2 Chern-Simons Hadronic Bag

The WWM relation establishes a one-to-one correspondence between a linear operator, ${\mbox{\boldmath {$D$}}}_\mu$ in our case, acting over a Hilbert space ${\cal H}$ of square integrable functions on $\mathbb{R}^D$, and a smooth function ${\cal A}_\mu(\, x\ , y\,)$ which is the anti-Fourier transform of ${\cal A}_\mu(\, k\ , z\,)$ in (5):

$${\cal A}_\mu(\, q\ , p\,)={1\over N}Tr_{{\cal H}} \left[\, {\mbox... ...oldmath {$p$}}}_i\, q^i -i\, {\mbox{\boldmath {$q$}}}_j\, p^j\,\right)\right]\$$ (7)

$${\cal A}_\mu(\, x\ , y\,)=\int d^Dq\, d^Dp\, {\cal A}_\mu(\, q\ , p\,)\, \exp\left(\, i\, q_i\, x^i +i\, p_j\, y^j\,\right),$$ (8)

where $Tr_{{\cal H}}$ means the sum over diagonal elements with respect to an orthonormal basis in ${\cal H}$. Under the WWM correspondence the matrix product turns into the Moyal product, or $\ast$-product, as follows:

$${\mbox{\boldmath {$U$}}}\, {\mbox{\boldmath {$V$}}}\rightarrow {\cal U}(\, x\ , y\,) \ast {\cal V}(\, x\ ,y\,) \equiv \exp\left[\, i\, {\hbar\over 2}\left(\, {\partial^2\over \partial... ..., {\cal V}(\, x^\prime \ ,y^\prime\,) \vert_{\stackrel{x^\prime=x}{y^\prime=y}}$$

$$\equiv \exp\left[\, i\, {\hbar\over 2}\,\omega^{ab} \, {\partial^2\over ... ... \xi^b } \, \right]{\cal U}(\,\sigma\,)\, {\cal V}(\,\xi\,) \vert_{\sigma=\xi},$$ (9)

where $\omega^{ab}$ is defined as

$$\omega^{ab}\equiv \left( \begin{array}{cc} 0 & -I_{D\times D}\\ I_{D\times D} &0 \end{array}\right)$$ (10)

and $a\,,\;b= i^1\ ,\dots, i^D\,,\; j^1\ ,\dots, j^D$. With the introduction of the non-commutative $\ast$-product it is possible to express the commutator between two matrices ${\mbox{\boldmath {$U$}}}$, ${\mbox{\boldmath {$V$}}}$ as the Moyal Bracket between their corresponding Weyl symbols ${\cal U}(x,y)$, ${\cal V}(x,y)$:

$${1\over \hbar}\, \left[\, {\mbox{\boldmath {$A$}}}\ , {\mbox{\boldmath {$A$}}}\,\right]\rightarrow \left\{\, {\cal U}\ , {\cal V}\,\right\}_{MB} \equiv {1\over \hbar}\left(\, {\cal U} \ast {\cal V} - {\cal V} \ast {\cal U}\,\right)$$

$$\equiv \, \omega^{ij}\, \partial_i\, {\cal U}\circ \partial_j\, {\cal V},$$ (11)

where we introduced the $\circ$-product which corresponds to the ``even'' part of the of the $\ast$-product [11]. In the limit of vanishing deformation parameter the Moyal bracket reproduces the Poisson bracket

$$\lim_{\hbar\to 0}\left\{\, {\cal U}\ , {\cal V}\,\right\}_{MB}= \left\{\, {\cal U}\ , {\cal V}\,\right\}_{PB}.$$ (12)

The last step of the mapping between matrix theory into a field model is carried out through the identification of the ``deformation parameter'' $\hbar$ with the inverse of $N$:

$$\lq\lq \hbar''\equiv {2\pi\over N}\; : \quad \lim_{N\to\infty} f(\, N \,)=\lim_{\hbar\to 0} f(\, \hbar \,)\ .$$ (13)

Consistently, the large-$N$ limit of the $\mathit{SU}(\, N\, )$ matrix theory, where the ${\mbox{\boldmath {$A$}}}_\mu$ quantum fluctuations freeze, corresponds to the quantum mechanical classical limit, $\hbar\to 0$, of the WWM corresponding field theory ⁴(from now on, we shall refer to the ``classical limit'' without distinguishing between the large-$N$ or small $\hbar$).
Finally, we can re-write the trace operation as an integration over a $2D$-dimensional manifold:

$${(2\pi)^4\over N^3}\, Tr_{{\cal H}}\, \rightarrow \int d^Dx\, d^Dy\, \equiv \int d^{2D}\sigma\ ,$$ (14)

and [11]

$${\cal F}_{\mu\nu}\equiv \left\{\, {\cal A}_\mu\ , {\cal A}_\... ...\partial_a\, {\cal A}_\mu\, \circ \, \partial_b\, {\cal A}_\nu$$ (15)

