# What is the physical significance of the dilaton in string theory?

Strings always have a dilaton in their spectrum. Its a scalar field (so presumably no spin), and so far a hypothetical particle. What is its physical significance?

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roughly speaking, it describes the (average?) scale of the compactification –  lurscher Jun 20 '12 at 22:39

The gravitational, massless, bosonic sector of the string effective action contains the metric tensor, and at least one more fundamental field φ, the dilaton. By comparing the Einstein’s (d+1) dimensional Einstein-Hilbert action with the effective tree level action mentioned before a relation between the effective string coupling (also fixing the G constant) and the dilaton field becomes manifest. The dilaton in string theory also can be rescaled to absorb trivial volume factors associated with compact spaces, but are also present in the non-compacted string models. The dilaton is a fundamental scalar field in closed string theory. The effective gravity equations in string theory includes the gravi-dilaton part that looks very similar to Brans-Dicke scalar-tensor theory of gravity (This is valid only at tree level). The dilaton field, as mention before, controls the string coupling constant so the genus expansion in string theory is directly related to the dilaton field and to corrections to General Relativity. There is also the possibility of a dilaton potential in noncritical dimensions. This creates the possibility that the dilaton field expectation value be fixed at a local minimum (probably in a non-perturbative regime), fixing the coupling between strings. The dilaton field is then an essential component of all superstrings models, and thus of the cosmological scenarios based on effective string actions. Chapter 9 in Maurizios Gasperini’s String Cosmology book is a very nice introduction to dilaton phenomenology and its importance in cosmology.

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In string theory, the dilaton field $\phi$ describes how strongly strings couple to one another. More precisely, in perturbation theory, the open string coupling constant is the exponential of the dilaton's expectation value $g_s = e^{\langle \phi\rangle}$.

The word dilaton is also used to refer to a quantum of the dilaton field. This is just a scalar particle.

In Kaluza-Klein theory, the word dilaton is sometimes used for the scalar field which arises when you compactify a metric, but the more word 'radion' is more standard; it's the diagonal component of the metric in the compactified direction. String theory mixes these ideas up to some degree. IIA string theory results from compactifying M-theory on a circle, and the dilaton field of IIA is the radion of M-theory on this circle.

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The dilaton in type IIA string theory is a measure of the local radial size of the 11th circular dimension in the M-theory description. The IIA perturbation series is an expansion around the zero radius limit, where the M-theory 2-brane wrapped on the circle becomes long and light, and then becomes the IIA string, so that the dilaton field is then completely local, and its geometrical origin is somewhat obscure.

The reason this is significant is because there is no dilaton in the M-theory, the low-energy excitations are the simplest maximally supersymmetric supermultiplet: an 11-dimensional graviton, a three-form gauge field, and a spinor. The 11-dimensional theory obeys the equivalence principle, but it isn't a string perturbation theory. For there to be a string perturbation theory, you need a long light string, a black hole charged with a two-form whose excitations become much lighter than anything else. The circle compactification does this, but also the torus compactification, or a line.

In type IIB string theory, the dilaton comes together with a partner which together parametrize the shape of the little torus of the M-theory compactification, which can be interpreted as T-dualized IIA. In heterotic strings, it's the distance between the Horava-Witten end-of-the-universe orbifold-walls in the 11-th dimension. In any way of looking at it, the dilaton comes from the reduction of dimensions where string theories emerge, so it isn't a fundamental feature of stringy gravity in any deep sense.

This phenomenon can be seen simply in plain Kaluza-Klein theory. When you compactify a 5 dimensional theory on a circle, in addition to a 4d graviton and a KK photon, you get a scalar from the 55 component of the metric tensor, which, by definition, measures the size of the extra dimension. This extra scalar has gravitational couplings from the reduction, and together with the graviton, it makes a 4-d scalar-tensor theory.

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