5 Our proposal

The string world-sheet is the target spacetime image $x^\mu=X^\mu(\, \sigma^0,\sigma^1\,)$ of the world manifold $\Sigma$ : $-\infty\le \sigma^0\le +\infty$, $0\le \sigma^1\le L$, $X^\mu$ belonging to the algebra $\mathcal{A}$ of $C^\infty$, of functions over $\Sigma$. Thus, to realize our program we must deform ${\mathcal{A}}$ to a non-commutative ``starred'' algebra by introducing a $\ast$-product. The general rule is to define the new product between two functions as (for a recent review see [12]):

$$f \ast g =f \,g + \hbar\, P_\hbar(\, f\ ,g \,) \ ,$$ (21)

where $P_\hbar(\, f\ ,g\, )$ is a bilinear map $P_\hbar : {\mathcal{A}}\times {\mathcal{A}}\rightarrow {\mathcal{A}}$. $\hbar$ is the deformation parameter, which, in our case, is defined as $\hbar\equiv {2\pi/ N}$. The Moyal product is defined as the deformed $\ast$-product

$$f(\sigma) \ast g(\sigma) \equiv \exp\left[ i {\hbar\over ... ...n} \right] f(\sigma) g(\xi) \Biggr\rceil_{\xi=\sigma} \ ,$$ (22)

where $\omega^{mn}$ is a non-degenerate, antisymmetric matrix, which can be locally written as

$$\omega^{mn}=\left( \begin{array}{cc} 0 & 1\\ -1 & 0 \end{array} \right) \ .$$ (23)

The Moyal product (22) takes a simple looking form in Fourier space

$$F(\sigma) \ast G(\sigma)=\int {d^{ 2}\xi\over (2\pi)} \exp\l... ...over 2}+ \xi \right) G\left( {\sigma \over 2}- \xi \right) \ ,$$ (24)

where $F$ and $G$ are the Fourier transform of $f$ and $g$. Let us consider the Heisenberg algebra

$$\left[\, \mathbf{K}\ , \mathbf{P}\, \right]=i\, \hbar \ ;$$ (25)

Weyl suggested, many years ago, how an operator ${\mathbf{O}}_F( \mathbf{K}, \mathbf{P} )$ can be written as a sum of algebra elements as

$$\mathbf{O}_F= {1\over (2\pi)}\int dp \, dk\, F\left(\, p\ ,k... ...ight) \exp\left( i p \,\mathbf{K} + i k \mathbf{P} \right) \ .$$ (26)

The Weyl map (26) can be inverted to associate functions, or more exactly symbols, to operators

$$F\left(\, q\ ,k \,\right)= \int {d\xi\over (2\pi)} \exp\left... ...\right) \Biggr\vert q - \hbar {\xi \over 2 } \Biggr\rangle \ ;$$ (27)

moreover it translates the commutator between two operators ${\mbox{\boldmath {$U$}}}$, ${\mbox{\boldmath {$V$}}}$ into the Moyal Bracket between their corresponding symbols ${\mathcal{U}}(\sigma)$, ${\mathcal{V}}(\sigma)$

$${1\over i \hbar} \left[ {\mbox{\boldmath {$U$}}}, {\mbox{\... ...\ast {\mathcal{V}} - {\mathcal{V}} \ast {\mathcal{U}} \right)$$

and the quantum mechanical trace into an integral over Fourier space. A concise but pedagogical introduction to the deformed differential calculus and its application to the theory of integrable system can be found in [13]. We are now ready to formulate the alleged relationship between the quenched model (19) and string model: the symbol of the matrix $\mathbf{A}^{(\mathrm{q})}_\mu$ is proportional to the string coordinates $X^\mu( \, \sigma^0\ , \sigma^{1}\, )$. Going through the steps discussed above the action $S^{(\mathrm{q})}_{\mathrm{YM/\Phi }}$ transforms into its symbol $W^{(\mathrm{q})}_{\mathrm{YM/\Phi }}$:

