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\begin{document}

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\section*{dilaton}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{action_functional_of_dilaton_gravity}{Action functional of dilaton gravity}\dotfill \pageref*{action_functional_of_dilaton_gravity} \linebreak
\noindent\hyperlink{global_cohomological_description}{Global cohomological description}\dotfill \pageref*{global_cohomological_description} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Generally in a context of [[Kaluza-Klein compactification]] a \emph{dilaton} is a [[field (physics)|fields]] in a [[theory (physics)|theory]] on a lower-dimensional [[spacetime]] which is a component of the field of [[gravity]] on a higher dimensional spacetime, in that it is part of the [[pseudo-Riemannian metric|metric]] of the [[fiber]]-spaces on which the KK-compactification takes place. Specifically for KK-compactification on a [[circle]] fiber ``the dilaton'' (or ``radion'') is the lowest [[Fourier series|Fourier]] mode of the metric of the circle, hence is the [[length]] (circumference) (or [[radius]], up to a factor) of the circle fiber.

The subtlety in [[Kaluza-Klein theory]] is that the dilaton should have small but approximately constant value in order to yields [[effective field theory]] gravity coupled to [[gauge theory]] in lower dimensions from pure gravity in higher dimensions. This is the problem of \emph{[[moduli stabilization]]}.

Specifically in [[string theory]], together with the field of [[gravity]] and the [[Kalb?Ramond field]], the \emph{dilaton field} is one of the three massless [[boson|bosonic fields]] that appears in [[effective QFT|effective background]] [[quantum field theory|quantum field theories]]. For [[type IIA string theory]] this may be interpreted as the Kaluza-Klein dilaton in the above sense, arising from [[11-dimensional supergravity]] ([[M-theory]]) compactified on a circle. Similarly for [[heterotic string theory]] and [[Horava-Witten theory]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{action_functional_of_dilaton_gravity}{}\subsubsection*{{Action functional of dilaton gravity}}\label{action_functional_of_dilaton_gravity}

Let $X$ be a [[compact space|compact]] [[smooth manifold]]. Write $Conf$ for the [[configuration space]] of [[pseudo-Riemannian metric]]s $g$ (the [[graviton]]) and of [[smooth function]]s $f$ (the \emph{dilaton} ) on $X$.

The [[action functional]] for [[dilaton gravity]] is

\begin{displaymath}
S : Conf \to \mathbb{R}
\end{displaymath}
\begin{displaymath}
S : (g,f) \mapsto \int_X e^{-f}(R_g dvol_g+ d f \wedge \star_g d f) 
  \,,
\end{displaymath}
where $R_g$ is the [[Riemann curvature]] scalar of $g$ and $\star_g$ the [[Hodge star operator]] and $dvol_g$ is the [[volume form]] of $g$.

For $f = 0$ this reduces to the [[Einstein-Hilbert action]]. For $f = const$ it is still a multiple of the Einstein-Hilbert action functional.

The [[gradient flow]] of this functional is [[Ricci flow]].

\hypertarget{global_cohomological_description}{}\subsubsection*{{Global cohomological description}}\label{global_cohomological_description}

The global nature of the gravitational field and the Kalb--Ramond field are well understood conceptually: the gravitational field is a [[pseudo-Riemannian metric]] and the Kalb--Ramond field is a cocycle in third integral [[differential cohomology]] (for instance realized by a cocycle in [[Deligne cohomology]] or by a [[bundle gerbe]] with connection).

In [[generalized complex geometry]], both these fields are shown to be unified as one single [[schreiber:∞-Lie algebroid valued differential forms|∞-Lie algebroid valued]] form field: a connection on a [[standard Courant algebroid]] (as described in more detail there).

While it was clear that the diaton field is locally just a real-valued function, is formal global identification has not been understood in an analogous manner for a long time.

But a proposal for a precise conceptual identification of the dilaton as a structure appearing in the context of [[generalized complex geometry]] is in

\begin{itemize}%
\item Mariana Gra\~n{}a, Ruben Minasian, Michela Petrini, Daniel Waldram, \emph{T-duality, generalized geometry and non-geometric backgrounds} (\href{http://arxiv.org/abs/0807.4527}{arXiv})

\end{itemize}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

The [[gradient flow]] of the [[action functional]] for [[dilaton gravity]] is essentially [[Ricci flow]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

[[!include fields and quanta - table]]

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

The derivation of dilaton gravity as part of the [[effective QFT]] of [[string theory]] is discussed for instance aroung page 911 of

\begin{itemize}%
\item [[Eric D'Hoker]], \emph{String theory} in [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], (eds.) \emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999. (\href{http://www.math.ias.edu/qft}{web version})

\end{itemize}
[[!redirects dilaton field]]

[[!redirects dilaton gravity]]



\end{document}