After this technical detour, let us come back to the two terms in the reduced action (2) which are mapped by the WWM correspondence into

$$W^{\mathrm{qYM}}_{\mathrm{red}} = -{1\over 4}\left({2\pi\over a}\right)^4 { N^4\over (2\pi)^3}\left... ...{\mathrm{YM}}} \int d^{2D}\sigma \,{\cal F}_{\mu\nu}\, \ast\, {\cal F}^{\mu\nu}$$

$$= -{1\over 16}\left({2\pi\over a}\right)^4 {\left({N\over 2\pi}\rig... ...[\, m}\, {\cal A}^{\mu}\circ \partial_{n\, ]}\, {\cal A}^{\nu}\, \omega^{mn}\ ,$$ (16)

$$W^{\mathrm{q\theta}}_{\mathrm{red}} = - {\theta\, g^2_{\mathrm{YM}}\over 16\pi^2} \left({2\pi\over a}\r... ...\sigma} \, \int d^{2D}\sigma\, {\cal F}_{\mu\nu}\, \ast\, {\cal F}_{\rho\sigma}$$

$$= -{ \theta\, g^2_{\mathrm{YM}} \over 64\pi^2} \left({2\pi\over a}\... ...\, m}{\cal A}_{\rho}\circ \partial_{n\, ]}\, {\cal A}_{\sigma}\, \omega^{mn}\ .$$ (17)

We could also write equation (16) without the $\ast$-product between the two Moyal brackets, due to the following property of integration over phase space in the absence of boundary [12], $\int d^4\sigma\, {\cal U} \,\ast\, {\cal V}\equiv \int d^4\sigma\, {\cal U} \, {\cal V}$ . Extra terms will appear whenever boundaries are present [13]. This is just what we expect to find, thus, we shall keep the $\ast$-product in (16) .
Before discussing the case $D=2$, it can be useful to show as the classical limit of $W^{\mathrm{qYM}}_{\mathrm{red}}$, with $D=1$ is related to the action of a bosonic string, which for simplicity we assumed to closed. Thus, only the term $W^{\mathrm{q\theta}}_{\mathrm{red}}$ has to be taken into account and gives:

$$W^{\mathrm{qYM}}_{\mathrm{red}}{}_{N>> 1} \approx -{1\over 16} \left(\, {2\pi\over a}\,\right)^4 {N^4\over (2\pi)^4... ...ega^{mn} \, \partial_{[\, m}\,{\cal A}^{\mu}\, \partial_{n\,]}\, {\cal A}^{\nu}$$

$$= -{1\over 16}\left(\, {2\pi\over a}\,\right)^4\, {\left(\,{N\over ... ...u}\, \right\}_{PB}\, \left\{\,{\cal A}^{\mu}\ , {\cal A}^{\nu}\, \right\}_{PB},$$ (18)

where, we took into account that both the $\ast$ and $\circ$ products collapse into the ordinary product in the classical limit [11]. Provided one appropriately re-scales the gauge field, (18) reproduces the Schild action for the relativistic, bosonic string [14], [15]. Let us remark that the Schild action is not invariant under reparametrization but under the more restricted group of area preserving diffeomorphisms. This result establishes the known correspondence between $\textit{SU}(\,\infty\,)$ and the group of area preserving diffeomorphisms [6].

$p$-brane Actions	brane Fields	Symmetry
$S _{\mathrm{DT}}\propto \int d^4\sigma \sqrt h \, h^{am}\, h^{bn} \, \partial_{... ...X^\mu\, \partial_{b\,]} X^\nu\, \partial_{[\, m} X_\mu\, \partial_{n\,]} X_\nu$	$X ^{\mu}(\sigma)$, $h _{mn} (\sigma)$	Reparametrization $+$ Conformal Invariance
$S _{\mathrm{Schild}}\propto \int d^4\sigma \, \left\{\, X^\mu\ , X^\nu\, \right\}_{PB}\, \left\{\, X_\mu\ , X_\nu\, \right\}_{PB}$	$X ^{\mu}(\sigma)$	Volume Preserving Diffeomorphisms
$S _{\mathrm{NG}}\propto \int d^4\sigma \, \sqrt{\, -det\left(\, \partial_ m\, X_\mu\, \partial_n\, X^\mu \,\right)\, }$	$X ^{\mu}(\sigma)$	Reparametrization Invariance
$S _{\mathrm{CS}}\propto \epsilon_{\lambda\mu\nu\rho}\, \int d^3\xi\, X^{\lambda}\, \left\{ \, X^\mu\ , X^\nu \ , X^{\rho} \,\right\}_{NPB}$	$X ^{\mu}(\xi)$	Reparametrization Invariance
Table 1: This table summarizes the various actions for the bulk three-brane and boundary two-brane we used in the paper.

