Vertex functions of type $D$ Nakajima quiver varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909798063144960 |
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| author | Dinkins, Hunter Jang, Jiwu |
| author_facet | Dinkins, Hunter Jang, Jiwu |
| contents | We study the quasimap vertex functions of type $D$ Nakajima quiver varieties. When the quiver varieties have isolated torus fixed points, we compute the coefficients of the vertex functions in the $K$-theoretic fixed point basis. We also give an explicit combinatorial description of zero-dimensional type $D$ quiver varieties and their vertex functions using the combinatorics of minuscule posets. Using Macdonald polynomials, we prove that these vertex functions can be expressed as products of $q$-binomial functions, which proves a degeneration of the conjectured 3d mirror symmetry of vertex functions. We provide an interpretation of type $D$ spin vertex functions as the partition functions of the half-space Macdonald processes of Barraquand, Borodin, and Corwin. This hints that the geometry of quiver varieties may provide new examples of integrable probabilistic models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_13937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Vertex functions of type $D$ Nakajima quiver varieties Dinkins, Hunter Jang, Jiwu Representation Theory High Energy Physics - Theory Mathematical Physics Algebraic Geometry Combinatorics We study the quasimap vertex functions of type $D$ Nakajima quiver varieties. When the quiver varieties have isolated torus fixed points, we compute the coefficients of the vertex functions in the $K$-theoretic fixed point basis. We also give an explicit combinatorial description of zero-dimensional type $D$ quiver varieties and their vertex functions using the combinatorics of minuscule posets. Using Macdonald polynomials, we prove that these vertex functions can be expressed as products of $q$-binomial functions, which proves a degeneration of the conjectured 3d mirror symmetry of vertex functions. We provide an interpretation of type $D$ spin vertex functions as the partition functions of the half-space Macdonald processes of Barraquand, Borodin, and Corwin. This hints that the geometry of quiver varieties may provide new examples of integrable probabilistic models. |
| title | Vertex functions of type $D$ Nakajima quiver varieties |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2502.13937 |