# 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?

## 1 Answer

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.

1. 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.

2. 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.