 

Moving to the case $D=2$, it can be useful to recall that the action for a $p$-brane can be written in several different forms [16]. With hindsight, we need to recall the conformally invariant four dimensional $\sigma$-model action introduced in [17]

$$S_{DT}=-{\mu^4_0\over 4}\, \int d^4\sigma\, \sqrt h \, h^{am... ...\,]} X^\nu\, \partial_{[\, m} X_\mu\, \partial_{n\,]} X_\nu\ ,$$ (19)

and the Chern-Simons membrane [18]

$$S_{CS} = -{\kappa\over 3!\times 4! }\, \epsilon_{\lambda\mu\nu\rho}\, \int... ...c}\, \partial_{ a}\, X^{\mu}\, \partial_b\, X^\nu\, \partial_{c}\, X^{\rho} \ ,$$

$$= -{\kappa\over 4! }\, \epsilon_{\lambda\mu\nu\rho}\, \int d^3\xi\, X^{\lambda}\, \left\{ \, X^\mu\ , X^\nu \ , X^{\rho} \,\right\}_{NPB}\ .$$ (20)

In equation (19) the indices inside square roots are anti-symmetrized and target spacetime indices are saturated by a flat metric $\eta_{\mu\nu}$. In this $\sigma$ model approach $X^\mu$ would be the coordinates of a $3$-brane in four dimensional target spacetime and the $\sigma^m$ are coordinates on the $4$-dimensional world volume. Moreover, $\eta_{\mu\rho}$ is the flat Minkowski metric tensor in target spacetime, while $h _{mn} (\sigma)$ is an independent, auxiliary, world-volume metric providing reparametrization invariance of the model. It can be worth to remind that, once the auxiliary metric is algebraically solved in terms of the induced metric, i.e. $h_{mn}\propto \partial_m\, X \cdot \partial_n\, X$, then, $S_{DT}$ turns into a Nambu-Goto type action. But, the Nambu-Goto action for a $3$-brane embedded in a four dimensional target spacetime is nothing but the world volume of the brane itself. Accordingly, the constant $\mu^4_0$ in front of it can be identified with the (constant) pressure inside the bag. Despite its non-trivial look, $S_{DT}$ does not describe transverse, propagating degrees of freedom, but only a constant energy density and pressure, non-dynamical, spacetime domain. All the dynamics is carried by the boundary of the domain, in a way which seems to satisfy the holographic principle [19] in a very strict sense: all the non-trivial dynamical degrees of freedom are confined to the membrane enclosing the bag. Among various kind of relativistic membranes the Chern-Simons one is a very interesting objects [18]. In the action $S_{CS}$ the three-volume element of the membrane is represented by the Nambu-Poisson brackets, while $\kappa$ is a constant with dimensions of energy per unit three-volume. The presence of the Nambu-Poisson bracket suggests a new kind of formulation of both classical and quantum mechanics for such an object, which is worth investigating by itself [20].
The formal structures of (16), (17) and (19), (20) are so similar that one expects some kind of relationship among these actions. On the other hand, we notice that while the $W^{\mathrm{qYM}}_{\mathrm{red}}$ action is defined over a flat phase space, $S_{DT}$ involves integration over a curved world-volume. In the latter case a Moyal deformation would be no longer valid, due to the lack of associativity of the $\ast$-product, and a Fedosov deformation quantization would be required [21]. However, the Weyl symmetry of the $S_{DT}$ suggests to restrict⁵the world metric to the conformally flat sector:

$$h_{mn}= e^{2\phi(\sigma)}\, \eta_{mn}\ .$$ (21)

We shall not consider in this paper the quantum conformal anomaly which could spoil the choice (21), and consider only equivalence between classical actions. Furthermore, we can implement the following relation between Poisson bracket and simplectic form in $4$-dimensional phase space:

$$\partial_{[\,a }\, X^\mu \partial_{b\,] }\, X^\nu= {1\over 4}\,\omega_{ab} \,\left\{\, X^\mu\ , X^\nu\, \right\}_{PB}\ .$$ (22)

Thus, we find that the $S_{\mathrm{DT}}$ action, in a conformally flat background geometry, looks like a generalized Schild action [14] for a three-brane:

$$S_{\mathrm{DT}}=-{\mu^4_0\over 16}\, \int d^4\sigma \, \left... ..., X_\mu\ , X_\nu\, \right\}_{PB} \equiv S_{\mathrm{Schild}}\ .$$ (23)

