Advances in Theoretical and Mathematical Physics

Volume 22 (2018)

Number 1

Non-Higgsable abelian gauge symmetry and $\mathrm{F}$-theory on fiber products of rational elliptic surfaces

Pages: 177 – 245



David R. Morrison (Departments of Mathematics and Physics, University of California at Santa Barbara)

Daniel S. Park (Simons Center for Geometry and Physics, State University of New York, Stony Brook, N.Y., U.S.A.; and NHETC and the Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, U.S.A.)

Washington Taylor (Center for Theoretical Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, Ma., U.S.A.)


We construct a general class of Calabi–Yau threefolds from fiber products of rational elliptic surfaces with section, generalizing a construction of Schoen to include all Kodaira fiber types. The resulting threefolds each have two elliptic fibrations with section over rational elliptic surfaces and blowups thereof. These elliptic fibrations generally have nonzero Mordell–Weil rank. Each of the elliptic fibrations has a physical interpretation in terms of a six-dimensional $\mathrm{F}$-theory model with one or more non-Higgsable abelian gauge fields. Many of the models in this class have mild singularities that do not admit a Calabi–Yau resolution; this does not seem to compromise the physical integrity of the theory and can be associated in some cases with massless hypermultiplets localized at the singular loci. In some of these constructions, however, we find examples of abelian gauge fields that cannot be “un-Higgsed” to a nonabelian gauge field without producing unphysical singularities that cannot be resolved. The models studied here can also be used to exhibit $\mathrm{T}$-duality for a class of little string theories.

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