Modularity of quantum invariants of Calabi-Yau threefolds

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Background: Calabi-Yau manifolds are central objects of research in both the mathematics of algebraic geometry and the physics of string theory. In the last few decades, subtle invariants of Calabi-Yau manifolds have arisen, often having parallel descriptions in math and physics. In math, the invariants are born from the geometry of various moduli spaces associated to the Calabi-Yau manifold; in physics, they arise out of various quantum field theories associated to the Calabi-Yau manifold. These invariants have turned out to have amazingly rich structure and surprisingly many connections to other branches of mathematics and have consequently become the objects of intense study in the last 15 years. The central instance of such invariants are the so-called Donaldson-Thomas invariants. Geometrically, these are subtle "counts" of sheaves on the manifold, in particular, they can count the ways that curves can sit inside and move around the manifold. Physically, these invariants correspond to counts of D-brane states in a certainly string theory. Roughly speaking, the counts tell about the particle spectrum of the associated quantum theory. In the last few years, a surprising and deep connection between Donaldson-Thomas theory and number theory has emerged. Through a series of computations and conjectures of several researchers, it has been noticed that the Donaldson-Thomas partition function is often given by a Jacobi modular form. The Donaldson-Thomas partition function of a Calabi-Yau manifold encodes all these geometric invariants into a single function, whereas Jacobi modular forms are functions with extraordinary symmetries which arise in number theory and have been studied in various forms for hundreds of years. This amazing connection between geometry and number theory appears to occur for a special class of Calabi-Yau manifolds, namely those which are elliptically fibered. The goal of this project is to refine and deepen our understanding of this conjectural phenomenon both by examining specific geometries and by developing new technology for computing partition functions. Having recently developed a new computational tool which is very effective for these sorts of geometries, I've seen tantalizing hints of how Jacobi forms emerge from the geometry. Fully understanding this phenomenon will shed new light on both the venerable subject of Jacobi modular forms and the newer subject of the geometry and physics of Calabi-Yau manifolds.