F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections

TL;DR

This paper constructs and classifies elliptic fibrations with two rational sections in F-theory, analyzing their singularities, matter charges, and explicit examples, advancing understanding of U(1)xU(1) gauge symmetries in string compactifications.

Contribution

It provides a systematic construction and classification of elliptic fibrations with two rational points, including explicit examples and analysis of matter charges and anomaly cancellation.

Findings

Classified elliptic fibrations with U(1)xU(1) symmetry over P^2.

Determined matter charges and multiplicities consistent with anomalies.

Constructed explicit toric examples with U(1)xU(1) and SU(5)xU(1)xU(1) gauge groups.

Abstract

We study F-theory compactifications with U(1)xU(1) gauge symmetry on elliptically fibered Calabi-Yau manifolds with a rank two Mordell-Weil group. We find that the natural presentation of an elliptic curve E with two rational points and a zero point is the generic Calabi-Yau onefold in dP_2. We determine the birational map to its Tate and Weierstrass form and the coordinates of the two rational points in Weierstrass form. We discuss its resolved elliptic fibrations over a general base B and classify them in the case of B=P^2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)xU(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over P^2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)xU(1) and SU(5)xU(1)xU(1) gauge symmetry. As a by-product, we prove the birational equivalence of the two elliptic fibrations with elliptic fibers in the two blow-ups Bl_(1,0,0)P^2(1,2,3) and Bl_(0,1,0)P^2(1,1,2) employing birational maps and extremal transitions.