Integrals of stable envelopes for cotangent bundles to Grassmannians

TL;DR

This work defines and analyzes the $\mathbb{C}^*_{\hbar}$-equivariant integral of cohomological stable envelopes for the cotangent bundle of the Grassmannian, $X = T^*Gr(k,n)$, using equivariant localization and a limit from the full torus. It proves a closed-form, purely combinatorial formula $F(p_I)$ for the integral, showing it is an integer times a power of $\hbar$, and recovers binomial coefficients in the $k=1$ case; for higher $k$ the resulting integers form rich combinatorial structures (e.g., the $Gr_2$-simplex) with new recurrence relations and connections to Narayana numbers. The paper also develops a path-interpretation conjecture via $\mathbb{V}(\lambda)$ that matches the integrals in the limit $a \to 0$, and discusses potential extensions to type A quiver and bow varieties, including when nonequivariant limits exist and their relation to Hanany–Witten equivalence. These results illuminate how non-equivariant limits of stable-envelope integrals encode curve-counting data in 3D mirror symmetry and suggest new combinatorial objects generalizing binomial coefficients.

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.

Figures (1)

• Figure 1: Moment graph of $\mathsf{T}=(\mathop{\mathrm{\mathbb{C}}}\nolimits^*)^4\times \mathop{\mathrm{\mathbb{C}}}\nolimits^*_\hbar$ action on $T^*Gr(2,4)$. The vertex $\langle ij\rangle$ represents the fixed point span$(e_i,e_j)$.

Theorems & Definitions (35)

• Example 1.1
• Theorem : Theorem \ref{['lim-calc']}
• Conjecture : \ref{['bow']}
• Proposition : Proposition (\ref{['chamber-dependence']})
• Theorem 2.1
• Example 2.1
• Proposition 2.1.1
• proof
• Proposition 2.1.2
• proof