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 $T_{x_i}^{\ast }C$, 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.