Title: Differential equations of the elliptic genus of Calabi-Yau manifolds.

Annotation: The elliptic genus of a manifold is an important generating function studied in topology, in the theory of automorphic forms, and in string theory. The elliptic genus of a complex manifold of dimension d with trivial first Chern class (e.g., of a Calabi-Yau manifold) turns out to be a Jacobi modular form of weight 0 and index d/2 with integer Fourier coefficients. The Jacobi form is a holomorphic function of two variables. The elliptic genus of a Calabi-Yau 3-fold is explicitly expressed in terms of the classical Jacobi theta functions. It follows immediately that the elliptic genus satisfies a first-degree differential equation with respect to the heat operator.

The main hero of the mini-course is the odd Jacobi function, discovered in the first half of the 19th century, for which we will rewrite the elliptic equation in terms of the integral Heisenberg group. The course is planned to start with a very brief introduction to modular and Jacobi forms with a clear and correct description of the Jacobi theta function in two variables. Next, we will analyze the definition of the elliptic genus and establish its automorphic properties using the Jacobi triple product formula, formal calculus with Chern roots of complex manifolds, and the Riemann-Roch-Hirzebruch theorem. In the third part of the course, we will derive explicit differential equations for the elliptic genus of Calabi-Yau manifolds of complex dimensions 2, 3, 4, and 5.