Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only.

We present new recursion relations for tree amplitudes in gauge theory that give very compact formulas. Our relations give any tree amplitude as a sum over terms constructed from products of two amplitudes of fewer particles multiplied by a Feynman propagator.The two amplitudes in each term are physical, in the sense that all particles are on-shell and momentum conservation is preserved. This is striking, since it is just like adding certain factorization limits of the original amplitude to build up the full answer. As examples, we recompute all known tree-level amplitudes of up to seven gluons and show that our recursion relations naturally give their most compact forms. We give a new result for an eight-gluon amplitude, A(1 + , 2 − , 3 + , 4 − , 5 + , 6 − , 7 + , 8 − ). We show how to build any amplitude in terms of three-gluon amplitudes only.

One-loop amplitudes of gluons in N = 4 gauge theory can be written as linear combinations of known scalar box integrals with coefficients that are rational functions. In this paper we show how to use generalized unitarity to basically read off the coefficients.The generalized unitarity cuts we use are quadruple cuts. These can be directly applied to the computation of four-mass scalar integral coefficients, and we explicitly present results in next-to-next-to-MHV amplitudes. For scalar box functions with at least one massless external leg we show that by doing the computation in signature (− − ++) the coefficients can also be obtained from quadruple cuts, which are not useful in Minkowski signature.As examples, we reproduce the coefficients of some one-, two-, and three-mass scalar box integrals of the seven-gluon next-to-MHV amplitude, and we compute several classes of three-mass and two-mass-hard coefficients of next-to-MHV amplitudes to all multiplicities.

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of N D-brane probes for both N → ∞ and finite N . The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The socalled "Plethystic Exponential" provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.

Via partial resolution of Abelian orbifolds we present an algorithm for extracting a consistent set of gauge theory data for an arbitrary toric variety whose singularity a D-brane probes. As illustrative examples, we tabulate the matter content and superpotential for a D-brane living on the toric del Pezzo surfaces as well as the zeroth Hirzebruch surface. Moreover, we discuss the non-uniqueness of the general problem and present examples of vastly different theories whose moduli spaces are described by the same toric data. Our methods provide new tools for calculating gauge theories which flow to the same universality class in the IR. We shall call it "Toric Duality."

Using the low limit of cosmic ages from globular cluster and the white dwarfs: t0 > 12Gyr, together with recent new high redshift supernova observations from the HST/GOODS program and previous supernova data, we give a considerable estimation of the equation of state for dark energy, with uniform priors as weak as 0.2 < Ωm < 0.4 or 0.1 < Ωmh 2 < 0.16. We find cosmic age limit plays a significant role in lowering the upper bound on the variation amplitude of dark energy equation of state. We propose in this paper a new scenario of dark energy dubbed Quintom, which gives rise to the equation of state larger than −1 in the past and less than −1 today, satisfying current observations. In addition we've also considered the implications of recent X-ray gas mass fraction data on dark energy, which favors a negative running of the equation of state.Age limits of our universe are among the earliest motivations for the existence of the mysterious dark energy. Namely, observations of the earliest galaxies could set a low limit on the age of the universe. In 1998, two groups [1, 2] independently showed the accelerating expansion of our universe basing on Type Ia Supernova (SNe Ia) observations of the redshift-distance relations. The recently released first year WMAP data [3] support strongly the concordance model with dark energy taking part of ∼ 2/3. The most recent discovery of 16 SNe Ia [4] with the Hubble Space Telescope during the GOODS ACS Treasury survey, together with former SNe Ia data alone could provide a strong hint for the existence of dark energy. Riess et al. [4] provided evidence at > 99% for the existence of a transition from deceleration to acceleration using supernova data alone.Despite our current theoretical ambiguity for the nature of dark energy, the prosperous observational data (e.g. supernova, CMB and large scale structure data and so on ) have opened a robust window for testing the recent and even early behavior of dark energy using some simple parameterization for its equation of state (e.g., Ref.[5] ) or even reconstruction of its recent density [6,7,8]. Both recent WMAP fit and more recent fit by Riess et al. find the behavior of dark energy is to great extent in consistency with a cosmological constant. In particular when the equation of state is not restricted to be a constant, the fit to observational data improves dramatically [9,10,11,12]. Huterer and Cooray [10] produced uncorrelated and nearly model-independent band power estimates (basing on the principal component analysis [13]) of the equation of state of dark energy and its density as a function of redshift, by fitting to the recent SNe Ia data they found marginal (2-σ) evidence for W (z) < −1 at z < 0.2, which is consistent with other results in the literature [7,9,10,11,14,15,16,17].The recent fit to first year WMAP and other CMB data, SDSS and 172 SNe Ia data [18] by Tegmark et al [19] provided the most complete and up-to-date fit. Although SNe Ia data accumulated more after that, Ref. [19] should still be a very profitable benchma...

Dimer models are 2-dimensional combinatorial systems that have been shown to encode the gauge groups, matter content and tree-level superpotential of the world-volume quiver gauge theories obtained by placing D3-branes at the tip of a singular toric Calabi-Yau cone. In particular the dimer graph is dual to the quiver graph. However, the string theoretic explanation of this was unclear. In this paper we use mirror symmetry to shed light on this: the dimer models live on a T 2 subspace of the T 3 fiber that is involved in mirror symmetry and is wrapped by D6-branes. These D6-branes are mirror to the D3-branes at the singular point, and geometrically encode the same quiver theory on their world-volume.

We propose a programme for systematically counting the single and multitrace gauge invariant operators of a gauge theory. Key to this is the plethystic function. We expound in detail the power of this plethystic programme for worldvolume quiver gauge theories of D-branes probing Calabi-Yau singularities, an illustrative case to which the programme is not limited, though in which a full intimate web of relations between the geometry and the gauge theory manifests herself. We can also use generalisations of Hardy-Ramanujan to compute the entropy of gauge theories from the plethystic exponential. In due course, we also touch upon fascinating connections to Young Tableaux, Hilbert schemes and the MacMahon Conjecture. * b.feng@imperial.ac.uk † ahanany@perimeterinstitute.ca ‡ hey@maths.ox.ac.uk