Quantum K-Theory with Level Structure

Abstract

Given a smooth, complex projective variety X, one can associate to it numerical invariants by taking holomorphic Euler characteristics of natural vector bundles on the moduli spaces of stable maps to X. The study of these invariants is called quantum K-theory. Since K-theory is closely related to representation theory, it is natural to revisit quantum K-theory from the representation theoretic point of view. One of the important concepts in representation theory is level. In this thesis, we introduce the notion of level in quasimap theory and refer to it as the level structure.

This thesis consists of two parts. In the first part, we define level structures in quasimap theory as certain determinant line bundles over moduli spaces of quasimaps. By twisting with these determinant line bundles, we define K-theoretic quasimap invariants with level structure. An important case of this construction is quantum K-theory with level structure. We study the basic properties of level structures and show that quantum K-theory with level structure satisfies the same axioms as the ordinary, i.e., Givental-Lee's, quantum K-theory. In the genus-0 case, the invariants are encoded in an important generating series: the J-function. We characterize the values of the J-function in quantum K-theory with level structure. As an application of this characterization, we prove a mirror theorem for toric varieties. One surprising finding is that the mirrors of some of the simplest examples are Ramanujan's mock theta functions.

In the second part, we study the Verlinde/Grassmannian correspondence, which is a K-theoretic generalization of Witten's result. It relates the Verlinde algebra, a representation theoretic object, with the quantum K-invariants of the Grassmannian with level structure. To prove this correspondence, an important observation is that the Verlinde invariants and quantum K-invariants of the Grassmannian can be defined using the same gauged linear sigma model but with different stability conditions. In this thesis, we study the delta-stability condition. In particular, we construct the moduli spaces of delta-stable parabolic N-pairs and prove that they are equipped with canonical perfect obstruction theories. Using virtual structure sheaves, we define Verlinde type invariants over these moduli spaces and prove that they do not change when we vary the stability parameter delta.