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 is named after

where it is discussed for the bosonic string and related to Liouville theory, and

where it is generalized to the superstring, but in fact (cf. Polyakov (2008), p. 3) it originates already in discussion of the spinning string in:

Further early discussion of the Polyakov action:

and further discussion of the related Liouville theory:

Detailed discussion of the relation to the Nambu-Goto action and the Dirac-Born-Infeld action


and mathematical details:

Discussion of the Polyakov action and Nambu-Goto action on worldsheets with boundary (i.e. in the generality of open strings) and cast in BV-BRST formalism:

See also

Last revised on June 19, 2023 at 10:21:57. See the history of this page for a list of all contributions to it.