Contents

Idea

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.

Details

Action functional of dilaton gravity

Let $X$ be a compact smooth manifold. Write $\mathrm{Conf}$ 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 ${\mathrm{dvol}}_g$ is the volume form of $g$.

For $f=0$ this reduces to the Einstein-Hilbert action. For $f=\mathrm{const}$ 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.

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)