  Based on the combinatorial description of the moduli spaces of curves
provided by Strebel differentials, Witten and Kontsevich have introduced
combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured
that these can be expressed in terms of Mumford-Morita-Miller classes. It is
argued that this link should be provided by a theorem of Di Francesco, Itzykson
and Zuber which relates the derivatives of the Witten-Kontsevich partition
function with respect to one set of variables to the derivatives with respect
to the other set of variables. Two things are shown. First of all that this
works in complex codimension 1. Secondly that in all the cases when it has been
possible to make the Di Francesco, Itzykson and Zuber correpondence explicit
this translates into identities of the type $$
\int_{W_{(m_0,m_1,m_2,\dots),n}}\prod\psi_i^{d_i}
=\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prod\psi_i^{d_i} $$
where the $X_{(m_0,m_1,m_2,\dots),n}$ are explicit polynomials in the
algebro-geometric classes and the $\psi_i$ are the Chern classes of the point
bundles, for any choice of $d_1,\dots,d_n$.
