2 Actions and their relations

As a starting point, let us consider an action smoothly interpolating between the ( area preserving ) Schild action and the fully ( reparametrization invariant ) Nambu-Goto action can be written by introducing an auxiliary world-sheet field $\Phi(\, \sigma\,)$ [2]:

$$I\left[\, \Phi\ , X\,\right]\equiv \frac{\mu_0}{2}\, \int_\... ..._{mn})\over \Phi(\,\sigma\,)} + \Phi(\, \sigma\,)\,\right]$$ (1)

where, $\gamma_{mn}\equiv \eta_{\mu\nu}\, \partial_m\, X^\mu\, \partial_n\, X^\nu$ is the induced metric on the string Euclidean world-sheet $x^\mu= X^\mu(\,\sigma\,)$, $\mathrm{sign}(\gamma_{mn})=(+\ ,+)$; finally, $\mu_0\equiv 1/2\pi\alpha^\prime$ is the string tension. We assigned the following dimension ( in natural units ) to the various quantities in (1):

$$\left[\, \sigma^m\,\right]= \mathrm{length} \ ,\qquad \left... ...mu\,\right]=\mathrm{length}\ ,\qquad \left[\, \Phi\,\right]=1$$ (2)

For later convenience, we recall the relation between $\mathrm{det}(\gamma_{mn})$ and the world manifold Poisson Bracket:

$$\mathrm{det}(\gamma_{mn})=\left\{\, X^\mu\ , X^\nu\,\right\}^2$$ (3)

$$\left\{\, X^\mu\ , X^\nu\,\right\}_{\mathrm{PB}}\equiv \epsilon^{mn}\,\partial_m\, X^\mu\, \partial_n\, X^\nu\,$$ (4)

The action (1) is reparametrization invariant provided the auxiliary field $\Phi(\, \sigma\,)$ transforms as a world-sheet scalar density:

$$\Phi(\, \sigma^\prime\,)=\left\vert {\partial \sigma\over \partial\sigma^\prime\, }\,\right\vert\Phi(\, \sigma\,)$$ (5)

Thus, by implementing reparametrization invariance $\Phi(\, \sigma\,)$ can be transformed to unity and the Schild action can be recovered as a ``gauge fixed'' form of (1):

$$I\left[\, \Phi=1\ , X\,\right]=\frac{\mu_0}{2}\, \int_\Sig... ...gamma_{mn}) + 1\,\right]=\mathrm{Schild} + \mathrm{const.}$$ (6)

where, the numerical constant is proportional to the area of the integration domain $\Sigma$.
On the other hand, by solving $\Phi(\, \sigma\,)$ in terms of $X$ from (1) one recovers, ``on-shell'', the Nambu-Goto action

$${\delta I\over \delta \Phi}=0\rightarrow \phi=\sqrt{\, det... ...w I=\mu_0\,\int_\Sigma d^2\sigma\,\sqrt{-det(\gamma_{mn})}$$ (7)

The inverse equivalence relation can be proven by starting from the Schild action [3]

$$I_S\equiv \mu_0\, \int_\Xi d^2\varphi\, \mathrm{det}\left[\, \gamma_{ab}(\varphi) \,\right]$$ (8)

and ``lifting'' the original world-sheet coordinates $\varphi^m$ to the role of dynamical variables by mean of reparametrization $\varphi^m\to \sigma^m=\sigma^m(\varphi)$ [4]:

$$I_{rep}\equiv \mu_0\, \int_\Sigma d^2\sigma\, \Phi^{-1} \m... ...\,\epsilon^{mn}\,\partial_m\, \phi^i\, \partial_n\, \phi^j$$ (9)

By variation of $I_{rep}$ with respect to $\varphi^i$ one gets the field equation

$$\epsilon_{ij}\,\epsilon^{mn} \partial_n\, \phi^j \partial_m... ...left[\, \gamma_{ab}(\sigma)\,\right] \over \Phi^2}\,\right)=0$$ (10)

Thus,

$${det\left[\, \gamma_{ab}(\sigma)\,\right] \over \Phi^2}= \mathrm{const.}\equiv {1\over 4\mu_0}$$ (11)

and the Nambu-Goto action (7) is obtained again.
This second option introduces a scalar doublet $\phi^i$, $i=1\ ,2$ and expresses the scalar density $\Phi$ as a ``composite'' object, rather than a fundamental one, or as a second ``integration measure'' [5]. In any case the final result is unchanged.
The action (1) is a special case of the general two-parameter family of p-brane actions [2]

$$I^p_n\equiv \frac{\mu_0^{(p+1)/2}}{n}\, \int_\Sigma d^{p+1... ...et}\gamma_{mn})^{n/2}\over e(\,\sigma\,)^n} + n-1\,\right]$$ (12)

where, in our notation $e(\,\sigma\,)$ has been replaced by $\Phi(\, \sigma\,)$ and $\mathrm{det}\gamma_{mn}$ is now the square of the Nambu-Poisson bracket $\left\{\, X^{\mu_1}\ , \dots\ ,X^{\mu_{p+1}}\,\right\}_{NPB}$.
For $n=2$ and $p=1$ we obtain (1), while for $n=1$ the auxiliary field decouples and we get the Nambu-Goto action.
In a naive, but not too much, way of thinking, one could trace back the correspondence between Yang-Mills gauge fields and Schild strings to the common quadratic form of both actions in the respective sets of ``field strengths'':

$${1\over 4}\,\left\{\, X^\mu\ , X^\nu\,\right\}_{\mathrm{PB}... ...er 4}\, \mathrm{Tr}\mathbf{F}_{\mu\nu}\, \mathbf{F}^{\mu\nu}$$ (13)