Partitions and quasimodular forms: Variations on the Bloch–Okounkov theorem

Abstract

There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets—certain normalized generating series—are quasimodular forms. This provides a tool for enumerative geometers to show that certain generating series of Gromov–Witten invariants or Hurwitz numbers are quasimodular forms.

In this thesis, our aim is to study graded algebras of functions on partitions such that all homogeneous elements of the algebra have quasimodular forms as q-brackets. That is, we give explicit constructions answering the following three main questions in the affirmative: (I) Are there other graded algebras than the algebra of shifted symmetric functions such that the q-brackets of its elements are quasimodular forms? (II) Given a congruence subgroup, is there an (even larger) algebra of functions for which the q-bracket is a quasimodular form for this subgroup? (III) What is the class of functions for which the q-bracket is not only a quasimodular form, but even a modular form?

The answer to the second and third question follows by studying the following question of independent interest: (IV) What is the modular or quasimodular behavior of the Taylor coefficients of meromorphic quasi-Jacobi forms?

This question brings us back to the origin of the results on the q-bracket in enumerative geometry. Namely, we show that the solutions to a differential equation originating from the study of K3 surfaces are quasi-Jacobi forms and describe their transformation.