Homological algebra of supersymmetry: Localization, Unitarization, Dualization

• Applicants: Dr. Ondrej Hulik, Ph.D.; 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

In particle physics, "supersymmetry'' refers to the idea that the properties and behavior of the basic building blocks of matter (fermions such as the electron) might not be independent from those of the basic force carriers (bosons such as the photon or the graviton), but instead be related in a way analogous to heads and tails of a penny. While the physical manifestation of this concept remains more elusive than ever, supersymmetric theories have had a deep and indelible impact in mathematics. The most prominent examples, such as mirror symmetry, Donaldson-Witten or Kapustin-Witten theory, come from certain simplifications of such theories known as "topological twists''. These avatars are associated with a particular property of the algebraic structure underlying supersymmetry, closely related to the famous Pauli exclusion principle by which two identical fermions can never be in one and the same state. Topological field theories are most powerful in combination with a second property of supersymmetric field theories known as "duality''. It entails that, when the twisted theory is not yet simple enough, it can be transposed to an a priori entirely different description in which it can be exactly solved. Over the years, many predictions that have been extracted from supersymmetric field theories by twist and duality have been verified to an amazing degree of mathematical accuracy. What remains dearly missing is a transfer of physical intuition, a structural understanding of the twisting procedure, and a microscopic derivation of dualities. Bridging this gap is the background of this project, which has been funded by DFG since 2023 and is now in its second phase. The initial focus was the relation between the so-called pure-spinor superfield formalism and the unitary representation theory of Lie superalgebras, and the development of the formalism in view of applications in curved spacetime. Results include a remarkable restructuring of representation theory, the discovery of a new wall-crossing phenomenon, and a highly fruitful soldering of new homological methods onto classical supergeometry. This will now be further extended in view of a systematic unitarization of the pure-spinor formalism, the incorporation of interactions, including dynamical supergravity, and the exploration of further consequences of all these newly discovered structures.

• DFG Programme: Research Grants