Topological modular forms and loop modules over lattice algebras

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Elliptic cohomology and topological modular forms were originally designed to provide a mathematical basis for conformal field theories in physics. The project’s objective is the construction of cocycles in elliptic cohomology in terms of sheaves of certain lattice vertex algebra modules. An example of such a cocycle is provided by the chiral de Rham complexes from which it is known how to recover the elliptic genus or the Witten genus. Using the noti- on of loop modules of Borisov and its interpretation by Kaparanov and Va- sserot, these elliptic objects should generalize from tangent bundles to all vector bundles as representatives of tmf(p)-Euler classes. The homotopy theory of these objects will be studied.