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Matter from geometry without resolution

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Abstract

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on ellipti- cally fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of \( {{\mathbb{Z}}^N} \), where N is the multiplicity of the codimension one singularity be- fore deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal’s recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E 6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D 4 to G 2 in an example. A companion mathematical paper will follow.

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, PA, 19104-6395, U.S.A.

    Antonella Grassi & Julius L. Shaneson

  2. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106-4030, U.S.A.

    James Halverson

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  1. Antonella Grassi
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Correspondence to James Halverson.

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Grassi, A., Halverson, J. & Shaneson, J.L. Matter from geometry without resolution. J. High Energ. Phys. 2013, 205 (2013). https://doi.org/10.1007/JHEP10(2013)205

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