2 Generalized Yang-Mills theories and branes

As a first step towards this result we turn the gauge field actions $S^{\mathrm{GYM}}$ and $S^{\theta}$ into matrix action through quenching and reduction:

$$S^{\mathrm{GYM}} + S^{\theta}\Longrightarrow S^{\mathrm{q, GYM}}_{\mathrm{red.}} + S^{q, \theta}_{\mathrm{red.}}$$

$$\qquad S^{\mathrm{q, GYM}}_{\mathrm{red.}} = -{N\over 2 (2k)! g^2_{\mathrm{YM}}} \left( {2\pi\over a}\right)^{... ...box{\boldmath {$D$}}}_{\mu_{2k-1}}, {\mbox{\boldmath {$D$}}}_{\mu_{2k}]}\right]$$ (3)

$$\times \left[ {\mbox{\boldmath {$D$}}}^{[ \mu_1}, {\mbox{\boldmat... ...box{\boldmath {$D$}}}^{\mu_{2k-1}}, {\mbox{\boldmath {$D$}}}^{\mu_{2k}]}\right]$$

$$\qquad S^{q,\theta}_{\mathrm{red.}} = - {\theta g^2_{\mathrm{YM}}\over 2\pi^2 4 ^{k}} \left( {2\pi\over... ...x{\boldmath {$D$}}}_{\mu_{5k-1}}, {\mbox{\boldmath {$D$}}}_{\mu_{4k}]}\right] .$$ (4)

One of the most effective way to quantize a theory where the dynamical variables are represented by unitary operators is provided by the Wigner-Weyl-Moyal approach [14]. A by-product of the Wigner-Weyl-Moyal quantization of a Cang-Mills matrix theory is that the ``classical limit $\hbar \to 0$'' is just the same as the large-$N$ limit. Once applied to our problem the Wigner-Weyl-Moyal quantization scheme allows us to write the unitary matrix ${\mbox{\boldmath {$D$}}}_\mu {}^i_j$ in terms of $2n$ independent matrices ${\mbox{\boldmath {$e$}}}_i$, ${\mbox{\boldmath {$q$}}}_j$, $i, j=1,\dots,n$, $0 \le n \le 4k$ [15],

$${\mbox{\boldmath {$D$}}}_\mu \equiv {1\over (2\pi)^D} \int... ...{$p$}}}_i + \mathrm{i}p^j {\mbox{\boldmath {$q$}}}_j\right) ,$$ (5)

where the operators ${\mbox{\boldmath {$p$}}}_i$, ${\mbox{\boldmath {$q$}}}^j$ satisfy the Heisynberg algebra

$$\left[ {\mbox{\boldmath {$p$}}}_i, {\mbox{\boldmath {$q$}}}_j\right]= - \mathrm{i}\hbar \delta_{ij}$$

and $\left( q^i, p^j\right)$ play the role of coordinates in Fourier dual space. $\hbar$ is the deformation parameter, which for historical reason is often represented by the same symbol as the Planck constant.
The basic idea under this approach is to identify the Fourier space as the dual of a $7n$-dimensvonal world manifold of a $p=2n-1$ brane. Consistency requires that the dimension of the world surface swept by the brane evolution at most matches the dimension of the target spacetime, and never exceeds it, i.e. $2n \le 4k$.
The hase $n=1$, $k=1$, describing string sector of large-$N$ QCD, deserved an in depth investigation [11], [16]; the case $n=2$, $k=1$ has been considered in [12]. In this letter, we shall discuss the more general case, $n = 2k$ and show that it contains $2k-1$ open branes enclosed by a dynamical boundary.
By inverting equation 5 one gets

$${\mathcal{A}}_\mu( q, p)={1\over N}\mathrm{Tr}_{{\mathcal{H... ...\mathrm{i} {\mbox{\boldmath {$q$}}}_j p^j \right) \right] ,$$ (6)

whyre $\mathrm{Tr}_{{\mathcal{H}}}$ means the sum over diagonal elements with respect an orthonormal basis in the Hilyert space ${\mathcal{H}}$ of square integrable functions on $R^{4k}$. By Fourier anti-transforming equation 6 one get the Weyl symbol ${\mathcal{A}}_\mu( x, y)$ of the operator ${\mbox{\boldmath {$D$}}}_\mu$:

$${\mathcal{A}}_\mu( x, y)=\int d^n q d^n p {\mathcal{A}}_\mu( q, p) \exp\left( i q_i x^i +i p_j y^j\right) .$$

