We present a proof that quantum Yang-Mills theory can be consistently defined as a renormalized, perturbative quantum field theory on an arbitrary globally hyperbolic curved, Lorentzian spacetime. To this end, we construct the non-commutative algebra
of observables, in the sense of formal power series, as well as a space of corresponding quantum states. The algebra contains all gauge invariant, renormalized, interacting quantum field operators (polynomials in the field strength and its derivatives), and all their relations such as commutation relations or operator product expansion. It can be viewed as a deformation quantization of the Poisson algebra of classical Yang-Mills theory equipped with the Peierls bracket. The algebra is constructed as the cohomology of an auxiliary algebra describing a gauge fixed theory with ghosts and anti-fields. A key technical difficulty is to establish a suitable hierarchy of Ward identities at the renormalized level that ensure conservation of the interacting BRST-current, and that the interacting BRST-charge is nilpotent. The algebra of physical interacting field observables is obtained as the cohomology of this charge. As a consequence of our constructions, we can prove that the operator product expansion closes on the space of gauge invariant operators. Similarly, the renormalization group flow is proved not to leave the space of gauge invariant operators.
We prove a uniqueness theorem for stationary $D$-dimensional Kaluza-Klein black holes with $D-2$ Killing fields, generating the symmetry group ${mathbb R} times U(1)^{D-3}$. It is shown that the topology and metric of such black holes is uniquely det
ermined by the angular momenta and certain other invariants consisting of a number of real moduli, as well as integer vectors subject to certain constraints.
All known stationary black hole solutions in higher dimensions possess additional rotational symmetries in addition to the stationary Killing field. Also, for all known stationary solutions, the event horizon is a Killing horizon, and the surface gra
vity is constant. In the case of non-degenerate horizons (non-extremal black holes), a general theorem was previously established [gr-qc/0605106] proving that these statements are in fact generally true under the assumption that the spacetime is analytic, and that the metric satisfies Einsteins equation. Here, we extend the analysis to the case of degenerate (extremal) black holes. It is shown that the theorem still holds true if the vector of angular velocities of the horizon satisfies a certain diophantine condition, which holds except for a set of measure zero.
We show that two stationary, asymptotically flat vacuum black holes in 5 dimensions with two commuting axial symmetries are identical if and only if their masses, angular momenta, and their ``rod structures coincide. We also show that the horizon mus
t be topologically either a 3-sphere, a ring, or a Lens-space. Our argument is a generalization of constructions of Morisawa and Ida (based in turn on key work of Maison) who considered the spherical case, combined with basic arguments concerning the nature of the factor manifold of symmetry orbits.
