(0,2) Mirror Symmetry and Heterotic Gromov-Witten Invariants

Shortly after the construction of the ten-dimensional heterotic string theories, it was realized that a compactification of these theories on Calabi-Yau manifolds could yield four-dimensional supersymmetric Poincaréinvariant vacua with the massless spectrum consisting of minimal supergravity coupled to a chiral non-abelian gauge theory. This was a remarkable development in theoretical physics, as it connected a heterotic string theory—believed to be a consistent theory of quantum gravity—to a chiral gauge theory remarkably similar to the Standard Model. Despite this beautiful relation, it was understood that a number of issues remained to be addressed. For example, it was difficult to produce either the Standard Model gauge group or a grand unified model with couplings that would lead to the Standard Model. Moreover, the construction was perturbative in two senses: the results required small string coupling and the large radius limit, the former being a statement about string perturbation theory, while the latter requiring the compactification geometry to be a smooth manifold with volume large compared to the string length. Both of these issues are intimately tied to the existence of moduli—parameters introduced by the compactification, such as Kähler and complex structures on the Calabi-Yau manifold. What is the structure of this moduli space? How do physical quantities depend on the moduli? Are there heterotic compactifications without a large radius limit? Can one obtain a Standard Model gauge group or a favorable grand unified theory? Does an understanding of these issues teach us something about non-perturbative effects in the heterotic string? The answers to these questions inevitably lead to new mathematical structures. Broadly speaking, the purpose of the workshop was to bring together researchers who are developing the mathematical structures and applying them to the physical questions. Major themes of the workshop were: