Barak Kol

Webs of (p,q) 5-branes, five dimensional field theories and grid diagrams

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.

E(7) symmetric area of the black hole horizon.

Extreme black holes with 1/8 of unbroken {ital N}=8 supersymmetry are characterized by the nonvanishing 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{sub 7(7)}. It vanishes in all cases when 1/4 or 1/2 of supersymmetry is unbroken. The supergravity nonrenormalization theorem for the area of the horizon in the {ital N}=8 case protects the unique {ital U}-duality invariant. {copyright} {ital 1996 The American Physical Society.}

The phase transition between caged black holes and black strings

Abstract Black-hole uniqueness is known to fail in higher dimensions, and the multiplicity of black hole phases leads to phase transitions physics in General Relativity. The black-hole black-string transition is a prime realization of such a system and its phase diagram has been the subject of considerable study in the last few years. The most surprising results seem to be the appearance of critical dimensions where the qualitative behavior of the system changes, and a novel kind of topology change. Recently, a full phase diagram was determined numerically, confirming earlier predictions for a merger of the black-hole and black-string phases and giving very strong evidence that the end-state of the Gregory–Laflamme instability is a black hole (in the dimension range 5 ⩽ D ⩽ 13 ). Here this progress is reviewed, illustrated with figures, put into a wider context, and the still open questions are listed.

Moduli, Scalar Charges, and the First Law of Black Hole Thermodynamics.

We show that under variation of moduli fields {phi} the first law of black hole thermodynamics becomes {ital dM}={kappa}{ital dA}/8{pi} +{Omega}{ital dJ}+{psi}{ital dq}+{chi}{ital dp}{minus}{Sigma}{ital d}{phi}, where {Sigma} are the scalar charges. Also the ADM mass is extremized at fixed {ital A}, {ital J}, ({ital p},{ital q}) when the moduli fields take the fixed value {phi}{sub fix}({ital p},{ital q}) which depend only on electric and magnetic charges. Thus the double-extreme black hole minimizes the mass for fixed conserved charges. We can now explain the fact that extreme black holes fix the moduli fields at the horizon {phi}={phi}{sub fix}({ital p},{ital q}): {phi}{sub fix} is such that the scalar charges vanish: {Sigma}{bold (}{phi}{sub fix},({ital p},{ital q}){bold )}=0. {copyright} {ital 1996 The American Physical Society.}

On exactly marginal deformations of N=4 SYM and type IIB supergravity on AdS(5) x S**5

= 4 supersymmetric Yang-Mills theory with gauge group SU(N) (N ≥ 3) is believed to have two exactly marginal deformations which break the supersymmetry to = 1. We discuss the construction of the string theory dual to these deformations, in the supergravity approximation, in a perturbation series around the AdS5 × S5 solution. We construct explicitly the deformed solution at second order in the deformation. We show that deformations which are marginal but not exactly marginal lead to a non-conformal solution with a logarithmically running coupling constant. Surprisingly, at third order in the deformation we find the same beta functions for the couplings in field theory and in supergravity, suggesting that the leading order beta functions (or anomalous dimensions) do not depend on the gauge coupling (the coefficient is not renormalized).

String webs and 1/4 BPS monopoles

Abstract We argue for the existence of many new 1 4 BPS states in N = 4 SU(Nc) Super-Yang-Mills theory with Nc ⩾ 3, by constructing them from supersymmetric string webs whose external strings terminate on parallel D3-branes. The masses of the string webs are shown to agree with the BPS bound for the corresponding states in SYM. We identify the curves of marginal stability, at which these states decay into other BPS states. We find the bosonic and fermionic zero-modes of the string webs, and thereby the degeneracy and spin content of some of the BPS states. States of arbitrarily high spin are predicted in this manner, all of which become massless at the conformal point. For Nc ⩾ 4 we find BPS states which transform in long multiplets, and are therefore not protected against becoming stable non-BPS states as moduli are varied. The mass of these extremal non-BPS states is constrained as they are connected to BPS states. Analogous geometric phenomena are anticipated.

Counting Schwarzschild and charged black holes

We review the arguments that fundamental string states are in one to one correspondence with black hole states. We demonstrate the power of the assumption by showing that it implies that the statistical entropy of a wide class of nonextreme black holes occurring in string theory is proportional to the horizon area. However, the numerical coefficient relating the area and entropy only agrees with the Bekenstein-Hawking formula if the central charge of the string is six which does not correspond to any known string theory. Unlike the current D-brane methods the method used in this paper is applicable for the case of Schwarzschild and highly non-extreme charged black holes.

Black-brane instability in an arbitrary dimension

The black-hole-black-string system is known to exhibit critical dimensions and therefore it is interesting to vary the spacetime dimension D, treating it as a parameter of the system. We derive the large D asymptotics of the critical, i.e. marginally stable, string following an earlier numerical analysis. For a background with an arbitrary compactification manifold we give an expression for the critical mass of a corresponding black brane. This expression is completely explicit for T n , the n-dimensional torus of an arbitrary shape. An indication is given that by employing a higher dimensional torus, rather than a single compact dimension, the total critical dimension above which the nature of the black-brane-black-hole phase transition changes from sudden to smooth could be as low as D ≤ 11.

Caged black holes: Black holes in compactified space-times. 2. 5-d numerical implementation

We describe the first convergent numerical method to determine static black hole solutions (with S^3 horizon) in 5d compactified spacetime. We obtain a family of solutions parametrized by the ratio of the black hole size and the size of the compact extra dimension. The solutions satisfy the demanding integrated first law. For small black holes our solutions approach the 5d Schwarzschild solution and agree very well with new theoretical predictions for the small corrections to thermodynamics and geometry. The existence of such black holes is thus established. We report on thermodynamical (temperature, entropy, mass and tension along the compact dimension) and geometrical measurements. Most interestingly, for large masses (close to the Gregory-Laflamme critical mass) the scheme destabilizes. We interpret this as evidence for an approach to a physical tachyonic instability. Using extrapolation we speculate that the system undergoes a first order phase transition.

5d field theories and M theory

5-brane configurations describing 5d field theories are promoted to an M-theory description a la Witten in terms of polynomials in two complex variables. The coefficients of the polynomials are the Coulomb branch. This picture resolves apparent singularities at vertices and reveals exponentially small corrections. These corrections ask to be compared to world line instanton corrections. From a different perspective this procedure may be used to define a diagrammatic representation of polynomials in two variables.