A Kind of Magic

We introduce the extended Freudenthal-Rosenfeld-Tits magic square based on six algebras: the reals $\mathbb{R}$, complexes $\mathbb{C}$, ternions $\mathbb{T}$, quaternions $\mathbb{H}$, sextonions $\mathbb{S}$ and octonions $\mathbb{O}$. The ternionic and sextonionic rows/columns of the magic square yield non-reductive Lie algebras, including $\mathfrak{e}_{7\scriptscriptstyle{\frac{1}{2}}}$. It is demonstrated that the algebras of the extended magic square appear quite naturally as the symmetries of supergravity Lagrangians. The sextonionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=3$ maximal $\mathcal{N}=16$, magic $\mathcal{N}=4$ and magic non-supersymmetric theories, obtained by dimensionally reducing the $D=4$ parent theories on a circle, with the graviphoton left undualised. In particular, the extremal intermediate non-reductive Lie algebra $\tilde{\mathfrak{e}}_{7(7)\scriptscriptstyle{\frac{1}{2}}}$ (which is not a subalgebra of $\mathfrak{e}_{8(8)}$) is the non-compact global symmetry algebra of $D=3$, $\mathcal{N}=16$ supergravity as obtained by dimensionally reducing $D=4$, $\mathcal{N}=8$ supergravity with $\mathfrak{e}_{7(7)}$ symmetry on a circle. The ternionic row (for appropriate choices of real forms) gives the non-compact global symmetries of the Lagrangian for the $D=4$ maximal $\mathcal{N}=8$, magic $\mathcal{N}=2$ and magic non-supersymmetric theories obtained by dimensionally reducing the parent $D=5$ theories on a circle. In particular, the Kantor-Koecher-Tits intermediate non-reductive Lie algebra $\mathfrak{e}_{6(6)\scriptscriptstyle{\frac{1}{4}}}$ is the non-compact global symmetry algebra of $D=4$, $\mathcal{N}=8$ supergravity as obtained by dimensionally reducing $D=5$, $\mathcal{N}=8$ supergravity with $\mathfrak{e}_{6(6)}$ symmetry on a circle.Comment: 38 pages. Reference added and minor corrections mad

Aspects of M-theory and quantum information

As the frontiers of physics steadily progress into the 21st century we should bear in mind that the conceptual edifice of 20th-century physics has at its foundations two mutually incompatible theories; quantum mechanics and Einstein’s general theory of relativity. While general relativity refuses to succumb to quantum rule, black holes are raising quandaries that strike at the very heart of quantum theory. M-theory is a compelling candidate theory of quantum gravity. Living in eleven dimensions it encompasses and connects the five possible 10-dimensional superstring theories. However, Mtheory is fundamentally non-perturbative and consequently remains largely mysterious, offering up only disparate corners of its full structure. The physics of black holes has occupied centre stage in uncovering its non-perturbative structure. The dawn of the 21st-century has also played witness to the birth of the information age and with it the world of quantum information science. At its heart lies the phenomenon of quantum entanglement. Entanglement has applications in the emerging technologies of quantum computing and quantum cryptography, and has been used to realize quantum teleportation experimentally. The longest standing open problem in quantum information is the proper characterisation of multipartite entanglement. It is of utmost importance from both a foundational and a technological perspective. In 2006 the entropy formula for a particular 8-charge black hole appearing in M-theory was found to be given by the ’hyperdeterminant’, a quantity introduced by the mathematician Cayley in 1845. Remarkably, the hyperdeterminant also measures the degree of tripartite entanglement shared by three qubits, the basic units of quantum information. It turned out that the different possible types of three-qubit entanglement corresponded directly to the different possible subclasses of this particular black hole. This initial observation provided a link relating various black holes and quantum information systems. Since then, we have been examining this two-way dictionary between black holes and qubits and have used our knowledge of M-theory to discover new things about multipartite entanglement and quantum information theory and, vice-versa, to garner new insights into black holes and M-theory. There is now a growing dictionary, which translates a variety of phenomena in one language to those in the other. Developing these fascinating relationships, exploiting them to better understand both M-theory and quantum entanglement is the goal of this thesis. In particular, we adopt the elegant mathematics of octonions, Jordan algebras and the Freudenthal triple system as our guiding framework. In the course of this investigation we will see how these fascinating algebraic structures can be used to quantify entanglement and define new black hole dualities

Brane orbits

We complete the classification of half-supersymmetric branes in toroidally compactified IIA/IIB string theory in terms of representations of the T-duality group. As a by-product we derive a last wrapping rule for the space-filling branes. We find examples of T-duality representations of branes in lower dimensions, suggested by supergravity, of which none of the component branes follow from the reduction of any brane in ten-dimensional IIA/IIB string theory. We discuss the constraints on the charges of half-supersymmetric branes, determining the corresponding T-duality and U-duality orbits.Comment: 34 pages, 3 figure

Explicit Orbit Classification of Reducible Jordan Algebras and Freudenthal Triple Systems

We determine explicit orbit representatives of reducible Jordan algebras and of their corresponding Freudenthal triple systems. This work has direct application to the classification of extremal black hole solutions of N = 2, 4 locally supersymmetric theories of gravity coupled to an arbitrary number of Abelian vector multiplets in D = 4, 5 space-time dimensions.Comment: 18 pages. Updated to match published versio

Global symmetries of Yang-Mills squared in various dimensions

Tensoring two on-shell super Yang-Mills multiplets in dimensions $D\leq 10$ yields an on-shell supergravity multiplet, possibly with additional matter multiplets. Associating a (direct sum of) division algebra(s) $\mathbb{D}$ with each dimension $3\leq D\leq 10$ we obtain formulae for the algebras $\mathfrak{g}$ and $\mathfrak{h}$ of the U-duality group $G$ and its maximal compact subgroup $H$, respectively, in terms of the internal global symmetry algebras of each super Yang-Mills theory. We extend our analysis to include supergravities coupled to an arbitrary number of matter multiplets by allowing for non-supersymmetric multiplets in the tensor product.Comment: 25 pages, 2 figures, references added, minor typos corrected, further comments on sec. 2.4 included, updated to match version to appear in JHE