On the spectrum of the Faddeev–Popov operator in topological background fields

In the Gribov–Zwanziger scenario the confinement of gluons is attributed to an enhancement of the spectrum of the Faddeev–Popov operator near eigenvalue zero. This has been observed in functional and also in lattice calculations. The linear rise of the quark–anti-quark potential and thus quark confinement on the other hand seems to be connected to topological excitations. To investigate whether a connection exists between both aspects of confinement, the spectrum of the Faddeev–Popov operator in two topological background fields is determined analytically in SU(2) Yang–Mills theory. It is found that a single instanton, which is likely irrelevant to quark confinement, also sustains only few additional zero-modes. A center vortex, which is likely important to quark confinement, is found to contribute much more zero-modes, provided the vortex is of sufficient flux. Furthermore, the corresponding eigenstates in the vortex case satisfy one necessary condition for the confinement of quarks.