We study non-BPS black hole solutions to ungauged supergravity with 8 supercharges coupled to vector multiplets in four and five dimensions. We identify a large class of five dimensional non-BPS solutions, which we call "almost BPS", that are supersymmetric on local patches and satisfy a first order flow governed by harmonic functions. By dimensional reduction, they give rise to new non-BPS solutions in four dimensions. These solutions allow for some nontrivial asymptotic moduli and multiple centres, similar to their globally supersymmetric cousins. We explicitly discuss a single centre and a two centre example.

An extended class of N = 2 locally supersymmetric invariants with higherderivative couplings based on full superspace integrals, is constructed. These invariants may depend on unrestricted chiral supermultiplets, on vector supermultiplets and on the Weyl supermultiplet. Supersymmetry is realized off-shell. A non-renormalization theorem is proven according to which none of these invariants can contribute to the entropy and electric charges of BPS black holes. Some of these invariants may be relevant for topological string deformations.

We analyze stationary BPS black hole solutions to 4d N = 2 abelian gauged supergravity. Using an appropriate near horizon ansatz, we construct rotating attractors with magnetic flux realizing a topological twist along the horizon surface, for any theory with a symmetric scalar manifold. An analytic flow to asymptotically locally AdS 4 is presented for a subclass of these near-horizon geometries, and an explicit new example of supersymmetric AdS 4 rotating black hole is discussed in detail. We further note that, upon tuning the gauging to special values, one can obtain solutions with different asymptotics, and in particular reductions of doubly spinning asymptotically AdS 5 black holes. Finally we present a proposal for the form of the BPS entropy function with rotation, which we expect to be holographically related to the refined twisted index of the dual theory.

A systematic off-shell reduction scheme from five to four space-time dimensions is presented for supergravity theories with eight supercharges. It is applicable to theories with higher-derivative couplings and it is used to address a number of open questions regarding BPS black holes in five dimensions. Under this reduction the 5D Weyl multiplet becomes reducible and decomposes into the 4D Weyl multiplet and an extra Kaluza-Klein vector multiplet. The emergence of the pseudoscalar field of the latter multiplet and the emergence of the 4D R-symmetry group are subtle features of the reduction. The reduction scheme enables to determine how a 5D supersymmetric Lagrangian with higher-derivative couplings decomposes upon dimensional reduction into a variety of independent 4D supersymmetric invariants, without the need for imposing field equations. In this way we establish, for example, the existence of a new N=2 supersymmetric invariant that involves the square of the Ricci tensor. Finally we resolve the questions associated with the 5D Chern-Simons terms for spinning BPS black holes and their relation to the corresponding 4D black holes.Comment: 35 pages; minor text changes and one more referenc