Regulating competitive balance in sport

Sport is an integral part of popular culture, capturing the hearts and minds of millions of people from every section of society 1. Maintaining competitive balance in sport is essential in preserving its long term survival. A higher uncertainty of outcome in contests is likely to result in greater spectator demand and increased revenues; whereas an imbalanced competition will eventually lead to a decline in attendance and the overall quality of the sport. The regulation of competitive balance in individual sports is difficult to implement, as success relies largely on the athlete’s individual skill and ability 2. In contrast, team sports can be influenced by a number of regulatory measures such as player drafts, salary caps, transfer windows, and player quotas. Analysis into Europe and North America’s regulation of competitive balance reveals a variety of measures used to foster greater equality of opportunity between teams competing with one another in sport. Unfortunately European sports are restricted somewhat, as access to sport is considered to be a social advantage and importance for the integration of citizens, and therefore cannot remain outside the scope of the fundamental principles of free movement. Although the European Union accepts the ‘specificity of sport’, it grants no exception from European Union law for the industry; a principle that has recently been reaffirmed in the decisions of Bosman 3 and Meca-Medina 4. Sports governing bodies must therefore ensure new rules and regulatory measures comply with European Union law; reducing the number of measures available to them

No-scale supersymmetry breaking vacua and soft terms with torsion

We analyze the conditions to have no-scale supersymmetry breaking solutions of type IIA and IIB supergravity compactified on manifolds of SU(3)-structure. The supersymmetry is spontaneously broken by the intrinsic torsion of the internal space. For type IIB orientifolds with O9 and O5-planes the mass of the gravitino is governed by the torsion class W_1, and the breaking is mediated through F-terms associated to descendants of the original N=2 hypermultiplets. For type IIA orientifolds with O6-planes we find two families of solutions, depending on whether the breaking is mediated exclusively by hypermultiplets or by a mixture of hypermultiplets and vector multiplets, the latter case corresponding to a class of Scherk-Schwarz compactifications not dual to any geometric IIB setup. We compute the geometrically induced mu-terms for D5, D6 and D9-branes on twisted tori, and discuss the patterns of soft-terms which arise for pure moduli mediation in each type of breaking. As for D3 and D7-branes in presence of 3-form fluxes, the effective scalar potential turns out to possess interesting phenomenological properties.Comment: 44 pages; several minor corrections and added reference

Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is $SU(3)/U(1)^2$. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.Comment: 32 pages, reference adde

Type IIA orientifold compactification on SU(2)-structure manifolds

We investigate the effective theory of type IIA string theory on six-dimensional orientifold backgrounds with SU(2)-structure. We focus on the case of orientifolds with O6-planes, for which we compute the bosonic effective action in the supergravity approximation. For a generic SU(2)-structure background, we find that the low-energy effective theory is a gauged N=2 supergravity where moduli in both vector and hypermultiplets are charged. Since all these supergravities descend from a corresponding N=4 background, their scalar target space is always a quotient of a SU(1,1)/U(1) x SO(6,n)/SO(6)xSO(n) coset, and is therefore also very constrained.Comment: 31 pages; v2: local report number adde

An index for the Dirac operator on D3 branes with background fluxes

We study the problem of instanton generated superpotentials in Calabi-Yau orientifold compactifications directly in type IIB string theory. To this end, we derive the Dirac equation on a Euclidean D3 brane in the presence of background fluxes. We propose an index which governs whether the generation of a superpotential in the effective 4d theory by D3 brane instantons is possible. Applying the formalism to various classes of examples, including the K3 x T^2/Z_2 orientifold, in the absence and presence of fluxes, we show that our results are consistent with conclusions attainable via duality from an M-theory analysis.Comment: Fermion coupling to five-form restored, conclusions of the paper unchange

Some No-go Theorems for String Duals of Non-relativistic Lifshitz-like Theories

We study possibilities of string theory embeddings of the gravity duals for non-relativistic Lifshitz-like theories with anisotropic scale invariance. We search classical solutions in type IIA and eleven-dimensional supergravities which are expected to be dual to (2+1)-dimensional Lifshitz-like theories. Under reasonable ansaetze, we prove that such gravity duals in the supergravities are not possible. We also discuss a possible physical reason behind this.Comment: 18 pages, Latex, flux conditions clarified (v2), brief summary of results added (v3

Low Energy Supersymmetry from Non-Geometry

We study a class of flux compactifications that have all the moduli stabilised, a high (GUT) string scale and a low (TeV) gravitino mass that is generated dynamically. These non-geometric compactifications correspond to type II string theories on SU(3)xSU(3) structure orientifolds. The resulting superpotentials admit, excluding non-perturbative effects, supersymmetric Minkowski vacua with any number of moduli stabilised. We argue that non-perturbative effects are present and introduce terms in the superpotential that are exponentially suppressed by the same moduli that appear perturbatively. These deform the supersymmetric Minkowski vacua to supersymmetric AdS vacua with an exponentially small gravitino mass. The resulting vacua allow for low scale supersymmetry breaking which can be realised by a number of mechanisms.Comment: 36pp; v2 references added, minor clarifications, JHEP versio

Heterotic Flux Attractors

We find attractor equations describing moduli stabilization for heterotic compactifications with generic SU(3)-structure. Complex structure and K\"ahler moduli are treated on equal footing by using SU(3)xSU(3)-structure at intermediate steps. All independent vacuum data, including VEVs of the stabilized moduli, is encoded in a pair of generating functions that depend on fluxes alone. We work out an explicit example that illustrates our methods.Comment: 37 pages, references and clarifications adde

D-branes on AdS flux compactifications

We study D-branes in N=1 flux compactifications to AdS_4. We derive their supersymmetry conditions and express them in terms of background generalized calibrations. Basically because AdS has a boundary, the analysis of stability is more subtle and qualitatively different from the usual case of Minkowski compactifications. For instance, stable D-branes filling AdS_4 may wrap trivial internal cycles. Our analysis gives a geometric realization of the four-dimensional field theory approach of Freedman and collaborators. Furthermore, the one-to-one correspondence between the supersymmetry conditions of the background and the existence of generalized calibrations for D-branes is clarified and extended to any supersymmetric flux background that admits a time-like Killing vector and for which all fields are time-independent with respect to the associated time. As explicit examples, we discuss supersymmetric D-branes on IIA nearly Kaehler AdS_4 flux compactifications.Comment: 43 pages, 2 pictures, 1 table; v2: added references, color to figure and corrected typo in (6.21b