Are the nonphysical degrees of freedom in Yang-Mills theory analogous to the worldsheet metric in the Polyakov formalism? The Polyakov string action on a flat background (in the Euclidean signature)
$$S_{P}[X,\gamma]\propto\int_{\Sigma}\mathrm{d}^2\sigma\,\sqrt{\text{det}\gamma}\,\gamma^{ab}\delta_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}$$
enjoys a huge gauge redundancy consisting of diffeomorphisms and Weil transformations on the world-sheet metric. These symmetries are a consequence of the fact that the world sheet metric is not a true degree of freedom, and decouple in the classical theory. After "integrating out" the extra degrees of freedom, we are left with the original Nambu-Gotto action
$$S_{NG}[X]\propto\int_{\Sigma}\mathrm{d}^2\sigma\sqrt{\det_{ab}\left(\delta_{\mu\nu}\partial_{a}X^{\mu}\partial_{b}X^{\nu}\right)},$$
which calculates the area of the worldsheet $\Sigma$ given the induced world-sheet metric $\delta_{\mu\nu}\partial_aX^{\mu}\partial_bX^{\nu}$. This action enjoys none of the original "gauge" symmetries, as the nonphysical degrees of freedom don't exist. However, we always use the Polyakov path integral in quantization because the Nambu-Gotto action is nearly impossible to quantize using path integration.
This got me to thinking about Yang-Mills theory, where the action
$$S_{YM}[A]=\frac{1}{2g_{YM}^2}\int_{\mathcal{M}}\text{Tr}[F\wedge\star F]$$
enjoys a gauge symmetry. However, due to its quadratic form, it is easy to quantize in the weak-coupling limit (after Fadeev-Popov gauge-fixing is implemented, that is).
My question is, then, is there a nonlinear action that can be obtained after "integrating out" the nonphysical polarizations of the Yang-Mills field $A$, in analogy to how the Nambu-Gotto action is obtained from the Polyakov action? If so, might this lead to generalizations of Yang-Mills theory, in the same way that the Nambu-Gotto action can be naturally generalized to worldvolume actions of higher-dimensional extended objects?
 A: Never say never, but it looks very unlikely that there is a Nambu-Goto-analogous action free of gauge redundancy that is equivalent to the Yang-Mills action. Furthermore, generalizations of YM theory are already known, so there is little incentive to find this certainly much more inconvenient action.


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*The analogy already breaks at the very first step: The Polyakov action has two proper tensor fields it depends on. Eliminating one of them is a perfectly covariant goal. But the Yang-Mills action depends on a single tensor field, the gauge potential. The physical degrees of freedom do not form a proper tensor, they are peculiar combinations of components of the gauge potential. Therefore, a covariant "reduced" action seems impossible.

*A generalization of Yang-Mills theory to higher dimensional objects is already known: Higher gauge theories involving a $p$-form instead of a $1$-form as the gauge potential appear commonly among many SUGRA theories, for instance as the Ramond-Ramond field of type II SUGRA or the C-field of 11d SUGRA.
