Building Blocks of Physical Theories from the Geometry of Quantization and BPS States

• Applicant: Professor Dr. Murad Alim
• Subject Area: Mathematics ; Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields
• Term: from 2016 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 289590681

The goal of the Emmy Noether project is to develop a mathematical and physical understanding of the building blocks of physical theories. The focus is on physical theories, whose content is governed by deformation families of Calabi-Yau (CY) manifolds, such as supersymmetric field theories and topological string theory. Topological string theory associates to a family of CY manifolds a partition function, which is defined as a perturbative asymptotic expansion in the string coupling. This partition function encodes all genus Gromov-Witten invariants of the underlying CY manifold. The mathematical definition of this partition function is obtained through a geometric quantization problem on the moduli space. At the same time, a non-perturbative completion of the topological string partition function is expected to encode the Donaldson-Thomas invariants of the underlying family, physically corresponding to invariants characterizing BPS states. The latter are subject to stability conditions and exhibit wall crossing phenomena. The goal of the Emmy Noether project is to achieve a mathematical understanding of the building blocks of both geometric quantization and BPS geometries and the precise relation between the two. Understanding and solving this problem in generality is a challenging yet very rewarding task both physically and mathematically.

• DFG Programme: Independent Junior Research Groups