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\begin{document}

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\section*{Polyakov action}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToNambuGotoAction}{Relation to Nambu-Goto action}\dotfill \pageref*{RelationToNambuGotoAction} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{Polyakov action functional} or \emph{energy functional} is a standard [[kinetic action|kinetic]] [[action functional]] for [[sigma models]] with [[worldvolume]] $(\Sigma,g)$ and [[target spacetime]] $(X,\mu)$ ([[pseudo-Riemannian manifold|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 [[classical field theory|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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let

\begin{itemize}%
\item $p \in \mathbb{N}$ (for [[p-brane]] dynamics);


\item $(X,\mu)$ a [[pseudo-Riemannian manifold]] ([[target spacetime]]);


\item $\Sigma$ a [[compact topological space|compact]] [[smooth manifold]] of dimension $(p+1)$ ([[worldvolume]]).


\item $[\Sigma,X]$ the [[diffeological space]] of [[smooth functions]] $\Sigma \to X$.


\item $Met(\Sigma)$ the [[moduli space]] of ([[pseudo-Riemannian metrics|pseudo]])[[Riemannian metrics]] $g$ on $\Sigma$.



\end{itemize}
The Polyakov action is the [[smooth function]]

\begin{displaymath}
S_{Pol}
  \;\colon\;
  [\Sigma,X]\times Met(\Sigma)
  \longrightarrow
  \mathbb{R}
\end{displaymath}
which on a map $\phi \colon \Sigma \to X$ and a metric $g$ on $\Sigma$ is given by the [[integral]]

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

\begin{displaymath}
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
\end{displaymath}
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.)

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToNambuGotoAction}{}\subsubsection*{{Relation to Nambu-Goto action}}\label{RelationToNambuGotoAction}

The Polyakov action with a suitable worldvolume cosmological constant term added is classically equivalent to the [[Nambu-Goto action]] (e.g. \hyperlink{Nieto01}{Nieto 01, section 2}).

This [[shell|on-shell]] equivalence is exhibited by the smooth function

\begin{displaymath}
\tilde S
  \;\colon\;
  [\Sigma,X] \times Met(\Sigma)\times \Gamma(T \Sigma \otimes T \Sigma \otimes Dens \Sigma)
  \longrightarrow \mathbb{R}
\end{displaymath}
which on a triple $(\phi,g,\Lambda)$ is given by

\begin{displaymath}
\tilde S(\phi,g,\Lambda)
  \coloneqq
  \int_\Sigma
  \left(
    dvol(g) + \Lambda( \vert d \phi \vert^2 - g)
  \right)
  \,,
\end{displaymath}
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

\begin{displaymath}
\vert d \phi \vert^2 - g = 0
\end{displaymath}
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

\begin{displaymath}
\tfrac{1}{2}g^{-1}dvol(g) - \Lambda = 0
\end{displaymath}
and substituting that back into $\tilde S$ gives

\begin{displaymath}
\tfrac{1}{2}\int_\Sigma 
  \left(
    \Vert d \phi\Vert^2
    +
    (p-1)
  \right)
  dvol(g)
\end{displaymath}
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]].)

\hypertarget{references}{}\subsection*{{References}}\label{references}

The Polyakov action was introduced in

\begin{itemize}%
\item [[Alexander Polyakov]],

\emph{Quantum geometry of bosonic strings} , Phys. Lett. B103 (1981) 207;

\emph{Quantum geometry of fermionic strings} , Phys. Lett. B103 (1981) 211



\end{itemize}
Detailed discussion of the relation to the [[Nambu-Goto action]] and the [[Dirac-Born-Infeld action]] is in

\begin{itemize}%
\item J. A. Nieto, \emph{Remarks on Weyl invariant p-branes and Dp-branes} (\href{http://arxiv.org/abs/hep-th/0110227}{arXiv:hep-th/0110227})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Polyakov_action}{Polyakov action}}

\end{itemize}
[[!redirects Polyakov action functional]]

[[!redirects energy functional]]



\end{document}