376 research outputs found
Metric projective geometry, BGG detour complexes and partially massless gauge theories
A projective geometry is an equivalence class of torsion free connections
sharing the same unparametrised geodesics; this is a basic structure for
understanding physical systems. Metric projective geometry is concerned with
the interaction of projective and pseudo-Riemannian geometry. We show that the
BGG machinery of projective geometry combines with structures known as
Yang-Mills detour complexes to produce a general tool for generating invariant
pseudo-Riemannian gauge theories. This produces (detour) complexes of
differential operators corresponding to gauge invariances and dynamics. We
show, as an application, that curved versions of these sequences give geometric
characterizations of the obstructions to propagation of higher spins in
Einstein spaces. Further, we show that projective BGG detour complexes generate
both gauge invariances and gauge invariant constraint systems for partially
massless models: the input for this machinery is a projectively invariant gauge
operator corresponding to the first operator of a certain BGG sequence. We also
connect this technology to the log-radial reduction method and extend the
latter to Einstein backgrounds.Comment: 30 pages, LaTe
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
For curved projective manifolds we introduce a notion of a normal tractor
frame field, based around any point. This leads to canonical systems of
(redundant) coordinates that generalise the usual homogeneous coordinates on
projective space. These give preferred local maps to the model projective space
that encode geometric contact with the model to a level that is optimal, in a
suitable sense. In terms of the trivialisations arising from the special
frames, normal solutions of classes of natural linear PDE (so-called first BGG
equations) are shown to be necessarily polynomial in the generalised
homogeneous coordinates; the polynomial system is the pull back of a polynomial
system that solves the corresponding problem on the model. Thus questions
concerning the zero locus of solutions, as well as related finer geometric and
smooth data, are reduced to a study of the corresponding polynomial systems and
algebraic sets. We show that a normal solution determines a canonical manifold
stratification that reflects an orbit decomposition of the model. Applications
include the construction of structures that are analogues of Poincare-Einstein
manifolds.Comment: 22 page
Einstein metrics in projective geometry
It is well known that pseudo-Riemannian metrics in the projective class of a
given torsion free affine connection can be obtained from (and are equivalent
to) the solutions of a certain overdetermined projectively invariant
differential equation. This equation is a special case of a so-called first BGG
equation. The general theory of such equations singles out a subclass of
so-called normal solutions. We prove that non-degerate normal solutions are
equivalent to pseudo-Riemannian Einstein metrics in the projective class and
observe that this connects to natural projective extensions of the Einstein
condition.Comment: 10 pages. Adapted to published version. In addition corrected a minor
sign erro
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Addressing the Food Loss and Waste Challenge – a WRAP perspective
Unsustainable production and consumption of food constitutes one of the biggest environmental threats to our planet. Eliminating food loss and waste to the largest extent possible – at all stages from producer to final consumer – stands out as an urgent and indispensable step towards more sustainable food systems. In fact, recent research shows that tackling food waste is the third most effective intervention to reduce greenhouse gas emissions, the most important priority of our time (Hawken 2017). The United Nations Sustainable Development Goal (SDG) 12.3 sets out a specific target on food waste to halve per capita global food waste at the retail and consumer levels and reduce food losses along production and supply chains, including post-harvest losses, by 2030. In order to measure global progress towards SDG 12.3, two indices have been proposed: the Food Waste Index (Global Innovation Exchange 2018) and the Food Loss Index (Fabi and English 2018). Successfully achieving SDG 12.3 requires new thinking, new partnerships and new actions to reduce resource use, and increase the efficiency of the production, preservation, processing and distribution of food at the producer, intermediary, processor and wholesale level. It needs wider education, increased awareness, and behavioural change among citizens, retailers, and policy makers across the globe. The goal is to produce more food to feed the world’s expanding population, while reducing land use, fertilizer applications and critically dramatically reducing greenhouse gas emissions (Flanagan et al. 2019). To help deliver this critical target, Champions 12.3 has been formed (Champions 12.3 2016). It is a unique coalition of executives from governments, businesses, international organizations, research institutions, and civil society dedicated to inspiring ambition, mobilizing action, and accelerating progress toward achieving SDG Target 12.3. It has produced a trajectory for delivering 12.3, what needs to happen and by when that provides the critical “roadmap for change” (Champions 12.3 2017a). In this paper we provide the perspective of WRAP (the Waste and Resources Action Programme) on the economic, social and environmental case for action, what research shows works in driving change and how these activities might be scaled to deliver SDG 12.3. WRAP is a not for profit organization, based in the UK and working in more than 20 countries worldwide, that aims to help people and planet thrive. WRAP is a leader in tackling food loss and waste effectively and supporting international food loss and waste prevention projects – including Champions 12.3. Since 2007, WRAP has been a partner in many global food loss and waste projects and initiatives and has co-authored key reports. This includes EU projects such as FUSIONS (2016) and REFRESH (2020a), as well as the development of the Food Loss and Waste Accounting and Reporting Standard (World Resources Institute 2016). In the UK, WRAP, food businesses and other partners have delivered large-scale interventions to reduce food waste across supply chains, and households for more than ten years (since 2007), supported by UK Governments and by businesses and enabled by a series of collaborative public-private partnerships. WRAP’s work in the UK with its partners has helped reduce food by 27% or 1.7 Mt/y saving food worth £5 billion/ year. Cumulatively the total food waste reduction has been 18.5 Mt worth US$50 billion (WRAP 2020a). This paper highlights the importance of tackling food loss and waste, using specific recent examples from the UK and Mexico. Second, we discuss the business case for addressing food loss and waste. Thirdly we highlighting two approaches that research shows can be particularly effective at driving change at scale, and we conclude by proposing a three-point plan for tackling food waste to deliver SDG 12.3 over the next 10 years
A sub-product construction of Poincare-Einstein metrics
Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably
related, we give an explicit construction of a Poincar\'e-Einstein
(pseudo-)metric with conformal infinity the conformal class of the product of
the initial metrics. We show that these metrics are equivalent to ambient
metrics for the given conformal structure. The ambient metrics have holonomy
that agrees with the conformal holonomy. In the generic case the ambient metric
arises directly as a product of the metric cones over the original Einstein
spaces. In general the conformal infinity of the Poincare metrics we construct
is not Einstein, and so this describes a class of non-conformally Einstein
metrics for which the (Fefferman-Graham) obstruction tensor vanishes.Comment: 23 pages Minor correction to section 5. References update
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