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.
Let
(for p-brane dynamics);
a compact smooth manifold of dimension (worldvolume).
the diffeological space of smooth functions .
the moduli space of (pseudo)Riemannian metrics on .
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.)
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 Polyakov action with “cosmological constant” .
(So the case where this cosmological constant correction disappears is corresponding to the string.)
The Polyakov action was introduced in:
Detailed discussion of the relation to the Nambu-Goto action and the Dirac-Born-Infeld action is in
See also
Last revised on December 21, 2022 at 11:44:14. See the history of this page for a list of all contributions to it.