Remarks on the Geometry of Wick Rotation in QFT and its Localization on Manifolds

The geometric aspect of Wick rotation in quantum field theory and its localization on manifolds are explored. After the explanation of the notion and its related geometric objects, we study the topology of the set of landing $W$ for Wick rotations and its natural stratification. These structures in two, three, and four dimensions are computed explicitly. We then focus on more details in two dimensions. In particular, we study the embedding of $W$ in the ambient space of Wick rotations, the resolution of the generic metric singularities of a Lorentzian surface $\Sigma$ by local Wick rotations, and some related $S^1$-bundles over $\Sigma$.