Integrals of stable envelopes for cotangent bundles to Grassmannians
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| Format: | Preprint |
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2025
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| _version_ | 1866915877197185024 |
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| author | Crawford, Matthew Kartik, Pavan Lance, Reese |
| author_facet | Crawford, Matthew Kartik, Pavan Lance, Reese |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_21573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Integrals of stable envelopes for cotangent bundles to Grassmannians Crawford, Matthew Kartik, Pavan Lance, Reese Algebraic Geometry High Energy Physics - Theory Combinatorics Representation Theory 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. |
| title | Integrals of stable envelopes for cotangent bundles to Grassmannians |
| topic | Algebraic Geometry High Energy Physics - Theory Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2510.21573 |