New Calabi-Yau Geometries in String Theory and Supersymmetry

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Since the first superstring revolution, string theory has seen a free flow of ideas from physics to mathematics and back again. Its future depends on expanding this interface, bringing together ideas as well as people. My work sets up a key framework for new and emerging physics, taking highly structured physical systems and using the language of geometry to constrain, classify, and evolve physical theories. Research at the interface of theoretical particle physics and mathematics generally falls into two categories, one whose focus is on specific physical questions, including string phenomenology and supersymmetry applications to collider physics. The second category involves mathematicians strongly inspired by theoretical particle physics, its structures and methods, and especially string theory and supersymmetric field theory. My research brings these groups together by studying string dualities and supersymmetry, and the critical role played by Calabi-Yau geometry.******A number of important string dualities are formulated in terms of fibration structures on Calabi-Yau geometries. Over the past six years, my research program has built a dictionary between period integrals and algebraic submanifolds of fibered Calabi-Yau manifolds. By proving effective forms of the famous Hodge Conjecture for the fibers, I provide an entirely new approach to understanding both fibered Calabi-Yau manifolds and their moduli spaces "from the inside out."******My research cuts across a large swath of the physics of string theory, including: the "non-geometric" Heterotic compactifications stemming from the Clingher-Doran-Malmendier-Morrison program in Heterotic/F-theory duality; unification of Calabi-Yau manifolds and Landau-Ginzburg models obtained by mirroring Calabi-Yau fibrations and Tyurin degenerations; the classification of orientifold theories on elliptic curves and elliptic fibered Calabi-Yau manifolds via new variants of topological KR-theory describing their BPS spectra; and the characterization of the modularity properties of the generating functions of vertical D4-D2-D0 bound states on smooth K3 surface fibered Calabi-Yau threefolds.******I have recently uncovered a link between the physics of supersymmetry and Calabi-Yau geometry. The dimensional reduction of supermultiplets to the world-line, which strips away their spatial dimensions, is encoded by a colored bipartite graph known as an Adinkra. My work shows that there is a super Riemann surface naturally associated with each Adinkra such that the Adinkra graph is embedded into the surface as a dimer model. This “geometrization” of supersymmetric representation theory provides a fundamental connection between supermultiplets and mirror symmetry, a line of research in theoretical physics which led to my appointment as the first ever Visiting Campobassi Professor of Physics at the University of Maryland.