Fermions and Kaluza-Klein vacuum decay: a toy model

We address the question of whether fermions with a twisted periodicity condition suppress the semiclassical decay of the $M^4\times S^1$ Kaluza–Klein vacuum. We consider a toy $(1{+}1)$-dimensional model with twisted fermions in a cigar-shaped Euclidean background geometry and calculate the fermion determinant. We find that the determinant is finite, contrary to expectations. We regard this as an indication that twisted fermions do not stabilize the Kaluza–Klein vacuum.