Euler characteristic of stable envelopes

TL;DR

This paper establishes a formula connecting the equivariant Euler characteristic of K-theoretic stable envelopes to the index vertex in the cotangent bundle of the full flag variety, revealing a mirror symmetry-based relationship.

Contribution

It provides the first explicit formula relating stable envelopes' Euler characteristics to the index vertex, demonstrating the rationality of the index vertex as a power series expansion.

Findings

The index vertex is the power series expansion of a rational function.

The formula is derived from 3d mirror self-symmetry.

Expected generalization to other mirror-related varieties.

Abstract

In this paper we prove a formula relating the equivariant Euler characteristic of $K$-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.