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Super Yang-Mills, division algebras and triality

A. Anastasiou, L. Borsten, M. J. Duff, L. J. Hughes, S. Nagy

TL;DR

This work constructs a unified division-algebra framework for $D=n+2$ SYM theories with $\mathcal{N}$ supersymmetries by pairing division algebras $(\mathds{A}_n,\mathds{A}_{n\mathcal{N}})$, and presents a master Lagrangian defined over $\mathds{A}_{n\mathcal{N}}$-valued fields from which all cases arise through Cayley-Dickson halving and Fano-plane truncations. It identifies triality algebras as the total on-shell symmetry and uses imaginary $\mathds{A}_{n\mathcal{N}}$-valued auxiliary fields to close the SUSY algebra off-shell, noting that maximal theories fail to close due to octonionic non-associativity. The paper details how $D=10,\mathcal{N}=1$ is octonionic and how dimensional reduction to $D=6,4,3$ proceeds via algebraic halving and truncation, yielding the $\mathcal{N}=1,2,4,8$ sequences in lower dimensions. It also proposes a broader program to assemble a cascading 'magic pyramid' of theories and connects octonionic structure to potential roles in M-theory and string dualities.

Abstract

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.

Super Yang-Mills, division algebras and triality

TL;DR

This work constructs a unified division-algebra framework for SYM theories with supersymmetries by pairing division algebras , and presents a master Lagrangian defined over -valued fields from which all cases arise through Cayley-Dickson halving and Fano-plane truncations. It identifies triality algebras as the total on-shell symmetry and uses imaginary -valued auxiliary fields to close the SUSY algebra off-shell, noting that maximal theories fail to close due to octonionic non-associativity. The paper details how is octonionic and how dimensional reduction to proceeds via algebraic halving and truncation, yielding the sequences in lower dimensions. It also proposes a broader program to assemble a cascading 'magic pyramid' of theories and connects octonionic structure to potential roles in M-theory and string dualities.

Abstract

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.

Paper Structure

This paper contains 20 sections, 142 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Normed Division Algebras
  3. Spacetime Fields in $D=n+2$
  4. The Isomorphism $\mathfrak{so}(1,1+n)\cong\mathfrak{sl}(2,\mathds{A})$
  5. Spinors and Pauli Matrices
  6. Vectors and 2-forms
  7. Little Groups
  8. Symmetries of Super Yang-Mills Theories in $D=n+2$
  9. $D=10,\space\mathcal{N}=1$
  10. From $D=10,\space\mathcal{N}=1$ to $D=6,\space\mathcal{N}=1,2$
  11. From $D=10,\space\mathcal{N}=1$ to $D=4,\space\mathcal{N}=1,2,4$
  12. From $D=10,\space\mathcal{N}=1$ to $D=3,\space\mathcal{N}=1,2,4,8$
  13. Triality Algebras
  14. Lagrangians of Super Yang-Mills Theories in $D=n+2$
  15. $\mathcal{N}=1$ Theories
  16. ...and 5 more sections

Figures (2)

  • Figure 1: The Fano plane Baez:2001dm. Each oriented line corresponds to a quaternionic subalgebra. For example, $e_2e_3=e_5$ and cyclic permutations; odd permutations go against the direction of the arrows on the Fano plane and we pick up a minus sign, e.g. $e_3e_2=-e_5$.
  • Figure 2: The Dynkin diagram of SO(8)