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Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.

For canonical positive kinetic terms in global supersymmetry, $V={\sum }_k|W_k{|}^2$ has mixed Hessian matrix $V_{i\bar{j}}={\sum }_k{W}_{ki}\overline{W_{kj}}$ and holomorphic block $V_{ij}={\sum }_k\overline{W_k}{W}_{kij}$. The $2n$ real scalar squared-mass sum is $2\mathrm{tr}(m^{†}m)$ for the chiral-superfield fermion mass matrix $m=W_{ij}$. Each Weyl spinor contributes two states, giving the same fermion weighted sum. Thus their boson-minus-fermion supertrace vanishes even with F-term breaking. The holomorphic Hessian splits real scalar masses without changing their sum. Noncanonical Kähler metrics, supergravity, and radiative corrections require different formulas.

A vectorlike gauge spectrum pairs complex gauge group representations $R$ and $\bar{R}$ among independent left-handed Weyl spinors. Such a pair permits a gauge-invariant fermion mass and occurs in a full charged hypermultiplet of four-dimensional $\mathcal{N}=2$ extended supersymmetry.

For mostly-minus metric signature, the Weyl sigma matrices are ${\sigma }^{\mu }=(I,{\sigma }_1,{\sigma }_2,{\sigma }_3)$ and ${\bar{\sigma }}^{\mu }=(I,-{\sigma }_1,-{\sigma }_2,-{\sigma }_3)$, where ${\sigma }_i$ are the Pauli matrices. The Pauli matrix multiplication law provesA real Lorentz four-vector is equivalently a Hermitian matrix $X=x^{0}I+x^i{\sigma }_i$ with $\det X=(x^0{)}^2-|\mathbf{x}{|}^2$. The transformation $X\mapsto MX{M}^{†}$ for $M\in \mathrm{SL}(2,\mathbb{C})$ preserves this determinant and gives the Lorentz spinor double cover. The indices ${\sigma }_{\alpha \dot{\beta }}^{\mu }$ therefore convert a vector index into one undotted and one dotted Weyl spinor index. They are the invariant tensors in the mixed supercharge anticommutator $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$.