# nLab dilaton

Contents

## Surveys, textbooks and lecture notes

#### Gravity

gravity, supergravity

# Contents

## Idea

Generally in a context of Kaluza-Klein compactification a dilaton is a fields in a 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 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 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 moduli stabilization.

Specifically in string theory, together with the field of gravity and the Kalb?Ramond field?, the dilaton field is one of the three massless bosonic fields that appears in effective background 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.

## Details

### Action functional of dilaton gravity

Let $X$ be a compact smooth manifold. Write $Conf$ for the configuration space of pseudo-Riemannian metrics $g$ (the graviton) and of smooth functions $f$ (the dilaton ) on $X$.

The action functional for dilaton gravity is

$S : Conf \to \mathbb{R}$
$S : (g,f) \mapsto \int_X e^{-f}(R_g dvol_g+ d f \wedge \star_g d f) \,,$

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.

### 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 ∞-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

• Mariana Graña, Ruben Minasian, Michela Petrini, Daniel Waldram, T-duality, generalized geometry and non-geometric backgrounds (arXiv)

## Applications

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

field (physics)

standard model of particle physics

force field gauge bosons

scalar bosons

flavors of fundamental fermions in the
standard model of particle physics
generation of fermions1st generation2nd generation3d generation
quarks
up-typeup quarkcharm quarktop quark
down-typedown quarkstrange quarkbottom quark
leptons
chargedelectronmuontauon
neutralelectron neutrinomuon neutrinotau neutrino

hadron (bound states of the above quarks)

solitons

minimally extended supersymmetric standard model

superpartners

bosinos:

dark matter candidates

Exotica

auxiliary fields

## References

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

David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

Last revised on July 15, 2017 at 09:40:40. See the history of this page for a list of all contributions to it.