Towards Non-equilibrium Field Theories*

In our study, we emphasized that index theorems are written in terms of characteristic classes. These characteristic classes describe the topology of the model under consideration. Through an examination of supersymmetric/topological path integrals, we aimed to understand the emergence of these topological properties from a path integral framework. This, in turn, offers insights into the manifestation of topological quantities in non-equilibrium cases.

We began by calculating interactions between positive and negative branes on the Schwinger-Keldysh contour. Subsequently, we integrated the Keldysh rotation into these calculations and explored a new form of the Schwinger-Keldysh contour. Then we tried to solve the problem by investigating Witten's iε prescription. And, finally, we settled on approaching the problem by first solving problems in the index theorem and from supersymmetric worldline path integrals.