The spectrum of topological modular forms (TMF) has become an object of great interest in homotopy theory, both because it provides the natural target of the Witten genus and because it encodes information about the height 2 part of the stable homotopy groups of spheres.
Building on the work of Borisov–Libgober and Ando–French–Ganter, one can associate to a Calabi–Yau manifold a two-variable elliptic genus taking values in Jacobi forms.
In this talk, I will introduce the spectrum of topological Jacobi forms, constructed using the S^1-equivariant TMF developed by Lurie and Gepner–Meier, and discuss its basic properties and its application to cyclic equivariant TMF.
Recent results by Lin–Yamashita and Carmeli–Luecke show that the topological Jacobi form admits an E_\infty-orientation MSU to TJF_\infty, which refines the two-variable elliptic genus originally constructed by Borisov–Libgober and Ando–French–Ganter.