Primitive axial algebras of Jordan type

An axial algebra over the field $\mathbb F$ is a commutative algebra generated by idempotents whose adjoint action has multiplicity-free minimal polynomial. For semisimple associative algebras this leads to sums of copies of $\mathbb F$. Here we consider the first nonassociative case, where adjoint minimal polynomials divide $(x-1)x(x-\eta)$ for fixed $0\neq\eta\neq 1$. Jordan algebras arise when $\eta=\frac{1}{2}$, but our motivating examples are certain Griess algebras of vertex operator algebras and the related Majorana algebras. We study a class of algebras, including these, for which axial automorphisms like those defined by Miyamoto exist, and there classify the $2$-generated examples. For $\eta \neq \frac{1}{2}$ this implies that the Miyamoto involutions are $3$-transpositions, leading to a classification.Comment: 41 pages; comments welcom

Bullion production in imperial China and its significance for sulphide ore smelting world-wide

Gold and silver production was of major importance for almost all ancient societies but has been rarely studied archaeologically. Here we present a reconstruction of a previously undocumented technology used to recover gold, silver and lead at the site of Baojia in Jiangxi province, China dated between the 7th and 13th centuries AD. Smelting a mixture of sulphidic and gossan ores in a relatively low temperature furnace under mildly reducing conditions, the process involved the use of metallic iron to reduce lead sulphide to lead metal, which acted as the collector of the precious metals. An experimental reconstruction provides essential information, demonstrating both the significant influence of sulphur on the silicate slag system, and that iron reduction smelting of lead can be carried out at a relatively low temperature. These new findings are relevant for further studies of lead and precious metal smelting slags world-wide. The technological choices of ancient smelters at this site are then discussed in their specific geographical and social-economic settings

Holographic representation of local bulk operators

The Lorentzian AdS/CFT correspondence implies a map between local operators in supergravity and non-local operators in the CFT. By explicit computation we construct CFT operators which are dual to local bulk fields in the semiclassical limit. The computation is done for general dimension in global, Poincare and Rindler coordinates. We find that the CFT operators can be taken to have compact support in a region of the complexified boundary whose size is set by the bulk radial position. We show that at finite N the number of independent commuting operators localized within a bulk volume saturates the holographic bound.Comment: 36 pages, LaTeX, 4 eps figure

Protecting the conformal symmetry via bulk renormalization on Anti deSitter space

The problem of perturbative breakdown of conformal symmetry can be avoided, if a conformally covariant quantum field phi on d-dimensional Minkowski spacetime is viewed as the boundary limit of a quantum field Phi on d+1-dimensional anti-deSitter spacetime (AdS). We study the boundary limit in renormalized perturbation theory with polynomial interactions in AdS, and point out the differences as compared to renormalization directly on the boundary. In particular, provided the limit exists, there is no conformal anomaly. We compute explicitly the "fish diagram" on AdS_4 by differential renormalization, and calculate the anomalous dimension of the composite boundary field phi^2 with bulk interaction Phi^4.Comment: 40 page

How to remove the boundary in CFT - an operator algebraic procedure

The relation between two-dimensional conformal quantum field theories with and without a timelike boundary is explored.Comment: 18 pages, 2 figures. v2: more precise title, reference correcte

General graviton exchange graph for four point functions in the AdS/CFT correspondence

In this note we explicitly compute the graviton exchange graph for scalar fields with arbitrary conformal dimension \Delta in arbitrary spacetime dimension d. This results in an analytical function in \Delta as well as in d.Comment: 14 pages, 2 figure

Tainted ores and the rise of tin bronzes in Eurasia, c. 6500 years ago

The earliest tin bronze artefacts in Eurasia are generally believed to have appeared in the Near East in the early third millennium BC. Here we present tin bronze artefacts that occur far from the Near East, and in a significantly earlier period. Excavations at Pločnik, a Vinča culture site in Serbia, recovered a piece of tin bronze foil from an occupation layer dated to the mid fifth millennium BC. The discovery prompted a reassessment of 14 insufficiently contextualised early tin bronze artefacts from the Balkans. They too were found to derive from the smelting of copper-tin ores. These tin bronzes extend the record of bronze making by c. 1500 years, and challenge the conventional narrative of Eurasian metallurgical development

The classification of non-local chiral CFT with c<1

All non-local but relatively local irreducible extensions of Virasoro chiral CFTs with c<1 are classified. The classification, which is a prerequisite for the classification of local c<1 boundary CFTs on a two-dimensional half-space, turns out to be 1 to 1 with certain pairs of A-D-E graphs with distinguished vertices.Comment: 13 pages. v3: additional material (concerning the Hilbert spaces) adde