Rationality of the K-theoretical capped vertex function for Nakajima quiver varieties

TL;DR

This paper proves the rationality of the capped vertex function with descendents for Nakajima quiver varieties, extending previous proofs and revealing trivial monodromy, with implications for wall-crossing formulas.

Contribution

It generalizes the rationality proof of the capped vertex function to all Nakajima quiver varieties using tautological classes, and establishes a GIT wall-crossing formula.

Findings

Proves rationality of the capped vertex function for all Nakajima quiver varieties.

Shows the monodromy of the capped vertex function is trivial.

Provides a GIT wall-crossing formula involving quantum difference and fusion operators.

Abstract

In this paper we prove the rationality of the capped vertex function with descendents for arbitrary Nakajima quiver varieties with generic stability conditions. We generalise the proof given by Smirnov to the general case, which requires to use techniques of tautological classes rather than the fixed-point basis. This result confirms that the "monodromy" of the capped vertex function is trivial, which gives a strong constraint for the monodromy of the capping operators. We will also provide a GIT wall-crossing formula for the capped vertex function in terms of the quantum difference operators and fusion operators.