The adjoint representation is the representation of a gauge group that acts on its own Lie algebra. For [[non-abelian|SU\((N_c)\)]] it has dimension \(N_c^2-1\) (eight for SU(3)), the same number as the generators \(T^a\). The gluons transform in the adjoint, and so does the scalar \(A_0\) of EQCD.

The generators in the adjoint are given directly by the structure constants,

\[ (T^a_{\text{adj}})_{bc} = -i f^{abc}, \]

so an adjoint field \(\Phi=\Phi^aT^a\) transforms by conjugation,

\[ \Phi \ \to\ g(x)\,\Phi\,g^\dagger(x),\qquad g(x)=e^{i\alpha^aT^a}. \]

Equivalently the covariant derivative acts through a commutator, \(D_\mu\Phi=\partial_\mu\Phi-ig[A_\mu,\Phi]\).

Because the gauge field \(A_\mu=A_\mu^aT^a\) transforms as \(A_\mu\to g(A_\mu+\tfrac{i}{g}\partial_\mu)g^\dagger\), its homogeneous part is exactly the adjoint conjugation. Hence the gluons carry color charge in the adjoint \(\mathbf 8\) — the reason they self-interact, the hallmark of a non-abelian theory. Quarks, by contrast, sit in the fundamental \(\mathbf 3\).

Under dimensional reduction the static time component \(A_0\) of the gluon field becomes a 3D scalar in the adjoint representation — the field whose Debye mass \(m_E\sim gT\) defines EQCD. Integrating this adjoint scalar out leaves the pure-gauge magnetic theory MQCD that fixes the spatial string tension.