First-Quantized Fermions in Compact Dimensions

We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author [1], [2]. We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.