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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 of string theory.


Action functional of dilaton gravity

Let XX be a compact smooth manifold. Write ConfConf for the configuration space of pseudo-Riemannian metrics gg (the graviton) and of smooth functions ff (the dilaton ) on XX.

The action functional for dilaton gravity is

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

where R gR_g is the Riemann curvature scalar of gg and g\star_g the Hodge star operator and dvol gdvol_g is the volume form of gg.

For f=0f = 0 this reduces to the Einstein-Hilbert action. For f=constf = 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)


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

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

Revised on June 13, 2013 00:44:04 by Urs Schreiber (