On First-Quantized Fermions in Compact Dimensions

TL;DR

This paper examines the path integral formulation for fermions in compact dimensions, highlighting the necessity of Wilson loops and their interpretation as fermion condensates, with implications for string theory.

Contribution

It introduces a detailed analysis of fermionic path integrals in compact target spaces, emphasizing the role of Wilson loops and their reinterpretation in string theory contexts.

Findings

Path integrals require nonvanishing Wilson loops in compact dimensions.

Wilson loops can be interpreted as condensates of world-line fermions.

Implications for fermionic string theory path integrals are discussed.

Abstract

We discuss the path integral representation for the fermionic particles and strings and concentrate at the problems arising when some target-space dimensions are compact. An example of partition function for fermionic particle at finite temperature or with one compact target-space dimension is considered in detail. It is demonstrated that the first-quantized path integral requires, in general, presence of nonvanishing "Wilson loops" and modulo some common problems for real fermions in Grassmannian formulation one can try to reinterpret them in terms of condensates of the world-line fermions. The properties of corresponding path integrals in string theory are also discussed.