2013/09/30 by Α. Αναστασίου, A. Anastasiou, L. Borsten +4 · 37 citations
Mathematics · Physics and Astronomy · #Algebra over a field #Algebraic and Geometric Analysis #Algebraic number #Arithmetic #Black Holes and Theoretical Physics #Dimension (graph theory) #Division (mathematics) #Division algebra #Geometry #Homogeneous space #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Reduction (mathematics) #Spacetime #Subalgebra #Triality #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/jhep08(2014)080
published in Journal of High Energy Physics 2014(8) (Springer Nature) · 24 pages, 2 figures. Updated to match published version. References added
openalex publication_date 2014/08/01 · arxiv created 2014/09/06 · arxiv updated 2014/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ( \mathbbAn , \mathbbAnN ) of division algebras, \mathbbAn ⊆ \mathbbAnN = ℝ,ℂ,ℍ,\mathbbO , where the subscripts denote the dimension of the algebras. We present a master Lagrangian, defined over \mathbbAnN -valued fields, which encapsulates all cases. Each possibility is obtained from the unique ( \mathbbO,\mathbbO ) (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 \mathbbAnN -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.