nLab dilaton

Idea

Generally in a context of Kaluza-Klein compactification a dilaton is a field 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 a small but approximately constant value in order to yield 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.

Related concepts

• Liouville theory

• JT-gravity

fields and particles in particle physics

and in the standard model of particle physics:

force field gauge bosons

• photon - electromagnetic field (abelian Yang-Mills field)

• W, Z, B-boson - electroweak field (Yang-Mills field)

• gluon - strong nuclear force (Yang-Mills field)

• graviton - field of gravity

• infraparticle

scalar bosons

• Higgs boson, inflaton (scalar field)

matter field fermions (spinors, Dirac fields)

flavors of fundamental fermions in the standard model of particle physics:
generation of fermions	1st generation	2nd generation	3d generation
quarks ($q$)
up-type	up quark ($u$)	charm quark ($c$)	top quark ($t$)
down-type	down quark ($d$)	strange quark ($s$)	bottom quark ($b$)
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino
bound states:
mesons	light mesons: pion ($u d$) ρ-meson ($u d$) ω-meson ($u d$) f1-meson a1-meson	strange-mesons: ϕ-meson ($s \bar s$), kaon, K*-meson ($u s$, $d s$) eta-meson ($u u + d d + s s$) charmed heavy mesons: D-meson ($u c$, $d c$, $s c$) J/ψ-meson ($c \bar c$)	bottom heavy mesons: B-meson ($q b$) ϒ-meson ($b \bar b$)
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

(also: antiparticles)

effective particles

• Goldstone bosons

hadrons (bound states of the above quarks)

• meson

  • scalar meson

    • σ-meson

    • pion

    • kaon

    • D-meson

    • B-meson

  • vector meson

    • ω-meson

    • ρ-meson

    • f1-meson

    • a1-meson

    • b1-meson

    • h1-meson

    • K*-meson

  • tensor meson

  • quarkonium

    • charmonium

    • ϒ-meson

• exotic meson

  • XYZ meson

  • tetraquark

• baryon

  • nucleon

    • proton

    • neutron

    • chemical element

      • carbon

      • nitrogen

  • Lambda baryon

  • pentaquark

solitons

• Skyrmion

• caloron

• instanton

in grand unified theory

• leptoquark

• Z'-boson

minimally extended supersymmetric standard model

superpartners

bosinos:

• gravitino - field of supergravity (Rarita-Schwinger field)

• gaugino - super Yang-Mills field

• gluino

sfermions:

• squark

dark matter candidates

• WIMP

• axion

Exotica

• preon

• graviphoton

• dilaton

• monopole

• dual graviton

• giant graviton

auxiliary fields

• ghost field

• antifield

• antighost field, Nakanishi-Lautrup field

References

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

• Eric D'Hoker, String theory in Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey,

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)

Relation of the string theory-dilaton to the double dimensional reduction of the quantum M2-brane:

• Krzysztof A. Meissner, Hermann Nicolai, Fundamental Membranes and the String Dilaton [arXiv:2208.05822]