Maximally Supersymmetric AdS4 Vacua in N=4 Supergravity

We study AdS backgrounds of N=4 supergravity in four space-time dimensions which preserve all sixteen supercharges. We show that the graviphotons have to form a subgroup of the gauge group that consists of an electric and a magnetic SO(3)_+ x SO(3)_-. Moreover, these N=4 AdS backgrounds are necessarily isolated points in field space which have no moduli.Comment: 11 page

Causal Sets: Quantum gravity from a fundamentally discrete spacetime

In order to construct a quantum theory of gravity, we may have to abandon certain assumptions we were making. In particular, the concept of spacetime as a continuum substratum is questioned. Causal Sets is an attempt to construct a quantum theory of gravity starting with a fundamentally discrete spacetime. In this contribution we review the whole approach, focusing on some recent developments in the kinematics and dynamics of the approach.Comment: 10 pages, review of causal sets based on talk given at the 1st MCCQG conferenc

Branes from Light: Embeddings and Energetics for Symmetric kk-Quarks in N=4\mathcal{N}=4 SYM

We construct the D3-brane dual to a $k$-quark of $\mathcal{N}=4$ super-Yang-Mills theory in the totally symmetric representation of $SU(N)$, undergoing arbitrary motion. Our method of construction generalizes previous work by Mikhailov, and proceeds by shooting light rays inward from the anti-de Sitter boundary, to trace out the brane embedding. We expect this method to have wider relevance, and provide evidence for this by showing that it correctly reproduces the known D5-brane embeddings dual to totally antisymmetric $k$-quarks. As an application of our solutions, we compute the energy of the D3-brane and extract from it the $k$-quark's intrinsic energy and rate of radiation. The result matches expectations based on previous calculations, and makes contact with the exact Bremsstrahlung function for the fundamental representation.Comment: 1+28 pages, 3 figures. v2: references adde

Supersymmetric embedding of antibrane polarization

We study the supersymmetry breaking induced by probe anti-D3-branes at the tip of the Klebanov-Strassler throat geometry. Antibranes inside this geometry polarize and can be described by an NS5-brane wrapping an $S^2$. When the number of antibranes is small compared to the background flux a metastable state exists that breaks supersymmetry. We present a manifestly supersymmetric effective model that realizes the polarized metastable state as a solution, spontaneously breaking the supersymmetry. The supersymmetric model relies crucially on the inclusion of Kaluza-Klein (matrix) degrees of freedom on the $S^2$ and two supersymmetric irrelevant deformations of ${\cal N}=4$ super-Yang-Mills (SYM), describing a large number of supersymmetric D3-branes in the IR. We explicitly identify the massless Goldstino and compute the spectrum of massive fluctuations around the metastable supersymmetry-breaking minimum, finding a Kaluza-Klein tower with masses warped down from the string scale. Below the Kaluza-Klein scale the massive tower can be integrated out and supersymmetry is realized nonlinearly. We comment on the effect of the Kaluza-Klein modes on the effective description of de Sitter vacua in string theory and inflationary model building.Comment: v1: 9 pages, v2: added references, changed title slightly and fixed typo

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Magnetic properties of dense holographic QCD

We investigate the Sakai-Sugimoto model at nonzero baryon chemical potential in a background magnetic field both in the confined phase and in the deconfined phase with restored chiral symmetry. In this case the 8-brane Chern-Simons term becomes important. In the confined phase it generates a gradient of the pseudo-scalar "pion", which carries a non-vanishing baryon charge. Above a critical value of the chemical potential there is a second order phase transition to a mixed phase which includes also ordinary baryonic matter. However, at fixed baryon charge density the matter is purely "pion"-gradient above a critical magnetic field. In the deconfined chiral-symmetric phase at nonzero chemical potential the magnetic field induces an axial current. We also compute the magnetization of the baryonic matter and find that it is paramagnetic in all three phases but with nonlinear behavior at large magnetic field.Comment: 20 pages, 19 figures; v2 figure and reference added; v3 corrected treatment of boundary terms in CS action leading to small quantitative adjustments; v4 explanation of boundary terms improved, to appear in PR

Chiral Symmetry Breaking and External Fields in the Kuperstein-Sonnenschein Model

A novel holographic model of chiral symmetry breaking has been proposed by Kuperstein and Sonnenschein by embedding non-supersymmetric probe D7 and anti-D7 branes in the Klebanov-Witten background. We study the dynamics of the probe flavours in this model in the presence of finite temperature and a constant electromagnetic field. In keeping with the weakly coupled field theory intuition, we find the magnetic field promotes spontaneous breaking of chiral symmetry whereas the electric field restores it. The former effect is universally known as the "magnetic catalysis" in chiral symmetry breaking. In the presence of an electric field such a condensation is inhibited and a current flows. Thus we are faced with a steady-state situation rather than a system in equilibrium. We conjecture a definition of thermodynamic free energy for this steady-state phase and using this proposal we study the detailed phase structure when both electric and magnetic fields are present in two representative configurations: mutually perpendicular and parallel.Comment: 50 pages, multiple figures, minor typo fixed, references adde