nLab Polyakov action

Idea

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.

Definition

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

$$S_{Pol} \;\colon\; [\Sigma,X]\times Met(\Sigma) \longrightarrow \mathbb{R}$$

which on a map $\phi \colon \Sigma \to X$ and a metric $g$ on $\Sigma$ is given by the integral

$$S_{Pol}(\phi,g) \coloneqq \int_\Sigma \Vert d\phi\Vert^2 dvol_g$$

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

$$S_{Pol}(\phi,g) = \int_{\Sigma} \sqrt{det g} \left( g^{a b} \eta_{\mu \nu} (\partial_a \phi^\mu) (\partial_b \phi^\nu) \right) d \sigma^0 \cdots d \sigma^p$$

with a sum over repeated indices understood. Here $det g$ denotes the absolute value of the determinant of $(g_{a b})$ (often written $-det g$ in the pseudo-Riemannian case.)

Properties

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

$$\tilde S \;\colon\; [\Sigma,X] \times Met(\Sigma)\times \Gamma(T \Sigma \otimes T \Sigma \otimes Dens \Sigma) \longrightarrow \mathbb{R}$$

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

$$\tilde S(\phi,g,\Lambda) \coloneqq \int_\Sigma \left( dvol(g) + \Lambda( \vert d \phi \vert^2 - g) \right) \,,$$

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

$$\vert d \phi \vert^2 - g = 0$$

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

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

and substituting that back into $\tilde S$ gives

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

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

(So the case where this cosmological constant correction disappears is $p = 1$ corresponding to the string.)

Related concepts

• minimal surface

• bosonic string

• Liouville theory

• Nambu-Goto action

References

The Polyakov action was introduced in:

• Alexander Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207-210 [doi:10.1016/0370-2693(81)90743-7, pdf]

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

• J. A. Nieto, Remarks on Weyl invariant p-branes and Dp-branes (arXiv:hep-th/0110227)

See also

• Wikipedia, Polyakov action