Integrals of stable envelopes for cotangent bundles to Grassmannians

TL;DR

This paper computes integrals of stable envelopes for cotangent bundles of Grassmannians, revealing combinatorial formulas for certain integers that relate to mirror symmetry and curve counting phenomena.

Contribution

It introduces a new combinatorial formula for integrals of stable envelopes in cotangent bundles of Grassmannians, connecting to mirror symmetry and quiver varieties.

Findings

Derived a combinatorial formula for integers in stable envelope integrals.

Connected non-equivariant limits to curve counting in mirror symmetry.

Explored combinatorics of integers for higher Grassmannian cases.

Abstract

We consider cohomological stable envelopes for a natural torus action $\mathsf{T}$ on $X=T^*Gr(k,n)$, introduced by Maulik-Okounkov. We define the $\mathbb{C}^*_\hbar$-equivariant integral of the stable envelope using equivariant localization over the subtorus $\mathbb{C}^*_\hbar\subset\mathsf{T}$, and compute the integral as a non-equivariant limit of the localization over the full torus, $\mathsf{T}$. The integral of such a class is an integer times a power of $\hbar$, and the main result of this paper is a combinatorial formula for these integers. In 3d mirror symmetry, these non-equivariant limits are expected to reflect some curve counting phenomena on the 3d mirror dual, $X^\vee$. When $k=1$, we obtain the binomial coefficients, and we study some of the combinatorics of the integers for higher $k$, which haven't appeared in the literature before. We give some conjectures and interpretations on extending this phenomena to type A quiver and bow varieties.