We place further restriction on the possible topology of stationary asymptotically flat vacuum black holes in 5 spacetime dimensions. We prove that the horizon manifold can be either a connected sum of Lens spaces and "handles" S 1 × S 2 , or the quotient of S 3 by certain finite groups of isometries (with no "handles"). The resulting horizon topologies include Prism manifolds and quotients of the Poincare homology sphere. We also show that the topology of the domain of outer communication is a cartesian product of the time direction with a finite connected sum of R 4 , S 2 × S 2 's and CP 2 's, minus the black hole itself. We do not assume the existence of any Killing vector beside the asymptotically timelike one required by definition for stationarity.

We give a complete, self-contained, and mathematically rigorous proof that Euclidean Yang-Mills theories are perturbatively renormalisable, in the sense that all correlation functions of arbitrary composite local operators fulfil suitable Ward identities. Our proof treats rigorously both all ultraviolet and infrared problems of the theory and provides, in the end, detailed analytical bounds on the correlation functions of an arbitrary number of composite local operators. These bounds are formulated in terms of certain weighted spanning trees extending between the insertion points of these operators. Our proofs are obtained within the framework of the Wilson-Wegner-Polchinski-Wetterich renormalisation group flow equations, combined with estimation techniques based on tree structures. Compared with previous mathematical treatments of massless theories without local gauge invariance [R. Guida and Ch. Kopper, arXiv:1103.5692; J. Holland, S. Hollands, and Ch. Kopper, Commun. Math. Phys. 342 (2016) 385] our constructions require several technical advances; in particular, we need to fully control the BRST invariance of our correlation functions.

We derive a novel formula for the derivative of operator product expansion (OPE) coefficients with respect to a coupling constant. The formula only involves the OPE coefficients themselves, and no further input, and is in this sense self-consistent. Furthermore, unlike other formal identities of this general nature in quantum field theory (such as the formal expression for the Lagrangian perturbation of a correlation function), our formula is completely well-defined from the start, i.e. requires no further UV-renormalization. This feature is a result of a cancelation of UV-divergences between various terms in our identity. Our proof, and an analysis of the features, of our identity is given for the example of massive, Euclidean ϕ 4 theory in 4 dimensional Euclidean space, and relies heavily on the framework of the renormalization group flow equations. It is valid to arbitrary, but finite orders in perturbation theory. The final formula, however, makes no explicit reference to the renormalization group flow, nor to perturbation theory, and we conjecture that it also holds non-perturbatively. The identity can be applied constructively because it gives recursive algorithm for the computation of OPE coefficients to arbitrary (finite) perturbation order in terms of the zeroth order coefficients corresponding to the underlying free field theory, which in turn are trivial to obtain. We briefly illustrate the relation of this method to more standard methods for computing the OPE in some simple examples.