# nLab Polyakov action

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## 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

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 Nambu goto action with “cosmological constant$(p-1)$.

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

## References

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