2.1 Spacetime Quenching $\longrightarrow$ IKKT-type model

In order to provide the reader a self-contained derivation of our model, let us briefly review how the quenching approximation works in a simple toy model [16]. Consider the following two-dimensional model of matrix non-relativistic quantum mechanics

$$S\equiv \int d^2x\, \mathrm{Tr}\,\left[\, {1\over 2}\, \left... ...right)^2 -V\left(\, \mbox{\textit{\bf {}M}}\,\right)\,\right],$$ (15)

where $\mbox{\textit{\bf {}M}}(\,x^0\ , x^1\,)$ is an Hermitian, $2\times2$ matrix and $V\left(\,\mbox{\textit{\bf {}M}}\,\right)$ is an appropriate potential term and we suppose that the system described by $\mbox{\textit{\bf {}M}}$ is enclosed in a (one-dimensional) ``spatial box'' of size $a$. Quenching is an approximation borrowed from the theory of spin glasses where only ``slow'' momentum modes are kept to compute the spectrum of the model. Slow modes are described by the eigenvalues of the linear momentum matrix $\mbox{\textit{\bf {}P}}$, while the off-diagonal, ``fast'' modes can be thought as being integrated out. Thus, $\mbox{\textit{\bf {}M}}(\, x\,)$ can be written as

$$\mbox{\textit{\bf {}M}}(\, x^0\ , x^1\,)= \exp\left(\, i\,\m... ..., x^0\,)\exp\left(\, -i\,\mbox{\textit{\bf {}P}}\,x^1\,\right)$$ (16)

and the spatial derivative $\partial_1 \mbox{\textit{\bf {}M}}(\,x^0\ , x^1\,)$ becomes the the commutator of $\mbox{\textit{\bf {}P}}$ and $\mbox{\textit{\bf {}M}}$

$$\partial_1 \mbox{\textit{\bf {}M}}=i\,\left[\,\mbox{\textit{\bf {}P}}\ , \mbox{\textit{\bf {}M}}\,\right]$$ (17)

so that the action becomes

$$S\equiv a\, \int dx^0\, \mathrm{Tr}\,\left[\, {1\over 2}\, \... ...right]^2 -V\left(\,\mbox{\textit{\bf {}M}}\,\right)\,\right],$$ (18)

in which the the $x^1$ dependence of the original matrix field has been removed.
Quenching can be applied to a Yang-Mills gauge theory by taking into account that in the large-$N$ limit $SU( N )\to U( N )$ and the group of spacetime translations fits into the diagonal part of $U( \infty )$. By neglecting off-diagonal components, spacetime dependent dynamical variables can be shifted to the origin by means of a translation operator $\mbox{\textit{\bf {}U}}(x)$: since the translation group is Abelian one can choose the matrix $\mbox{\textit{\bf {}U}}(x)$ to be a plane wave diagonal matrix [17]

$$\mbox{\textit{\bf {}U}}_{ab}(x)=\delta_{ab}\exp\left( i q^a{}_\mu x^\mu \right) \ ,$$ (19)

where $q^a {}_\mu$ are the eigenvalues of the four-momentum $\mbox{\textit{\bf {}q }}_\mu$. Then

$$\mbox{\textit{\bf {}A}}_\mu(x)=\exp\left(-i\mbox{\textit{\bf... ...) \mbox{\textit{\bf {}A}}_\mu^{(0)} \mbox{\textit{\bf {}U}}(x)$$

and in view of the equality

$$\mbox{\textit{\bf {}D}}_ \mu \mbox{\textit{\bf {}A}}_\nu = i... ...mbox{\textit{\bf {}A}}_\nu \right] \mbox{\textit{\bf {}U}}(x),$$

which when antisymmetrized yields

$$\mbox{\textit{\bf {}D}}_{[ \mu} \mbox{\textit{\bf {}A}}_{\nu... ... {}A}}_\nu^{(\mathrm{q})} \right] \mbox{\textit{\bf {}U}}(x) ,$$

we can see that the translation is compatible with the covariant differentiation, so that

$$\mbox{\textit{\bf {}F}}_{\mu\nu}(x)= \exp\left(-i\mbox{\text... ...{\textit{\bf {}F}}_{\mu\nu}^{(0)} \mbox{\textit{\bf {}U}}(x) .$$

Once the original gauge field theory is turned into a constant matrix model, we still need to dispose of the spacetime volume integration. The gluon field is spatially confined inside a volume $V_H$ comparable with the typical size of an hadron. Thus, for any finite time interval $T$ we can replace the four-volume integral by

$$\int_0^T dt\int_{V_H} d^3x\quad\longrightarrow \quad T\, V_H$$

and the quenched action becomes

$$S^{(\mathrm{q})}_{\mathrm{YM-red.}}= T\, V_H {N\over g_0{}^2... ...\mbox{\textit{\bf {}A}}_\nu^{(\mathrm{q})} \right] \right) \ ,$$ (20)

which is the first order formulation of the IKKT-type action in four spacetime dimensions [12]. The usual second order formulation is readily obtained by solving for $\mbox{\textit{\bf {}F}}_{\mu\nu}^{(0)}$ in terms of $\mbox{\textit{\bf {}A}}_\mu^{(\mathrm{q})}$

$$\mbox{\textit{\bf {}F}}_{\mu\nu}^{(0)}= i \left[ \mbox{\text... ...thrm{q})} , \mbox{\textit{\bf {}A}}_\nu^{(\mathrm{q})} \right]$$

and substituting back this result into (20)

$$S^{(\mathrm{q})}_{\mathrm{YM-red.}}\rightarrow S^{IKKT}_{(4)... ...mbox{\textit{\bf {}A}}_\nu^{(\mathrm{q})} \right] \right)^2\ .$$ (21)

The string-like excitations of this model and the relation between large-$N$ gauge symmetry and area-preserving diffeomorphism have been investigated in several papers [6]. More recently, we found that not only strings are present in the large-$N$ spectrum of (21) but also spacetime filling, bag-like objects [8], for which a non-trivial boundary dynamics was found through the addition of topological terms to the original Yang-Mills action. Here, we would like to explore a different route leading in a more straightforward way to a dynamical brane action. From this purpose we need to introduce a different quenching approximation, which we discuss in the next section.