Residual-Dimension Obstructions for Null-Projected Supersymmetric Yang–Mills Theories

We classify null-projected supersymmetric Yang--Mills theories with a Majorana--Weyl gaugino in signature (s,t), defined by five conditions on the gaugino constraint, the transformation rules, the residual gauge field, the translation bracket and the absorption of the cubic gaugino terms. Such a theory built on \(k\) null vectors exists if and only if \(s+t-2k=10\) and \(s-t\in\{8,0,-8\}\), or \(s+t-2k=2\) and \(s=t\). The key step is a converse of the cubic gamma-matrix identity: symmetric gamma matrices that satisfy it act on spinors of dimension 2(d-2). Hence the signatures (9 + 8n + q) + (1 + q) with n ≥ 1 and 0 ≤ q ≤ 3 proposed by Rios, Marrani and Chester, among them 27+3, carry no such theory. Neither another contraction of the source, nor any number of gauginos, mixed or not, nor a gauge bracket that is not a Lie bracket changes this. In the single-time signatures (9+8n)+1 no unitary massless multiplet without tensor charges consists of scalars, spinors and vectors.