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.

Jeffrey Herschel Giansiracusa 2005-06-27