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black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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.
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
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.
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
The gradient flow of the action functional for dilaton gravity is essentially Ricci flow.
standard model of particle physics
photon - electromagnetic field (abelian Yang-Mills field)
matter field fermions (spinors, Dirac fields)
hadron (bound states of the above quarks)
minimally extended supersymmetric standard model
superpartner gauge field fermions
Exotica
The derivation of dilaton gravity as part of the effective QFT of string theory is discussed for instance aroung page 911 of