String Compactifications: From Geometry to Effective Field Theory

This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech. String theory is a proposal for a fundamental theory of how nature operates in which the roles of physics and 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 funds a collaborative program of research to explore the physics that arises from string theory. Because string theory predicts extra unseen dimensions beyond length, width, and height, these extra dimensions must be "curled up" in ways that render them too small to be detected with current experiments. However, the physics that string theory predicts depends crucially on the geometric properties of these curled-up dimensions. The goals of this work include strengthening the links between string theory and current progress in particle physics, in part by bounding and characterizing the geometries associated with these curled-up dimensions. Experience shows that when strong physical requirements are expressed in the language of geometry, 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 providing new insights into fundamental physics. Professors Anderson, Gray and Sharpe will also involve junior scientists in this project, including a postdoctoral researcher and several graduate students. Their efforts will include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary between both fields. In all of these aspects of student training and professional dialog, Professors Anderson, Gray and Sharpe are committed to actively encouraging the inclusion of members of under-represented groups into the frontline of progress in the sciences. More specifically, Professors Anderson, Gray, and Sharpe will investigate a new class of symmetries (known as higher-form symmetries), and two of the most flexible frameworks for four-dimensional compactifications of string theory, namely heterotic string theory and F-theory. They will also investigate a striking application known as decomposition. This is an observation that quantum field theories with certain higher-form symmetries are equivalent to disjoint unions of other quantum field theories. Within heterotic string theory, novel geometric tools will be used to compute previously undetermined aspects of the effective theory, including the N=1 matter field Kahler potential and physically normalized Yukawa couplings (including non-perturbative contributions). The goal of this study will be to understand the masses and interactions of particles within string compactification. Furthermore, new tools will be developed to study the physics of topology-changing transitions within heterotic string theory. In the context of F-theory, new results in the geometry of elliptic fibrations will be used to study the possible boundedness of the set of smooth Calabi-Yau varieties, heterotic/F-theory duality and 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.