String Compactifications: From Geometry to Effective Field Theory

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This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech.String theory is a proposed framework of physics that encompasses both quantum mechanics (the theory describing the behavior of subatomic particles) and gravity (the force dictating the large-scale behavior of the Universe). In string theory, physics and the mathematical subject of geometry are intrinsically intertwined. While the questions that string theory attempts to answer are physical, the path to those answers frequently involves cutting-edge challenges in modern mathematics. This award will fund a collaborative program of research to explore the physics that arises from studying aspects of string theory. The goals of this work include developing new foundational tools for the subject of string phenomenology, which is the attempt to extract the observable consequences of string theory for experimental physics. As part of their research, Professors Anderson, Gray, and Sharpe aim to further bound and characterize the types of geometrical shapes that arise in string theory. Experience shows that when strong physical requirements are expressed in geometrical terms, they can open the door to new and unexpected results in both physics and mathematics. As a result, research in this area advances the national interest by promoting the progress of basic science. Professors Anderson, Gray, and Sharpe will involve junior scientists in this project, including a postdoctoral researcher and several graduate students who will take part in the collaborative research. Their efforts will also include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary of both fields, all while actively encouraging the inclusion of under-represented groups into the frontline of progress in the sciences.More specifically, two of the most flexible frameworks for four-dimensional compactifications of string theory --- namely heterotic string theory and F-theory --- will be investigated. Within heterotic string theory, novel geometric tools will be used to compute previously undetermined aspects of the effective theory. These include the N=1 matter field Kahler potential, physically normalized Yukawa couplings, and vacuum values of the perturbative superpotential. The non-perturbative contributions to Yukawa couplings will also be computed via quantum sheaf cohomology, a generalization of ordinary quantum cohomology. Within F-theory, new results in the geometry of elliptic fibrations will be used to study the properties of singular Calabi-Yau manifolds and their links to Hitchin systems, as well as to study the implications of the ubiquity of multiply fibered manifolds for string dualities, effective theories, and possible boundedness of the set of smooth Calabi-Yau varieties. Recent progress in heterotic/F-theory duality will be used to extract new features of the effective theories describing F-theory compactifications, including the explicit four-dimensional field-dependent form of flux contributions to the superpotential. Finally, Professors Anderson, Gray, and Sharpe will also apply some of their recent insights into the global structure of moduli spaces of SCFTs to study possible swampland conjectures.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.