Heterotic sigma models via formal geometry and BV quantization

• Main Author: Ladouce, James
• Other Authors: Szczesny, Matt
• Language: en_US
• Published: 2021
• Subjects: Mathematics; Anomaly; BC beta gamma system; BV quantization; Formal geometry; Heterotic sigma model; Supersymmetry
• Online Access: https://hdl.handle.net/2144/43183

Nonlinear sigma-models in physics have been a source of interesting and important ideas in geometry, topology, and algebra. One such model is the curved beta gamma system. This purely bosonic model studies maps from a Riemann surface to a target complex manifold X. The solutions to the classical equations of motion are holomorphic maps. An extension of this model - the so-called heterotic model, incorporates fermionic fields valued in a holomorphic vector bundle E on the complex manifold. In this thesis, I study this extended model within the framework of effective field theory and BV quantization developed by Kevin Costello. Building on earlier work of Gorbounov-Gwilliam-Williams in the purely bosonic case, my approach uses tools of Gelfand-Kazhdan formal geometry and derived deformation theory to extract obstructions to quantization (anomalies) and identify these with characteristic classes of the target manifold. Specifically, I show that the obstruction to solving the Quantum Master Equation can be identified with the class ch_2 (TX)-ch_2(E), and the obstruction to the quantizing equivariantly with respect to holomorphic vector fields on the source Riemann surface can be identified with c_1 (TX) - c_1(E). By analyzing the theory where the source is an elliptic curve, an explicit geometric construction of the partition function is given.