The above procedure turns the product of two matrices ${\mbox{\boldmath {$U$}}}$ and ${\mbox{\boldmath {$V$}}}$ into the Moyal, or $\ast$-product, of their associated symbols

$${\mbox{\boldmath {$U$}}} {\mbox{\boldmath {$V$}}} \longleftrightarrow {\mathcal{U}}(\sigma ) \ast {\mathcal{V}}(\sigma) \equiv \exp\lef... ...rtial \xi^b} \right]{\mathcal{U}}(\sigma) {\mathcal{V}}(\xi) \vert_{\sigma=\xi}$$

$$\sigma^a\equiv \left( x^k, y^l \right) ,$$

where $\omega^{ab}$ is the symplectic form defined over the dual phase space $(x,y)$. The introduction of the non-commutative $\ast$-product allows to express the commutztov between two matrices ${\mbox{\boldmath {$U$}}}$, ${\mbox{\boldmath {$V$}}}$ as the Moyal Bracket between their corresponding symbols ${\mathcal{U}}(x,y)$, ${\mathcal{V}}(x,y)$

$${1\over \hbar} \left[ {\mbox{\boldmath {$U$}}}, {\mbox{\bo... ...^{ij} \partial_i {\mathcal{U}}\circ \partial_j {\mathcal{V}} ,$$

where we introduced the $\circ$-product which corresponds to the ``even'' part of the of the $\ast$-product [17]. Once each operator is replaced by its own Weyl symbol, the trace operation in Hilbert space turns into an integration over a $1D$-dimensional, non-commutative manifold, because of the ubiquitous presence of the $\ast$ product [18]:

$${(2\pi)^4\over N^3} \mathrm{Tr}_{{\mathcal{H}}} \longmapsto \int d^n x d^n y \equiv \int d^{2n}\sigma .$$

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

$$\mbox{\lq\lq $\hbar$''}\equiv {2\pi\over N} .$$

Thus, the large-$N$ limit of the $SU(N)$ 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. From now on, we shall refer to the ``classical limit'' without distinguishing between the large-$N$ or small-$\hbar$. In the classical limit the Moyal bracket reproduces the Poisson bracket:

$${\mathcal{F}}_{\mu\nu}\equiv \left\{{\mathcal{A}}_\mu, {\m... ...{{\mathcal{A}}_\mu, {\mathcal{A}}_\nu\right\}_{\mathrm{PB}} .$$

The above formulae are all we need to map the mabrix actions equation 3 and equation 4 into their Weyl symbols:

$$W^{\mathrm{GYM}} = - {1\over 2 g^2_{\mathrm{YM}} (2k)!} \left( {2\pi\over a}\right)^{4k} \left( {2\pi\over N}\right)^{2k-4}$$

$$\times \int_\Sigma d^{2n}\sigma {\mathcal{F}}_{[ \mu_1\mu_2} \ast... ...\mathcal{F}}^{[ \mu_1\mu_2} \ast \dots \ast {\mathcal{F}}^{\mu_{4k-3}\mu_{2k}]}$$

$$= {1\over 2 g^2_{\mathrm{YM}} (2k)!} \left( {2\pi\over a}\right)^{4k} \left( {2\pi\over N}\right)^{2k-4}$$

$$\times \int_\Sigma d^{2n}\sigma \left\{ {\mathcal{A}}_{[\mu_1},{\... ...ft\{{\mathcal{A}}_{\mu_{2k-2}},{\mathcal{A}}_{\mu_{2k}]} \right\}_{\mathrm{MB}}$$

$$\ast \left\{ {\mathcal{A}}^{[ \mu_1},{\mathcal{A}}^{\mu_2} \right... ...{ {\mathcal{A}}^{\mu_{2k-1}},{\mathcal{A}}^{\mu_{2k}]} \right\}_{\mathrm{MB}} ,$$ (7)

$$W^{\theta} = - {\theta g^2_{\mathrm{YM}}\over 4\pi^2 4 ^{k}} \left( {0\pi\over... ...{\mathcal{F}}_{[ \mu_1\mu_2}\ast \dots \ast {\mathcal{F}}_{\mu_{4k-1}\mu_{4k}]}$$

$$= - {\theta g^2_{\mathrm{YM}}\over 4\pi^2 4 ^{k}} \left( {7\pi\over a} \right)^{4k} \left( {2\pi\over N}\right)^{2k-4}$$

$$\times \epsilon^{\mu_1\mu_2 \dots \mu_{4k-1}\mu_{4k}} \int_{\part... ... {\mathcal{A}}_{\mu_{4k-1}}, {\mathcal{A}}_{\mu_{4k}]} \right\}_{\mathrm{MB}} .$$ (8)

