Lectures on K-theoretic computations in enumerative geometry

TL;DR

This paper explores advanced K-theoretic computations in enumerative geometry, proving key conjectures and revealing new properties in quantum K-theory of Nakajima quiver varieties and threefolds.

Contribution

It proves the main conjecture from arXiv:hep-th/0412021 and the conjecture from arXiv:1404.2323 for smooth curves, and establishes large framing vanishing and the relation of shift operators to qKZ.

Findings

Proof of main conjecture in quantum K-theory

Establishment of large framing vanishing property

Identification of shift operators with qKZ operators

Abstract

These are notes from my lectures on quantum K-theory of Nakajima quiver varieties and K-theoretic Donaldson-Thomas theory of threefolds given at Columbia and Park City Mathematics Institute. They contain an introduction to the subject and a number of new results. In particular, we prove the main conjecture of arXiv:hep-th/0412021 and the conjecture of arXiv:1404.2323 in the simplest case of reduced smooth curves. We also prove the the absence of quantum corrections to the capped vertex with descendents for sufficiently large framing (and polarization), which is a property we call large framing vanishing. The shift operators for minuscule shift are shown to be given by qKZ operators, which is a K-theoretic analog of the result of arXiv:1211.1287.