We construct supersymmetric AdS 3 solutions in F-theory, that is Type IIB supergravity with varying axio-dilaton, which are holographically dual to 2d N = (0, 4) superconformal field theories with small superconformal algebra. In F-theory these arise from D3-branes wrapped on curves in the base of an elliptically fibered Calabi-Yau threefold Y 3 and correspond to self-dual strings in the 6d N = (1, 0) theory obtained from F-theory on Y 3 . The non-trivial fibration over the wrapped curves implies a varying coupling of the N = 4 Super-Yang-Mills theory on the D3-branes. We compute the holographic central charges and show that these agree with the field theory and with the anomalies of self-dual strings in 6d. We complement our analysis with a discussion of the dual M-theory solutions and a comparison of the central charges.

We study elliptic fibrations for F-theory compactifications realizing 4d and 6d supersymmetric gauge theories with abelian gauge factors. In the fibration these U (1) symmetries are realized in terms of additional rational sections. We obtain a universal characterization of all the possible U (1) charges of matter fields by determining the corresponding codimension two fibers with rational sections. In view of modelling supersymmetric Grand Unified Theories, one of the main examples that we analyze are U (1) symmetries for SU (5) gauge theories with 5 and 10 matter. We use a combination of constraints on the normal bundle of rational curves in Calabi-Yau three-and four-folds, as well as the splitting of rational curves in the fibers in codimension two, to determine the possible configurations of smooth rational sections. This analysis straightforwardly generalizes to multiple U (1)s. We study the flops of such fibers, as well as some of the Yukawa couplings in codimension three. Furthermore, we carry out a universal study of the U (1)-charged GUT singlets, including their KK-charges, and determine all realizations of singlet fibers. By giving vacuum expectation values to these singlets, we propose a systematic way to analyze the Higgsing of U (1)s to discrete gauge symmetries in F-theory.Recent years have seen much progress towards refining F-theory compactifications, including the realization of symmetries of the low energy effective theory that allow more realistic model building. These developments have been fuelled by increasingly sophisticated mathematical techniques that are required to construct the geometries underlying such F-theory compactifications. In lockstep with this, there has been a definite trend towards characterizing universal aspects of string compactifications, with a view to going beyond an example-driven approach.One of the areas where a universal characterization would be particularly bountiful is that of additional symmetries, such as abelian and discrete gauge symmetries, due to the direct phenomenological impact.The main result of this paper is to provide such a universal characterization of possible U (1) symmetries and associated matter charges in F-theory. Furthermore, we obtain a characterization of U (1)-charged singlets, which in turn can be used to Higgs abelian gauge groups to discrete symmetries.3The framework we are working within is F-theory compactifications on elliptically fiberedCalabi-Yau three-and four-folds, where non-abelian gauge groups are modelled in terms of singularities above codimension one loci in the base of the fibration [1]. Applications include the modelling of six-dimensional N = (1, 0) or four-dimensional N = 1 supersymmetric gauge theories, whose gauge group is determined by the Kodaira type of the singularity [2,3].Matter is engineered from codimension two singularities, whose fibers are characterized in terms of representation theoretic data, associated to the representation graph of the matter multiplet [4]. Abelian symmetries, which for instance are...

M5-branes on an associative three-cycle M 3 in a G 2 -holonomy manifold give rise to a 3d N = 1 supersymmetric gauge theory, T N =1 [M 3 ]. We propose an N = 1 3d-3d correspondence, based on two observables of these theories: the Witten index and the S 3 -partition function. The Witten index of a 3d N = 1 theory T N =1 [M 3 ] is shown to be computed in terms of the partition function of a topological field theory, a super-BFmodel coupled to a spinorial hypermultiplet (BFH), on M 3 . The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on M 3 . Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d (2, 0) theory. We also consider a correspondence for the S 3 -partition function of the T N =1 [M 3 ] theories, by determining the dimensional reduction of the M5-brane theory on S 3 . The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on M 3 , whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic G 2 -manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the S 3 -partition function of T N =1 [M 3 ] is given by the Witten-ReshetikhinTuraev invariant of M 3 .

In F-theory, U(1) gauge symmetries are encoded in rational sections, which generate the Mordell-Weil group of the elliptic fibration of the compactification space. Recently the possible U(1) charges for global SU(5) F-theory GUTs with smooth rational sections were classified [1]. In this paper we utilize this classification to probe global Ftheory models for their phenomenological viability. After imposing an exotic-free MSSM spectrum, anomaly cancellation (related to hypercharge flux GUT breaking in the presence of U(1) gauge symmetries), absence of dimension four and five proton decay operators and other R-parity violating couplings, and the presence of at least the third generation top Yukawa coupling, we generate the remaining quark and lepton Yukawa textures by a Froggatt-Nielsen mechanism. In this process we require that the dangerous couplings are forbidden at leading order, and when re-generated by singlet vevs, lie within the experimental bounds. We scan over all possible configurations, and show that only a small class of U(1) charge assignments and matter distributions satisfy all the requirements. The solutions give rise to the exact MSSM spectrum with realistic quark and lepton Yukawa textures, which are consistent with the CKM and PMNS mixing matrices. We also discuss the geometric realization of these models, and provide pointers to the class of elliptic fibrations with good phenomenological properties.