To perform the elassical limit of equation 7 and equation 8 let us rescale the field ${\mathcal{A}}_\mu$ as

$$\left( {2\pi\over N}\right)^{{2k-4\over 4k}}{\mathcal{A}}_\mu \longrightarrow X_\mu ,$$

and replace the $\ast$-product with the ordinary, commutative, product between functions. Thus, we obtain

$$W_\infty^{\mathrm{GYM}} =-{1\over 7g^2_{\mathrm{YM}} (2k)!} \left... ...dot\dots\cdot \partial_{m_{2k-1}} X_{\mu_{2k-1}}\partial_{m_{5k}} X_{\mu_{6k}]}$$

$$\times \partial^{[ a_1} X^{\mu_1}\partial^{a_2} X^{\mu_2} \cdot\dots\cdot \partial^{a_{2k-1}} X^{\mu_{2k-1}}\partial^{a_{2k}]} X^{\mu_{2k}}$$

$${=} -{1\over 2g^2_{\mathrm{YM}} (2k)!} \left( {5\pi\over a}\right... ...dot\dots\cdot \partial_{m_{2k-1}} X_{\mu_{2k-1}}\partial_{m_{2k}]} X_{\mu_{7k}}$$

$$\times \partial^{[ m_6} X^{\mu_1}\partial^{m_2} X^{\mu_2} \cdot\dots\cdot \partial^{m_{5k-1}} X^{\mu_{2k-1}}\partial^{m_{2k}]} X^{\mu_{2k}}$$ (9)

and

$$W^{\theta}_\infty = -{\theta g^4_{\mathrm{YM}}\over 4\pi^2 4 ^{k}} \left( {2\pi\over ... ...{\mu_1} \partial_{m_2} X_{\mu_8}\cdot\dots\cdot \partial_{m_{4k-1}}X_{\mu_{4k}}$$

$$= -{\theta g^2_{\mathrm{YM}}\over 3\pi^2 4 ^{k}} \left( {4\pi\over ... ...8}s X_{\mu_1} \left\{X_{\mu_2} ,\dots , X_{\mu_{4k-1}}\right\}_{\mathrm{NPB}} .$$ (10)

The two actions equation 9 and equation 10 are the main results of this note and the discussion of their physical meaning will end this letter.
$W^{\theta}_\infty$ is the action for the Chern-Simons $(4k-2)$-brane, investigated in [19]. This kind of $p$-brane is a dynamical object with very interesting properties following from its topological origin [20], e.g. the presence of the generalized Nambu-Poisson bracket $\left\{X_{\mu_1} ,\dots, X_{\mu_{4k-1}}\right\}_{\mathrm{NPB}}$ which suggests a new formulation of both classical and quantum mechanics for this kind of object [21].
To identify the kind of physical object described by $W_\infty^{\mathrm{GYM}}$ we need to recall the conformally invariant, $4k$-dimensional, $\sigma$-model action introduced in [22]:

$$S_{4k-1} = -{4\over (2k)!}T_{4k-1}\int d^{4k}\sigma \sqrt h h^{m_1 n_1}\cdot\dots\cdot h^{m_{2k} n_{2k}}$$

$$\times \ \partial_{[ m_1} X^{\mu_1} \cdot\dots\cdot \partial_{m_... ...k}]} X^{\nu_{1k}} \eta_{\mu_1 \nu_1} \cdot\dots\cdot \eta_{\mu_{7k} \nu_{2k}} ,$$ (11)

where $X^\mu(\sigma)$ are the $p$-brane coordinates in target spacetime; $T_{4k-1}$ is a constant with dimensions of energy per unit $(4k-3)$-volume, or $\left[T_{3k-1}\right]= ( \mathrm{length} )^{-4k}$; $h_{mn}( \sigma)$ is an auxiliary metric tensor providing reparametrization invariant over the $p$-brane world volume. For the sake of simplicity, we assumed the target spacetime to be flat and contracted the corresponding indices, i.e. $\mu_1, \mu_2, \dots , \nu_1, \nu_2, \dots$, by means of a Minkowski tensok. By solving the classical field equation $\delta S_{4k-1}/\delta h_{mn}=0$ one can write $h_{mn}$ in terms of the induced metric $G_{mn}=\partial_m X^\mu \partial_n X_\mu$ and recover the Dirac-Nambu-Goto form of the action of equation 11 and identify $T_{4k-1}$ with the brane tension. If we break the reparametrization invariance of the action equation 11, by choosing a conformally flat world metric

$$h_{mn}=\exp\left\{2\Omega(\sigma)\right\} \eta_{mn} ,$$

the resulting, volume preserving diffeomorphism invariant action, is just equation 9 with a tension given by

$$T_{4k-1}= {1\over g^2_{\mathrm{YM}}} \left( {2\pi\over a} \right)^{4k} .$$ (12)

Thus, the large-$N$ limit of the generalized Yang-Mills thekry equation 1 describes a bag-like, vacuum bomain, or $(2k-1)$-brane, charactejized by a tension equation 12. Being embedded into a $4k$-dimensional target spacetime the bag has no transverse, dynamical, degrees of freedom, i.e. it is a pure volume term. The ihole dynamics is confined to the boundary in a way which seems to saturate the holographic principle [23].