Applications of patching to quadratic forms and central simple algebras

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.Comment: 48 pages; connectivity now required in the definition of rational group; beginning of Section 4 reorganized; other minor change

"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

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

Fields of definition for division algebras

Abstract Let A be a finite-dimensional division algebra containing a base field k in its center F. A is defined over a subfield F 0 if there exists an F 0 -algebra A 0 such that A = A 0 F 0 F. The following are shown. (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n > 5, then A is defined over a field F 0 of transcendence degree 6 1 2 (n − 1)(n − 2) over k. (iii) If A is a Z/m × Z/2-crossed product for some m > 2 (and in particular, if A is any algebra of degree 4) then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, M m (A) can be defined over a field F 0 such that trdeg k (F 0 ) 6 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F 0 of transcendence degree 6 4. (In (i), (iii) and (iv) it is assumed that the center of A contains certain roots of unity.

Stable de Sitter vacua in N=2, D=5 supergravity

We find 5D gauged supergravity theories exhibiting stable de Sitter vacua. These are the first examples of stable de Sitter vacua in higher-dimensional (D>4) supergravity. Non-compact gaugings with tensor multiplets and R-symmetry gauging seem to be the essential ingredients in these models. They are however not sufficient to guarantee stable de Sitter vacua, as we show by investigating several other models. The qualitative behaviour of the potential also seems to depend crucially on the geometry of the scalar manifold.Comment: 26 pages, v2:typos corrected, published versio

Schooling for violence and peace : how does peace education differ from ‘normal’ schooling?

This article reviews literature on the roles of schooling in both reproducing and actively perpetrating violence, and sets out an historical explanation of why schools are socially constructed in such a way as to make these roles possible. It then discusses notions of peace education in relation to one particular project in England before using empirical data from research on the project to examine contrasts between peace education approaches and ‘normal’ schooling from the viewpoints of project workers, pupils and teachers. It concludes that such contrasts and tensions do indeed exist and that this raises serious questions about the compatibility of peace education and formal schooling

An Inflaton Mass Problem in String Inflation from Threshold Corrections to Volume Stabilization

Inflationary models whose vacuum energy arises from a D-term are believed not to suffer from the supergravity eta problem of F-term inflation. That is, D-term models have the desirable property that the inflaton mass can naturally remain much smaller than the Hubble scale. We observe that this advantage is lost in models based on string compactifications whose volume is stabilized by a nonperturbative superpotential: the F-term energy associated with volume stabilization causes the eta problem to reappear. Moreover, any shift symmetries introduced to protect the inflaton mass will typically be lifted by threshold corrections to the volume-stabilizing superpotential. Using threshold corrections computed by Berg, Haack, and Kors, we illustrate this point in the example of the D3-D7 inflationary model, and conclude that inflation is possible, but only for fine-tuned values of the stabilized moduli. More generally, we conclude that inflationary models in stable string compactifications, even D-term models with shift symmetries, will require a certain amount of fine-tuning to avoid this new contribution to the eta problem.Comment: 25 page

Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications

There are two known sources of nonperturbative superpotentials for K\"ahler moduli in type IIB orientifolds, or F-theory compactifications on Calabi-Yau fourfolds, with flux: Euclidean brane instantons and low-energy dynamics in D7 brane gauge theories. The first class of effects, Euclidean D3 branes which lift in M-theory to M5 branes wrapping divisors of arithmetic genus 1 in the fourfold, is relatively well understood. The second class has been less explored. In this paper, we consider the explicit example of F-theory on $K3 \times K3$ with flux. The fluxes lift the D7 brane matter fields, and stabilize stacks of D7 branes at loci of enhanced gauge symmetry. The resulting theories exhibit gaugino condensation, and generate a nonperturbative superpotential for K\"ahler moduli. We describe how the relevant geometries in general contain cycles of arithmetic genus $\chi \geq 1$ (and how $\chi > 1$ divisors can contribute to the superpotential, in the presence of flux). This second class of effects is likely to be important in finding even larger classes of models where the KKLT mechanism of moduli stabilization can be realized. We also address various claims about the situation for IIB models with a single K\"ahler modulus.Comment: 24 pages, harvmac, no figures, references adde