Two Approaches towards elliptic cohomology

2023.03.24（星期五），15:00-16:00

点：腾讯会议：682-969-703密码：0324

要：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. In the talk I will present two approaches to study elliptic cohomology.

One is an idea indicated by Witten that the elliptic cohomology of a space is related to the circle-equivariant K-theory of the free loop space of it. Motivated by this, I constructed quasi-elliptic cohomology during my PhD. It is closely related to Tate K-theory. I formulate the stringy power operation of this theory. Applying that I prove the finite subgroups of Tate curve can be classified by the Tate K-theory of symmetric groups modulo a certain transfer ideal. Recently, together with Young, we construct twisted Real quasi-elliptic cohomology as the twisted KR-theory of loop groupoids. The theory systematically incorporates loop rotation and reflection. After establishing basic properties of the theory, we construct Real analogues of the stringy power operation of quasi-elliptic cohomology as well as its twisted elliptic Pontryagin character, to further study its relation with field theories.

The other 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.