nLab Polyakov action



Riemannian geometry

String theory



The Polyakov action functional or energy functional is a standard kinetic action functional for sigma models with worldvolume (Σ,g)(\Sigma,g) and target spacetime (X,μ)(X,\mu) (pseudo)Riemannian manifolds. Its critical points are the harmonic maps from Σ\Sigma to XX

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 Σ\Sigma in YY. 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

S Pol:[Σ,X]×Met(Σ) S_{Pol} \;\colon\; [\Sigma,X]\times Met(\Sigma) \longrightarrow \mathbb{R}

which on a map ϕ:ΣX\phi \colon \Sigma \to X and a metric gg on Σ\Sigma is given by the integral

S Pol(ϕ,g) Σdϕ 2dvol g S_{Pol}(\phi,g) \coloneqq \int_\Sigma \Vert d\phi\Vert^2 dvol_g

where the derivative dϕΓ(T *Σϕ *TX)d \phi \in \Gamma(T^\ast \Sigma\otimes \phi^\ast T X), and where the norm \Vert - \Vert is given jointly by the metrics gg and μ\mu of Σ\Sigma and XX.

When both Σ\Sigma and XX are covered by single coordinate charts (σ a) a=0 p(\sigma^a)_{a = 0}^p and (x μ)(x^\mu), then this reads

S Pol(ϕ,g)= Σdetg(g abη μν( aϕ μ)( bϕ ν))dσ 0dσ p S_{Pol}(\phi,g) = \int_{\Sigma} \sqrt{det g} \left( g^{a b} \eta_{\mu \nu} (\partial_a \phi^\mu) (\partial_b \phi^\nu) \right) d \sigma^0 \cdots d \sigma^p

with a sum over repeated indices understood. Here detgdet g denotes the absolute value of the determinant of (g ab)(g_{a b}) (often written detg-det g 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

S˜:[Σ,X]×Met(Σ)×Γ(TΣTΣDensΣ) \tilde S \;\colon\; [\Sigma,X] \times Met(\Sigma)\times \Gamma(T \Sigma \otimes T \Sigma \otimes Dens \Sigma) \longrightarrow \mathbb{R}

which on a triple (ϕ,g,Λ)(\phi,g,\Lambda) is given by

S˜(ϕ,g,Λ) Σ(dvol(g)+Λ(|dϕ| 2g)), \tilde S(\phi,g,\Lambda) \coloneqq \int_\Sigma \left( dvol(g) + \Lambda( \vert d \phi \vert^2 - g) \right) \,,

where now |dϕ| 2\vert d\phi \vert^2 denotes the square norm only with respect to the metric on XX.

Because, on the one hand, the equations of motion induced by S˜\tilde S for variation of Λ\Lambda are

|dϕ| 2g=0 \vert d \phi \vert^2 - g = 0

and substituting that constraint back into S˜\tilde S gives the Nambu-Goto action. On the other hand, the equations of motion induced by S˜\tilde S for variation of gg are

12g 1dvol(g)Λ=0 \tfrac{1}{2}g^{-1}dvol(g) - \Lambda = 0

and substituting that back into S˜\tilde S gives

12 Σ(dϕ 2+(p1))dvol(g) \tfrac{1}{2}\int_\Sigma \left( \Vert d \phi\Vert^2 + (p-1) \right) dvol(g)

which is the Polyakov action with “cosmological constant(p1)(p-1).

(So the case where this cosmological constant correction disappears is p=1p = 1 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.