FRG: Collaborative Research: Homotopy Renormalization of Topological Field Theories

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Mathematics inspired by physics has successfully provided the background and the language for most sophisticated areas of modern physics. In this project the principal investigators aim to create new algebraic and geometric tools helping to formulate ideas and methods of quantum physics in a precise mathematical way. Specifically, the team will combine its members' past experience and achievements to construct new Topological Quantum Field Theories for manifolds with additional structures. Inspired by Witten's Chern-Simons theory and invariants of 3-manifolds that are known now as Witten-Reshetikhin-Turaev invariants, the principal investigators will explore their new Field Theories with the objective to find physical definitions for the resulting topological invariants of manifolds. This work is stimulated by several fundamental examples and will open the door to new research avenues in algebra, topology, geometry, mathematical physics, and related areas of mathematics. The broader impacts of the project belong to two main categories: mentoring and outreach. The members of the research team are currently mentoring a total of twelve PhD students. They will advise graduate students on projects related to the main objectives of the grant. The outreach component is to organize several workshops and conferences aimed at developing communication and collaborative research between participants of the project, establishing scientific connections with other mathematicians, as well as fostering broader applications of this work.In 1999, Turaev introduced Homotopy Quantum Field Theories (HQFTs), which are generalizations of Topological Quantum Field Theories (TQFTs) studied by Schwartz, Witten, and Atiyah. HQFTs produce topological invariants of manifolds furnished with extra data that add supplementary topology/geometry to the context of TQFTs. Most of the theory of quantum invariants and HQFTs involves monoidal categories which have certain additional properties like being semi-simple. In various collaborations started in 2005, the team members developed a theory of re-normalized Quantum Invariants that derives non-trivial topological invariants from non-semi-simple categories. In this project the principal investigators will further use renormalization to develop non-trivial HQFTs, based on the examples coming from the theory of unrestricted quantum group. These new HQFTs should share the strength and the new features of the re-normalized Quantum Invariants. The principal investigators will further search for a physical interpretation of these new invariants.