MQCD is the three-dimensional effective field theory obtained from EQCD by integrating out the adjoint scalar \(A_0\) (whose Debye mass \(m_E\sim gT\) is heavy on the magnetic scale). It is the final, ultrasoft effective theory in the dimensional reduction chain and it governs the spatial string tension.

MQCD is nothing but pure 3D SU(3) Yang–Mills:

\[ \mathcal{L}_{MQCD}=\tfrac12\,\mathrm{Tr}\,F_{ij}^2, \qquad F_{ij}=\partial_i A_j-\partial_j A_i - i g_M[A_i,A_j]. \]

The single parameter is the magnetic coupling, matched to EQCD as

\[ g_M^2 = g_E^2\big[1+\mathcal{O}(g)\big] = g^2(T)\,T\big[1+\mathcal{O}(g)\big]. \]

In three dimensions \(g_M^2\) has the dimension of mass, so it sets the only scale of the theory.

Pure 3D Yang–Mills is confining: it has a mass gap and a linearly rising potential. A spatial Wilson loop therefore obeys an area law with a string tension fixed by dimensional analysis,

\[ \sqrt{\sigma_M}=c\,g_M^2 . \]

The pure number \(c\) is measured on the 3D lattice:

\[ c=\frac{\sqrt{\sigma_M}}{g_M^2}=0.554(4)\quad(\text{SU(3)}). \]

Because a purely spatial Wilson loop only involves the magnetic field \(A_i\), the 4D spatial string tension equals the MQCD string tension:

\[ \sqrt{\sigma_s(T)}=\sqrt{\sigma_M}=c\,g_M^2\simeq 0.554\,g^2(T)\,T . \]

This is the quantitative prediction tested by Cheng et al. (arXiv:0806.3264); see Why spatial string tension and dimensional reduction. MQCD also embodies the Linde infrared problem: the ultrasoft magnetic sector is inherently non-perturbative and is accessible only through such a 3D lattice theory.