Witten classes

There are $n$ natural line bundles $L_i$ over $\mathcal{M}_{g,n}$ (or $\overline{\mathcal{M}}_{g,n}$); the fibre at $C$ of $L_i$ is the cotangent space $T^*_{x_i}C$, where $x_i$ is the $i^{th}$ marked point in $C$. The Witten class $\psi_i$ is then the first Chern class of $L_i$. These live naturally in the cohomology of either the compactified or non-compactified moduli space.