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F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections

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Abstract

We study F-theory compactifications with U(1)×U(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 \( \mathcal{E} \) 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 Weier- strass 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 = \( \mathbb{P} \) 2. A thorough analysis of the generic codimension two singularities of these elliptic Calabi-Yau manifolds is presented. This determines the general U(1)×U(1)-charges of matter in corresponding F-theory compactifications. The matter multiplicities for the fibration over \( \mathbb{P} \) 2 are determined explicitly and shown to be consistent with anomaly cancellation. Explicit toric examples are constructed, both with U(1)×U(1) and SU(5)×U(1)×U(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) \( \mathbb{P} \) 2(1, 2, 3) and Bl (0,1,0) \( \mathbb{P} \) 2(1, 1, 2) employing birational maps and extremal transitions.

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Authors and Affiliations

  1. Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, 19104-6396, U.S.A.

    Mirjam Cvetič, Denis Klevers & Hernan Piragua

  2. Center for Applied Mathematics and Theoretical Physics, University of Maribor, Maribor, Slovenia

    Mirjam Cvetič

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  1. Mirjam Cvetič
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  2. Denis Klevers
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Correspondence to Denis Klevers.

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ArXiv ePrint: 1303.6970

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Cvetič, M., Klevers, D. & Piragua, H. F-theory compactifications with multiple U(1)-factors: constructing elliptic fibrations with rational sections. J. High Energ. Phys. 2013, 67 (2013). https://doi.org/10.1007/JHEP06(2013)067

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