The alleged correspondence between $W^{\mathrm{qYM}}_{\mathrm{red}}$ and $S_{\mathrm{DT}}$ is clear. By choosing $D=2$ in (16), we find

$$W^{\mathrm{qYM}}_{\mathrm{red}}{}_{N>> 1} \approx -{1\over 16} \left(\, {2\pi\over a}\,\right)^4 \, \left(\, { N\ov... ...mega^{mn} \, \partial_{[\, m}\,{\cal A}^{\mu}\,\partial_{n\,]}\, {\cal A}^{\nu}$$

$$= -{1\over 16}\left(\, {2\pi\over a}\,\right)^4 \, \left(\, { N\ove... ...\, \right\}_{PB}\, \left\{\,{\cal A}^{\mu}\ , {\cal A}^{\nu}\, \right\}_{PB}\ ;$$ (24)

$$W^{\mathrm{q\theta}}_{\mathrm{red}} = - {\theta\, g^2_{\mathrm{YM}}\over 16\pi^2} \left({2\pi\over a}\r... ...o\sigma} \, \int_\Sigma d^{4}\sigma\, {\cal F}_{\mu\nu}\, {\cal F}_{\rho\sigma}$$

$$= -{ \theta\, g^2_{\mathrm{YM}} \over 64\pi^2} \left({2\pi\over a}\... ...rtial_{m}\, {\cal A}_{\rho}\, \partial_{n}\, {\cal A}_{\sigma}\, \omega^{mn\,]}$$

$$= -{ \theta\, g^2_{\mathrm{YM}} \over 32\pi} \left({2\pi\over a}\ri... ...\{\, {\cal A}_{\nu}\ , {\cal A}_{\rho}\ , {\cal A}_{\sigma}\,\right\}_{NPB} \ .$$ (25)

By rescaling the fields according with⁶

$${\cal A}_\mu \, \longrightarrow \left(\,{2\pi\over N}\,\right)^{1/4}\, X_\mu \ ,$$ (27)

$${\cal F}_{\mu\nu}\,\longrightarrow\, \left(\,{2\pi\over N} \,\right)^{1/2} \, \left\{\, X_\mu\ , X_\nu\, \right\}_{PB}\ ,$$ (28)

we get

$$W^{\mathrm{qYM}}_{\mathrm{red}} + W^{\mathrm{q\theta}}_{\mathrm{red}} = -{1\over 16}\left(\, {2\pi\over a}\,\right)^4 \, {1\over g^2_{\ma... ... { X}_{\nu}\, \right\}_{PB}\, \left\{\,{ X}^{\mu}\ , { X}^{\nu}\, \right\}_{PB}$$

$$- { \theta\, g^2_{\mathrm{YM}} \over 32\pi} \left({2\pi\over a}\rig... ...X}_{\mu}\, \left\{\, { X}_{\nu}\ , { X}_{\rho}\ , { X}_{\sigma}\,\right\}_{NPB}$$ (29)

which matches the sum of the actions $S_{DT}$ and $S_{CS}$ provide we identify

$$\mu_0^4\equiv {1\over 4\pi }\left(\, {2\pi\over a}\,\right)^4 \, {4\pi\over g^2_{\mathrm{YM}}}\ ,$$ (30)

$$\kappa \equiv { \theta\, g^2_{\mathrm{YM}} \over 32\pi}\left({2\pi\over a}\right)^4\ .$$ (31)

As a consistency check of our dynamically generated bag pressure consider equation (30) in the strong coupling regime, where it is conventionally assumed $g^2_{\mathrm{YM}}/4\pi \simeq 0.18$. If we identify the inverse lattice spacing $2\pi/a$ with the $QCD$ scale $\Lambda_{QCD}\simeq 200\, MeV$ then, equation (30) provides

$$\mu_0^4\equiv {1\over 4\pi }\left(\, 200\, MeV\,\right)^4 \, {1\over 0.18}\ .$$ (32)

The actual value of $\mu_0$ is close to $\Lambda_{QCD}$; this is not a bad result, compared with the phenomenological value $\mu_0\simeq 110 MeV$, if one takes into account the uncertainty on the value of $\Lambda_{QCD}$, i.e. $120\, MeV \, \le\, \Lambda_{QCD}\, \le\, 350\, MeV$ .

Figure 1: The figure shows the web of relationships among the various actions discussed in this note. It can be read as ``map'' to move from the Yang-Mills action, in the lower left corner, to complete bag action in the upper right corner.