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

TL;DR

This paper investigates the geometric structure of Wick rotations in quantum field theory, analyzing their topological and stratification properties on manifolds across different dimensions, with detailed focus on two-dimensional cases.

Contribution

It provides a detailed geometric and topological analysis of Wick rotations on manifolds, including explicit computations and local resolutions of metric singularities in low dimensions.

Findings

Explicit computation of the topology of Wick rotation sets in 2, 3, and 4 dimensions

Analysis of the embedding of Wick rotation sets in ambient spaces

Resolution of metric singularities via local Wick rotations in 2D

Abstract

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 $\Sigma$.