The Polyakov action functional or energy functional is a standard kinetic action functional for sigma models with worldvolume $(\Sigma,g)$ and target spacetime $(X,\mu)$ (pseudo)Riemannian manifolds. Its critical points are the harmonic maps from $\Sigma$ to $X$
With a suitable “worldvolume cosmological constant” added and with the worldvolume metric “integrated out”, then the Polyakov action is classically equivalent to the Nambu-Goto action functional, which is simply the “proper volume” function for images of $\Sigma$ in $Y$. But as opposed to the Nambu-Goto action the Polyakov action is quadratic in derivatives. Therefore it is lends itself better to perturbation theory of scattering amplitudes – where the kinetic contributions have to be Gaussian integrals – such as in worldline formalism for quantum field theory as well as in perturbative string theory.
On the other hand, the Nambu-Goto action lends itself better to generalizations such as the Dirac-Born-Infeld action for D-branes.
Let
$p \in \mathbb{N}$ (for p-brane dynamics);
$(X,\mu)$ a pseudo-Riemannian manifold (target spacetime);
$\Sigma$ a compact smooth manifold of dimension $(p+1)$ (worldvolume).
$[\Sigma,X]$ the diffeological space of smooth functions $\Sigma \to X$.
$Met(\Sigma)$ the moduli space of (pseudo)Riemannian metrics $g$ on $\Sigma$.
The Polyakov action is the smooth function
which on a map $\phi \colon \Sigma \to X$ and a metric $g$ on $\Sigma$ is given by the integral
where the derivative $d \phi \in \Gamma(T^\ast \Sigma\otimes \phi^\ast T X)$, and where the norm $\Vert - \Vert$ is given jointly by the metrics $g$ and $\mu$ of $\Sigma$ and $X$.
When both $\Sigma$ and $X$ are covered by single coordinate charts $(\sigma^a)_{a = 0}^p$ and $(x^\mu)$, then this reads
with a sum over repeated indices understood. Here $det g$ denotes the absolute value of the determinant of $(g_{a b})$ (often written $-det g$ in the pseudo-Riemannian case.)
The Polyakov action with a suitable worldvolume cosmological constant term added is classically equivalent to the Nambu-Goto action (e.g. Nieto 01, section 2).
This on-shell equivalence is exhibited by the smooth function
which on a triple $(\phi,g,\Lambda)$ is given by
where now $\vert d\phi \vert^2$ denotes the square norm only with respect to the metric on $X$.
Because, on the one hand, the equations of motion induced by $\tilde S$ for variation of $\Lambda$ are
and substituting that constraint back into $\tilde S$ gives the Nambu-Goto action. On the other hand, the equations of motion induced by $\tilde S$ for variation of $g$ are
and substituting that back into $\tilde S$ gives
which is the Nambu goto action with “cosmological constant” $(p-1)$.
(So the case where this cosmological constant correction disappears is $p = 1$ corresponding to the string.)
The Polyakov action was introduced in
Quantum geometry of bosonic strings , Phys. Lett. B103 (1981) 207;
Quantum geometry of fermionic strings , Phys. Lett. B103 (1981) 211
Detailed discussion of the relation to the Nambu-Goto action and the Dirac-Born-Infeld action is in
See also