Gribov ambiguity in asymptotically AdS three-dimensional gravity

In this paper the zero modes of the de Donder gauge Faddeev-Popov operator for three-dimensional gravity with negative cosmological constant are analyzed. It is found that the ${\mathrm{AdS}}_3$ vacuum produces (infinitely many) normalizable smooth zero modes of the Faddeev-Popov operator. On the other hand, it is found that the Bañados-Teitelboim-Zanelli black hole (including the zero mass black hole) does not generate zero modes. This differs from the usual Gribov problem in QCD where, close to the maximally symmetric vacuum, the Faddeev-Popov determinant is positive definite while “far enough” from the vacuum it can vanish. This suggests that the zero mass Bañados-Teitelboim-Zanelli black hole could be a suitable ground state of three-dimensional gravity with negative cosmological constant. Because of the kinematic origin of this result, it also applies for other covariant gravity theories in three dimensions with ${\mathrm{AdS}}_3$ as maximally symmetric solution, such as new massive gravity and topologically massive gravity. The relevance of these results for supersymmetry breaking is pointed out.