Critical string models
The Polyakov action functional or energy functional is a standard kinetic action functional for sigma models with worldvolume and target spacetime (pseudo)Riemannian manifolds. Its critical points are the harmonic maps from to
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 in . 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.
The Polyakov action is the smooth function
which on a map and a metric on is given by the integral
where the derivative , and where the norm is given jointly by the metrics and of and .
When both and are covered by single coordinate charts and , then this reads
with a sum over repeated indices understood. Here denotes the absolute value of the determinant of (often written in the pseudo-Riemannian case.)
Relation to Nambu-Goto action
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 is given by
where now denotes the square norm only with respect to the metric on .
Because, on the one hand, the equations of motion induced by for variation of are
and substituting that constraint back into gives the Nambu-Goto action. On the other hand, the equations of motion induced by for variation of are
and substituting that back into gives
which is the Nambu goto action with “cosmological constant” .
(So the case where this cosmological constant correction disappears is 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