Homological algebra of supersymmetry: Locality, Unitarity, Duality

• Applicants: Simone Noja, Ph.D.; Dr. Ingmar Saberi; Professor Dr. Johannes Walcher
• Subject Area: Mathematics ; Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 517493862

Supersymmetry has been a driving force behind many developments linking quantum theory and pure mathematics in the past several decades. Supersymmetric theories can serve to illustrate a range of physical and mathematical properties of quantum field theories in a controlled setting that is amenable to an exact analysis. At the same time, the solvable "BPS" subsectors of supersymmetric field and string theories encode the mathematical data of special geometric structures and their often unexpected relationships under "dualities." At first sight, the mathematical tools required for supersymmetric field theories are simple generalizations of those used to describe ordinary physical systems. In particular, supersymmetry itself can be understood either in a Lagrangian setting, emphasizing field locality, or from a Hamiltonian perspective, in which unitarity of particle representations is the organizing principle. Remarkable relationships to other areas of mathematics arise, however, because this is not the full story. Indeed, a closer look reveals many features unique to supersymmetric theories that require more abstract and sophisticated mathematical methods. Conversely, a restriction to the "predictions" of physical dualities does not constitute a satisfactory mathematical explanation of their microscopic origin. A case in point is the mathematical interpretation of holographic duality, which is beginning to emerge from a better understanding of holomorphic twists of supergravity; concrete physical applications of this line of work remain largely unexplored. Recently, we have contributed to the development of powerful tools from homological algebra for the construction and study of supersymmetric multiplets from the Lagrangian point of view. These tools abstract and generalize what is known as the "pure spinor superfield formalism." Concurrently, there has been remarkable progress in the unitary representation theory of super Lie algebras and super Lie groups, with striking new results about superconformal algebras, which are of central importance in field theory. This project aims to develop and join these two lines of research with an eye towards applications, especially to twisted holography. We expect the results to promote a broader perspective on supersymmetric field theory and to usher in new applications of supersymmetry to mathematics.

• DFG Programme: Research Grants