We continue to study 5d N = 1 supersymmetric field theories and their compactifications on a circle through brane configurations. We develop a model, which we call (p, q) Webs, which enables simple geometrical computations to reproduce the known results, and facilitates further study. The physical concepts of field theory are transparent in this picture, offering an interpretation for global symmetries, local symmetries, the effective (running) coupling, the Coulomb and Higgs branches, the monopole tensions, and the mass of BPS particles. A rule for the dimension of the Coulomb branch is found by introducing Grid Diagrams. Some known classifications of field theories are reproduced. In addition to the study of the vacuum manifold we develop methods to determine the BPS spectrum. Some states, such as quarks, correspond to instantons inside the 5-brane which we call strips. In general, these may not be identified with (p, q) strings. We describe how a strip can bend out of a 5-brane, becoming a string. A general BPS state corresponds to a Web of strings and strips. For special values of the string coupling a few strips can combine and leave the 5-brane as a string.

We show that under variation of moduli fields φ the first law of black hole thermodynamics becomes dM = κdA 8π + ΩdJ + ψdq + χdp − Σdφ, where Σ are the scalar charges. We also show that the ADM mass is extremized at fixed A, J, (p, q) when the moduli fields take the fixed value φ fix (p, q) which depend only on electric and magnetic charges. It follows that the least mass of any black hole with fixed conserved electric and magnetic charges is given by the mass of the double-extreme black hole with these charges. Our work allows us to interpret the previously established result that for all extreme black holes the moduli fields at the horizon take a value φ = φ fix (p, q) depending only on the electric and magnetic conserved charges: φ fix (p, q) is such that the scalar charges Σ(φ fix , (p, q)) = 0. PACS 04.70. Dy, 11.30.Pb, hep-th/9607108

Extreme black holes with 1/8 of unbroken N = 8 supersymmetry are characterized by the non-vanishing area of the horizon. The central charge matrix has four generic eigenvalues. The area is proportional to the square root of the invariant quartic form of E 7(7) . It vanishes in all cases when 1/4 or 1/2 of supersymmetry is unbroken. The supergravity non-renormalization theorem for the area of the horizon in N = 8 case protects the unique U-duality invariant.

In the presence of compact dimensions massive solutions of General Relativity may take one of several forms including the black-hole and the black-string, the simplest relevant background being R 3+1 × S 1 . It is shown how Morse theory places constraints on the qualitative features of the phase diagram, and a minimalistic diagram is suggested which describes a first order transition whose only stable phases are the uniform string and the black-hole. The diagram calls for a topology changing "merger" transition in which the black-hole evolves continuously into an unstable black-string phase. As evidence a local model for the transition is presented in which the cone over S 2 × S 2 plays a central role. Horizon cusps do not appear as precursors to black hole merger. A generalization to higher dimensions finds that whereas the cone has a tachyon function for d = 5, its stability depends interestingly on the dimension -it is unstable for d < 10, and stable for d > 10.

We find some lifts to M theory of orientifold and orbifold planes including the O1, O3 and O5 planes of Type IIB and their transformations under SL(2, Z) . The possible discrete torsion variants (or K theory classes) are explored, and are interpreted as arising from brane intersections with planes. We find new variants of the O0 and of an orbifold line (OF1) and determine their tensions in some cases. A systematic review of orientifolds, M orientifolds, and known M lifts, with some new clarifications is included together with a discussion of the role of T duality.