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

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