2.2 Spatial Quenching $\longrightarrow$ BFSS-type model

Instead of shifting $\mbox{\textit{\bf {}A}}_\mu( t,\vec x )$ to a single point, as we did in the final part of the previous section, we translate the matrix gauge field to a fixed time slice by means of the conserved three-momentum $\vec q$. As in the previously discussed case

$$\mbox{\textit{\bf {}A}}_i(t, \vec x) = \exp\left(-i\mbox{\textit{\bf {}q }}_i x^i \right) \mbox{\textit{... ...}}^\dagger(\vec x) \mbox{\textit{\bf {}A}}_i(t) \mbox{\textit{\bf {}U}}(\vec x)$$

$$\mbox{\textit{\bf {}A}}_t(t, \vec x) = \exp\left(-i\mbox{\textit{\bf {}q }}_i x^i \right) \mbox{\textit{... ...^\dagger(\vec x) \mbox{\textit{\bf {}A}}_t(t) \mbox{\textit{\bf {}U}}(\vec x) ,$$

the translation operation commutes with the covariant differentiation since

$$\mbox{\textit{\bf {}D}}_i \mbox{\textit{\bf {}A}}_n = i \mbo... ...ox{\textit{\bf {}A}}_n \right] \mbox{\textit{\bf {}U}}(\vec x)$$

implies

$$\mbox{\textit{\bf {}D}}_{[ m} \mbox{\textit{\bf {}A}}_{n ]} ... ...textit{\bf {}A}}_\nu(t) \right]\mbox{\textit{\bf {}U}}(\vec x)$$

and for the conjugate momentum we also get

$$\mbox{\textit{\bf {}U}}^\dagger(\vec x) \mbox{\textit{\bf {}... ..._n^{(\mathrm{q})}(t) \right] \mbox{\textit{\bf {}U}}(\vec x) .$$

Enclosing the system in a proper quantization volume $V_H$ while keeping the time integration free

$$\int_{V_H} d^3x\quad\longrightarrow \quad V_H$$

we get

$$S=V_H \int dt \left[ {g_0{}^2\over 2N} \mathrm{Tr} \left( \m... ...extit{\bf {}A}}^{(\mathrm{q})}_n \right] \right)^2 \right] \ ,$$

which is the action in the first order formulation; by substituting for $\mbox{\textit{\bf {}E}} ^{i} (t)$ its expression in terms of the vector potential we obtain, in the second order formulation, an action quite similar to the one for the bosonic sector of the BFSS model describing a system of $N$ $D0$-branes in the gauge $A_0=0$:

$$S^{\mathrm{BFSS}}=V_H {N\over g_0{}^2} \int dt \left[ {1\ove... ...tit{\bf {}A}}^{(\mathrm{q})}_n \right] \right)^2 \right] \ ;$$ (22)

the only difference is the range of the spatial indices: we are working in three rather than nine spatial dimensions. An action of the form (22) can also be obtained from monopole condensation and toroidal compactification [18].