ABSTRACT
This chapter proposes a symplectic invariant associated with Calabi-Yau manifolds, and discusses the particular case of the K3 manifold. The invariant counts the number of special states, BPS in the string theory associated to the Calabi-Yau manifold, and for K3 we relate this counting to the number of rational curves with double points on K3. As the number of such rational curves has only been computed up to genus equal to six, this provides an infinite set of conjectures in enumerative geometry. Many geometric invariants enjoy descriptions in terms of physical theories. Cohomology classes, Donaldson invariants, link invariants and questions in enumerative geometry have all arisen in quantum field theory and string theory. String theory, a fundamental theory of physics, has as its low-energy description a quantum field theory including gravity. The chapter considers in detail the example of a two-complex-dimensional Calabi-Yau manifold, K3, which can arise either through string theory compactifications or by considering a six-dimensional spacetime.
