139 research outputs found

### 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

### 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

### The pure BRST Einstein-Hilbert Lagrangian from the double-copy to cubic order

We construct the pure gravity Becchi-Rouet-Stora-Tyutin (BRST)
Einstein-Hilbert Lagrangian, to cubic order, using the BRST convolution product
of two Yang-Mills theories, in conjunction with the Bern-Carrasco-Johansson
(BCJ) double-copy.Comment: 20 page

### 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

### Super Yang-Mills, division algebras and triality

We give a unified division algebraic description of (D=3, N=1,2,4,8), (D=4,
N=1,2,4), (D=6, N=1,2) and (D=10, N=1) super Yang-Mills theories. A given
(D=n+2, N) theory is completely specified by selecting a pair (A_n, A_{nN}) of
division algebras, A_n, A_{nN} = R, C, H, O, where the subscripts denote the
dimension of the algebras. We present a master Lagrangian, defined over
A_{nN}-valued fields, which encapsulates all cases. Each possibility is
obtained from the unique (O, O) (D=10, N=1) theory by a combination of
Cayley-Dickson halving, which amounts to dimensional reduction, and removing
points, lines and quadrangles of the Fano plane, which amounts to consistent
truncation. The so-called triality algebras associated with the division
algebras allow for a novel formula for the overall (spacetime plus internal)
symmetries of the on-shell degrees of freedom of the theories. We use imaginary
A_{nN}-valued auxiliary fields to close the non-maximal supersymmetry algebra
off-shell. The failure to close for maximally supersymmetric theories is
attributed directly to the non-associativity of the octonions.Comment: 24 pages, 2 figures. Updated to match published version. References
adde

- …