$$S^{(\mathrm{q})}_{\mathrm{YM/\Phi }}\to W^{(\mathrm{q})}_{\m... ...rm{MB}}^2\, +{g_0{}^2\over N\,V\, }\, \Phi(\sigma)\,\right]\ .$$ (28)

and we rescale the Yang-Mills charge and field⁵as

$${N \over g_0{}^2} \quad \longmapsto \quad {1 \over g^2}\ ,\qquad \mathrm{Tr} \longmapsto \frac{1}{2\pi\sqrt V}\int_\Sigma d^2\sigma$$ (29)

$$A^\mu \quad \longmapsto \quad V^{-1/2}\, X^\mu\ ,\qquad \mathbf{F}^{(0)... ...\nu}\longmapsto \quad V^{-1/2}\left\{ X^\mu\ , X^\nu \right\}_{\mathrm{MB}} \ .$$ (30)

Finally, if $N \gg 1$ the Moyal bracket can be approximated by the Poisson bracket

$$\left\{ X^\mu\ , X^\nu \right\}_{\mathrm{MB}} \longmapsto \left\{ X^\mu\ , X^\nu \right\}_{\mathrm{PB}}$$

and (28) takes the form

$$W^{(\mathrm{q})}_{\mathrm{YM/\Phi }}\to S_{\mathrm{NG}}= {1\... ...igma) \,\right\}_{\mathrm{PB}}^2 + g^2 \,\Phi(\sigma)\,\right]$$ (31)

which is (1) provided we identify

$$\mu_0\longrightarrow\equiv {1\over 4\pi\, g\, \sqrt V }$$ (32)

According with the initial discussion we can establish the following, large-$N$, correspondence:

$$S^{(\mathrm{q})}_{\mathrm{BI-YM}}\approx S^{(\mathrm{q})}_{... ...g\, \sqrt V }\int_\Sigma d^2\sigma \,\sqrt{-det(\gamma_{mn})}$$ (33)

As a concluding remark, it may be worth mentioning that the approach discussed above can be extended in a straightforward way to a more general non-Abelian Born-Infeld action including topological terms. Instead of starting from (17), one can consider

$$S^{(\mathrm{q})}_{\mathrm{YM/\Phi\, }}= \frac{M^4\, V}{2} \mathrm{Tr}\left\{\, \mathbf{\Phi}^{-1} \left[\... ...I} +{N\over 4g_0^2 M^4 }\mathbf{F}_{\mu\nu}\, \mathbf{F}_{\mu\nu}\right.\right.$$

$$+ \left. \left. \left(\, {N\over 4g_0^2 M^4 }\,\right)^2 \,\left(\,... ...)^2 \right]- \mathbf{I} \right\} +{M^4\,V\over 2}\,\mathrm{Tr}\mathbf{\Phi} \ .$$ (34)

where, $\mathbf{I}$ is the $N\times N$ identity matrix and $M$ a new mass scale. The coefficient in front of the topological density has been assigned in analogy to the Abelian case, where no trace ambiguity occurs and the square root form is derived by expanding the determinant of the matrix $M_{\mu\nu}\equiv\delta_{\mu\nu} +\mathrm{const.}\times F_{\mu\nu}$.
The topological density contribution trivially vanishes in the large-$N$ limit:

$$\epsilon^{\lambda\mu\nu\rho}\, \mathbf{F}_{\lambda\mu}\, \m... ...\nu(\sigma)\ , X_\rho(\sigma) \,\right\}_{\mathrm{PB}}\equiv 0$$ (35)

A non-vanishing contribution from the topological term can only be obtained if strings are replaced by higher dimensional extended objects [9]. Thus, as far as strings are concerned, one finds

$$S^{(\mathrm{q})}_{\mathrm{YM/\Phi\, }}\to S^{(\mathrm{q})}_{... ...M^4\, V }\,\mathrm{det}[\,\gamma_{mn}\,]\,\right)} -1\,\right]$$ (36)