Abstract: An elliptic cohomology theory is an even periodic multiplicative generalized cohomology theory whose associated formal group is the formal completion of an elliptic curve. It is at the intersection of a variety of areas in mathematics, including algebraic topology, algebraic geometry, mathematical physics, representation theory and number theory. From different perspectives we have different interpretations of elliptic cohomology, which gives us different ways to study it.
One approach is via a representing object of elliptic cohomology. Other than elliptic spectrum, a good choice is its geometric object. For example, the geometric object of K-theory is vector bundle. As a higher version of K-theory, the geometric object of elliptic cohomology should be “2-vector bundle”. Analogous to the relation between vector bundles and group representations, a 2-representation of a 2-group is a 2-vector bundle at a point. We glue the local (equivariant) 2-vector bundles together by higher sheafification and obtain the 2-stack of (equivariant) 2-vector bundles. Currently I'm exploring further the relation between this model of 2-vector bundles and elliptic cohomology.
