Effective actions on finite cylinders

TL;DR

This paper analyzes spectral problems on finite cylinders, deriving explicit formulas for effective actions and eigenvalues for scalar and fermionic fields under various boundary conditions, revealing spectral asymmetries and boundary effects.

Contribution

It provides new explicit expressions for effective actions and eigenvalues for scalar and fermionic fields on finite cylinders, including general boundary conditions and spectral asymmetries.

Findings

Explicit elliptic function expressions for scalar effective actions.

General eigenvalue formulas for fermions with bag boundary conditions.

Discovery of spectral asymmetry between positive and negative Dirac modes.

Abstract

Some free--field spectral problems on a generalised cylinder are revisited. In two dimensions, conformal scalar effective actions for various boundary conditions are written in elliptic function terms and some special values given. Fermions are then discussed in arbitrary dimensions and an analysis of the most general local (`bag') boundary conditions is given leading to an intrinsic formula for the eigenvalues which interpolate between Neveu--Schwarz and Ramond. This is used to give reasonably explicit expressions for the effective action and other values of the zeta function. It is shown that these boundary conditions are transferred as a chemical potential to the boundary spectral problem. The existence of real exponential eigenmodes along the cylinder is pointed out and a curious asymmetry between the positive and negative Dirac spectra uncovered for large cylinder length.