Super Yang-Mills, division algebras and triality

TL;DR

This paper presents a unified algebraic framework using division algebras to describe various super Yang-Mills theories across different dimensions and supersymmetries, revealing deep symmetry structures.

Contribution

It introduces a master Lagrangian over division algebra-valued fields that unifies multiple super Yang-Mills theories and connects them through algebraic operations like dimensional reduction and truncation.

Findings

Unified division algebraic description of super Yang-Mills theories.

New formula for spacetime and internal symmetries using triality algebras.

Off-shell closure of non-maximal supersymmetry algebra with imaginary division algebra fields.

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.