The standard model of hot big-bang cosmology requires initial conditions which are problematic in two ways: (1) The early universe is assumed to be highly homogeneous, in spite of the fact that separated regions were causally disconnected (horizon problem); and (2) the initial value of the Hubble constant must be fine tuned to extraordinary accuracy to produce a universe as flat (i.e., near critical mass density) as the one we see today (flatness problem). These problems would disappear if, in its early history, the universe supercooled to temperatures 28 or more orders of magnitude below the critical temperature for some phase transition. A huge expansion factor would then result from a period of exponential growth, and the entropy of the universe would be multiplied by a huge factor when the latent heat is released. Such a scenario is completely natural in the context of grand unified models of elementary-particle interactions. In such models, the supercooling is also relevant to the problem of monopole suppression. Unfortunately, the scenario seems to lead to some unacceptable consequences, so modifications must be sought.

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Inflationary universe: A possible solution to the horizon and flatness problems

Modified Gravity and Cosmology

In the past few years one of the most exciting areas of research in physics has been the interdisciplinary field of cosmology and particle physics. The NSF's Institute for Theoretical Physics in Santa Barbara devoted a 6-month program and an intensive 1-week workshop to the subject. A brief review is given of both the workshop and this field which is attracting attention, in part, because the early Universe seems to be the only laboratory in which to study grand unification.

The early Universe

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

We construct three dimensional Chern-Simons-matter theories with gauge groups U(N) × U(N) and SU(N) × SU(N) which have explicit = 6 superconformal symmetry. Using brane constructions we argue that the U(N) × U(N) theory at level k describes the low energy limit of N M2-branes probing a C4/Zk singularity. At large N the theory is then dual to M-theory on AdS4 × S7/Zk. The theory also has a 't Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS4 × CP3. For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU(2) × SU(2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory.

N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals

Coleman-Weinberg mechanism

The Atiyah-Singer index theorem has recently been used to determine the number of parameters entering the general self-dual Euclidean Yang-Mills configuration with a given topological charge. We give a simple derivation of the form which the index theorem takes in this application.

Note on the Atiyah-Singer index theorem

The effects of pseudoparticle solutions to Yang-Mills field theories are discussed for gauges where the F/sub ..mu..//sub ..nu../ uniquely determine the potentials. Even though a unique vacuum is found, the physical consequences obtained by previous authors are shown to occur. It is then shown how the A/sub 0/ = 0 gauge may be reformulated to give a unique vacuum. The many vacuums found by previous authors are seen to be artifacts of a particular parametrization of function space.

Interpretation of pseudoparticles in physical gauges

The properties of BPS monopoles carrying nonabelian magnetic charges are investigated by following the behavior of the moduli space of solutions as the Higgs field is varied from a value giving a purely abelian symmetry breaking to one that leaves a nonabelian subgroup of the gauge symmetry unbroken. As the limit of nonabelian unbroken symmetry is reached, some of the fundamental abelian monopoles remain massive but acquire nonabelian magnetic charges. The BPS mass formula indicates that others should because massless in this limit. These do not correspond to distinct solitons, but instead are manifested as a ``nonabelian cloud'' surrounding the massive monopoles, with their position and phase degrees of freedom being transformed into parameters characterizing the cloud.

Erick J. Weinberg

Papers

Coleman-Weinberg mechanism

Note on the Atiyah-Singer index theorem

Interpretation of pseudoparticles in physical gauges

Massless and Massive Monopoles Carrying Nonabelian Magnetic Charges

Causality bounds and monopoles connected by strings.