Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by


$$
g\_{mn} = \left(
\begin{array}{cc}
 g\_{\mu\nu} & g\_{\mu 5} \\
 g\_{5\nu} & g\_{55} \\
\end{array}
\right) 
$$


where $g\_{\mu\nu}$ corresponds to the ordinary four dimensional metric and $g\_{\mu 5}$ is the ordinary four dimensional Maxwell gauge field, $g\_{55}$ is the dilaton field.


As there is one dilaton for one extra dimension, I naively would expect that the zero mass states of closed string theory, which can be written as


$$
\sum\limits\_{I.J} R\_{I.J} a\_1^{I\dagger} \bar{a}\_1^{I\dagger} ¦p^{+},\vec{p}\_T \rangle
$$


and the square matrix $R\_{I.J}$ can be separated into a symmetric traceless part corresponding to the graviton field, an antisymmetric part corresponding to a generalized Maxwell gauge field, and the trace which corresponds to the dilaton field.


Why is there only one dilaton field given by the trace of $R\_{I.J}$, instead of $22$ dilaton fields corresponding to $22$ extra dimensions of closed string theory which has critical dimension $D = 26$? For example, why are there not $22$ dilaton fields needed to parameterize the shape of the 22 extra dimensions if they are compactified?



Why does closed string theory have only one dilaton field instead of $22$?
You would actually *not* expect 22 dilatons. Let me try to explain.


As you have pointed out, a putative field theory limit of the closed bosonic string would consist of a metric, a 2-form (which is the potential for a 3-form) and a dilaton.


Let us assume that such a theory exists and let us dimensionally reduce to four dimensions à la Kaluza-Klein.


The 26-dimensional dilaton $\Phi$ gives a scalar field in four dimensions. The 26-dimensional metric $g\_{MN}$ gives the following four-dimesional fields: a metric $g\_{\mu\nu}$, 22 gauge fields $g\_{\mu n}$ and $\binom{23}2 = 253$ scalars $g\_{mn}$. Finally, the 26-dimensional 2-form $B\_{MN}$ gives the following: $\binom{22}{2} = 231$ scalars $B\_{mn}$, $22$ gauge fields $B\_{\mu n}$ and $\binom{4}{2} = 6$ scalars dual to the 2-forms $B\_{\mu\nu}$. (In four dimensions, 2-forms can be dualised to scalars: via $\star dB = d\varphi$.)


So from a 4-dimensional perspective the role of the “dilaton” is now the $22 \times 22$ symmetric matrix $g\_{mn}$.


This does not imply that you should expect any number of scalars in the massless sector of the closed string. The massless sector of the string theory corresponds to a field theory in 26 dimensions and in order to interpret it in 4-dimensional terms, requires dimensional reduction.


