Conforma Field Theory and Stringy Geometry

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This award funds the research activities of Professor Ilarion V. Melnikov at James Madison University. Quantum Field Theory is a powerful mathematical framework that bears on almost every aspect of contemporary physics. These include the spectacularly successful Standard Model of Particle Interactions, the quantum phenomena in exotic materials that hold out great promise for new technologies such as quantum computing or energy storage and transfer, as well as theories that aim to uncover the fundamental laws of nature, such as String Theory. There has been tremendous progress in Quantum Field Theory over the past thirty years, and many questions that seemed intractable have been given precise and incisive answers. This project will build on those results by tackling a number of timely questions in Quantum Field Theory within realms where many of the standard tools are misleading or inapplicable. The project will also address questions in two-dimensional Quantum Field Theory, relevant both for the physics of exotic two-dimensional materials and for the emergent geometry of spacetime in String Theory. As a result, research in this area advances the national interest by promoting the progress of science in one of its most fundamental directions: the discovery and understanding of new physical laws. This project will also have significant broader impacts. Crucially, it will involve undergraduate researchers, who will be given a unique opportunity to engage in theoretical physics research and have early exposure to the mathematical and physical ideas at the forefront of current explorations. This research will also be interdisciplinary, bringing together collaborations of physicists and mathematicians, and it will be presented in public lectures to convey the exciting progress in and enthusiasm for research in theoretical physics. More technically, Professor Melnikov's work will explore foundational questions in conformal field theory and geometry in two-dimensional quantum field theory relevant to string compactification. In this setting powerful symmetries and constraints from spacetime geometry will help to give precise quantitative insights into strongly coupled quantum field theory and string propagation in non-trivial spacetimes. The project will address three concrete problems: (i) the characterization of obstructions to marginal deformations of supersymmetric conformal field theories and their relation to spacetime physics of the heterotic string; (ii) the identification of marginal deformations of conformal theories with local operators in asymptotically free gauge theories that reduce to the conformal theory in the low-energy limit; and (iii) the discovery of new conformal theories that enlarge the landscape of string vacua. It will bear on long-standing issues in stringy geometry and the mathematics and physics of (0,2) mirror symmetry --- a significant generalization of the mirror correspondence. The concrete lessons learned will also have wider applicability in low-dimensional quantum field theory at strong coupling.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.