to the weights of 10 representation. The negative sign means that the negative of the weight corresponds to a effective curve.
• Figure 2: The chain of the Dynkin diagrams for the phase I. The number in the nodes denotes the multiplicity. The intersection structure cannot be inferred by simple group theoretic arguments about the weights, but requires an inspection of the resolution geometry.
• Figure 3: The chain of the Dynkin diagrams for $A_4 \rightarrow A_5 \rightarrow A_6$ and $A_4 \rightarrow A_5^{\prime} \rightarrow A_6$. $e_\Sigma$ denotes the singlet weight $\frac{1}{2}(e_1+e_2+e_3+e_4+e_5)$. The number in the node denotes the multiplicity.
• Figure 4: The chain of the Dynkin diagrams for the phase II. The number in the nodes denotes the multiplicity.
• Figure 5: The chain of the Dynkin diagrams for the phase III. The number in the node denotes the multiplicity.