Towards Lefschetz thimbles in Sigma models, I

TL;DR

This paper explores complex critical points and Lefschetz thimbles in two-dimensional sigma models, specifically $O(N)$ and ${f CP}^{N-1}$ models, to understand their path integral contours at finite volume and temperature.

Contribution

It identifies a large class of relevant complex critical points in $O(N)$ and ${f CP}^{N-1}$ models, focusing on zero instanton charge sectors and specific model cases.

Findings

Identified relevant complex critical points for $O(N)$ and ${f CP}^{N-1}$ models.

Analyzed path integral contours in finite volume and temperature settings.

Set the stage for future work on instanton charges and broader solution classes.

Abstract

We study two dimensional path integral Lefschetz thimbles, i.e. the possible path integration contours. Specifically, in the examples of the $O(N)$ and ${\bf CP}^{N-1}$ models, we find a large class of complex critical points of the sigma model actions which are relevant for the theory in finite volume at finite temperature, with various chemical potentials corresponding to the symmetries of the models. In this paper we discuss the case of the $O(2m)$ and the ${\bf CP}^{N-1}$ models in the sector of zero instanton charge, as well as some solutions of the $O(2m+1)$ model. The ${\bf CP}^{N-1}$-model for all instanton charges and a more general class of solutions of the $O(N)$-model with odd $N$ will be discussed in the forthcoming paper.