Homological Methods in Quantum Field Theory

• Applicant: Dr. Christoph Chiaffrino
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 536887995

The goal of this project is to find new applications of homological algebra to classical and quantized fieldtheories. The theory of homological algebras in particular is applied to homopy algebras and the Batalin-Vilkovisky (BV) formalism. Explicitly, I want to explore the following applications: Double field theory as the double copy of Yang-Mills theory: Yang-Mills theory can be described as a homotopy commutative algebra. It was shown that, to quadratic order in the coupling constant, that this algebra can be expanded to a homotopy BV algebra. It was further shown (to quadratic order in coupling), that the tensor product of two such homotopy BV algebras describes double field theory. The goal is to generalize this result to all orders. My main focus will be on finding a suitable tensor product of homotopy BV algebras. The tensor product of a pair of Yang-Mills theories is then a special case. Quantum expectation values via BV cohomology: It was shown, that in quantum mechanics the (ill defined) pathintegral can also be computed by looking at the cohomology of the BV Laplacian. Cohomology as a concept is well defined, in contrast to the path integral. In particular, a one-to-one map between cohomology and a choice of boundary conditions in the path integral was constructed. As a next step, I want to apply this idea to gauge field theories and test it there. The goal is to get rid of some of the mathematical inconsistencies of the path integral via this ansatz. AdS-CFT correspondence as a homotopy transfer: Field theories can be described as homotopy Lie algebras. A powerful tool is the homotopy transfer, which maps a theory to an equivalent one. In the case of a scalar field in an AdS background, it was shown that the tree level expectation values of the corresponding CFT on the AdS boundary can be obtained via homotopy transfer. In the project, I want to show the existence of this homotopy transfer for more complex theories, in particular gauge theories and perturbative gravity. Local homotopy algebras: Locality is a central requirement to any field theory. When descriging a field theory as a homotopy Lie algebra, we loose the notion of locality. My goal in the project is to generalize the definition of homotopy algebras, so that they can incorporate locality. From homotopy Lie to BV: It is known that the perturbative expansion of a BV theory gives a homotopy Lie algebra. One goal in the project is to show a similar relation in the other direction. The mechanism is known in finite dimensions. In the project I want to explore the infinite dimensional case.

• DFG Programme: Research Grants
• International Connection: France, USA
• Co-Investigators: Dr. Roberto Bonezzi, Ph.D.; Felipe Diaz-Jaramillo; Professor Dr. Olaf Hohm; Professor Dr. Ivo Sachs
• Cooperation Partners: Professor Dr. Owen Gwilliam; Professor Dr. Bruno Vallette