Quasirandomness in hypergraphs

An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. We give short and purely combinatorial proofs of the main equivalences in Towsner's result.Comment: 19 page

On the Effective Description of Large Volume Compactifications

We study the reliability of the Two-Step moduli stabilization in the type-IIB Large Volume Scenarios with matter and gauge interactions. The general analysis is based on a family of N=1 Supergravity models with a factorizable Kaehler invariant function, where the decoupling between two sets of fields without a mass hierarchy is easily understood. For the Large Volume Scenario particular analyses are performed for explicit models, one of such developed for the first time here, finding that the simplified version, where the Dilaton and Complex structure moduli are regarded as frozen by a previous stabilization, is a reliable supersymmetric description whenever the neglected fields stand at their leading F-flatness conditions and be neutral. The terms missed by the simplified approach are either suppressed by powers of the Calabi-Yau volume, or are higher order operators in the matter fields, and then irrelevant for the moduli stabilization rocedure. Although the power of the volume suppressing such corrections depends on the particular model, up to the mass level it is independent of the modular weight for the matter fields. This at least for the models studied here but we give arguments to expect the same in general. These claims are checked through numerical examples. We discuss how the factorizable models present a context where despite the lack of a hierarchy with the supersymmetry breaking scale, the effective theory still has a supersymmetric description. This can be understood from the fact that it is possible to find vanishing solution for the auxiliary components of the fields being integrated out, independently of the remaining dynamics. Our results settle down the question on the reliability of the way the Dilaton and Complex structure are treated in type-IIB compactifications with large compact manifold volumina.Comment: 23 pages + 2 appendices (38 pages total). v2: minor improvements, typos fixed. Version published in JHE

Metastable SUSY Breaking, de Sitter Moduli Stabilisation and K\"ahler Moduli Inflation

We study the influence of anomalous U(1) symmetries and their associated D-terms on the vacuum structure of global field theories once they are coupled to N=1 supergravity and in the context of string compactifications with moduli stabilisation. In particular, we focus on a IIB string motivated construction of the ISS scenario and examine the influence of one additional U(1) symmetry on the vacuum structure. We point out that in the simplest one-Kahler modulus compactification, the original ISS vacuum gets generically destabilised by a runaway behaviour of the potential in the modulus direction. In more general compactifications with several Kahler moduli, we find a novel realisation of the LARGE volume scenario with D-term uplifting to de Sitter space and both D-term and F-term supersymmetry breaking. The structure of soft supersymmetry breaking terms is determined in the preferred scenario where the standard model cycle is not stabilised non-perturbatively and found to be flavour universal. Our scenario also provides a purely supersymmetric realisation of Kahler moduli (blow-up and fibre) inflation, with similar observational properties as the original proposals but without the need to include an extra (non-SUSY) uplifting term.Comment: 38 pages, 8 figures. v2: references added, minor correction

"Big" Divisor D3/D7 Swiss Cheese Phenomenology

We review progress made over the past couple of years in the field of Swiss Cheese Phenomenology involving a mobile space-time filling D3-brane and stack(s) of fluxed D7-branes wrapping the "big" (as opposed to the "small") divisor in (the orientifold of a) Swiss-Cheese Calabi-Yau. The topics reviewed include reconciliation of large volume cosmology and phenomenology, evaluation of soft supersymmetry breaking parameters, one-loop RG-flow equations' solutions for scalar masses, obtaining fermionic (possibly first two generations' quarks/leptons) mass scales in the O(MeV-GeV)-regime as well as (first two generations') neutrino masses (and their one-loop RG flow) of around an eV. The heavy sparticles and the light fermions indicate the possibility of "split SUSY" large volume scenario.Comment: Invited review for MPLA, 14 pages, LaTe

Neutrino Masses, Baryon Asymmetry, Dark Matter and the Moduli Problem : A Complete Framework

Recent developments in string theory have led to "realistic" string compactifications which lead to moduli stabilization while generating a hierarchy between the Electroweak and Planck scales at the same time. However, this seems to suggest a rethink of our standard notions of cosmological evolution after the end of inflation and before the beginning of BBN. We argue that within classes of realistic string compactifications, there generically exists a light modulus with a mass comparable to that of the gravitino which generates a large late-time entropy when it decays. Therefore, all known mechanisms of generating the baryon asymmetry of the Universe in the literature have to take this fact into account. In this work, we find that it is still possible to naturally generate the observed baryon asymmetry of the Universe as well as light left-handed neutrino masses from a period of Affleck-Dine(AD) leptogenesis shortly after the end of inflation, in classes of realistic string constructions with a minimal extension of the MSSM below the unification scale (consisting only of right-handed neutrinos) and satisfying certain microscopic criteria described in the text. The consequences are as follows. The lightest left-handed neutrino is required to be virtually massless. The moduli (gravitino) problem can be naturally solved in this framework both within gravity and gauge mediation. The observed upper bound on the relic abundance constrains the moduli-matter and moduli-gravitino couplings since the DM is produced non-thermally within this framework. Finally, although not a definite prediction, the framework naturally allows a light right-handed neutrino and sneutrinos around the electroweak scale which could have important implications for DM as well as the LHC.Comment: 41 pages, no figures, journal version adde

Wavefunctions and the Point of E8 in F-theory

In F-theory GUTs interactions between fields are typically localised at points of enhanced symmetry in the internal dimensions implying that the coefficient of the associated operator can be studied using a local wavefunctions overlap calculation. Some F-theory SU(5) GUT theories may exhibit a maximum symmetry enhancement at a point to E8, and in this case all the operators of the theory can be associated to the same point. We take initial steps towards the study of operators in such theories. We calculate wavefunctions and their overlaps around a general point of enhancement and establish constraints on the local form of the fluxes. We then apply the general results to a simple model at a point of E8 enhancement and calculate some example operators such as Yukawa couplings and dimension-five couplings that can lead to proton decay.Comment: 46 page

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

Discovery of an Unbound Hyper-Velocity Star in the Milky Way Halo

We have discovered a star, SDSS J090745.0+024507, leaving the Galaxy with a heliocentric radial velocity of +853+-12 km/s, the largest velocity ever observed in the Milky Way halo. The star is either a hot blue horizontal branch star or a B9 main sequence star with a heliocentric distance ~55 kpc. Corrected for the solar reflex motion and to the local standard of rest, the Galactic rest-frame velocity is +709 km/s. Because its radial velocity vector points 173.8 deg from the Galactic center, we suggest that this star is the first example of a hyper-velocity star ejected from the Galactic center as predicted by Hills and later discussed by Yu & Tremaine. The star has [Fe/H]~0, consistent with a Galactic center origin, and a travel time of <80 Myr from the Galactic center, consistent with its stellar lifetime. If the star is indeed traveling from the Galactic center, it should have a proper motion of 0.3 mas/yr observable with GAIA. Identifying additional hyper-velocity stars throughout the halo will constrain the production rate history of hyper-velocity stars at the Galactic center.Comment: 4 pages, submitted to ApJ Letter