Quantum K-theory of flag varieties via non-abelian localization
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866916483677814784 |
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| author | Yan, Xiaohan |
| author_facet | Yan, Xiaohan |
| contents | In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big $\mathcal{J}$-function, for any partial flag variety. The reconstruction may start with any Weyl-group-invariant value of the well-understood big $\mathcal{J}$-function of an associated toric variety. We generalize the recursive method \cite{Givental:perm2}, based on torus fixed point localization, to deal with non-isolated one-dimensional toric orbits, through incorporating ``balanced broken orbits'' into consideration and subsequently proving a vanishing result of their contribution. Furthermore, we extend the study to twisted generating functions of the flag variety, and demonstrate properties including a non-abelian quantum Lefschetz theorem and a duality between level structures. In the end, we construct a K-theoretic mirror in terms of Jackson-type integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_06281 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantum K-theory of flag varieties via non-abelian localization Yan, Xiaohan Algebraic Geometry 14N35 In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big $\mathcal{J}$-function, for any partial flag variety. The reconstruction may start with any Weyl-group-invariant value of the well-understood big $\mathcal{J}$-function of an associated toric variety. We generalize the recursive method \cite{Givental:perm2}, based on torus fixed point localization, to deal with non-isolated one-dimensional toric orbits, through incorporating ``balanced broken orbits'' into consideration and subsequently proving a vanishing result of their contribution. Furthermore, we extend the study to twisted generating functions of the flag variety, and demonstrate properties including a non-abelian quantum Lefschetz theorem and a duality between level structures. In the end, we construct a K-theoretic mirror in terms of Jackson-type integrals. |
| title | Quantum K-theory of flag varieties via non-abelian localization |
| topic | Algebraic Geometry 14N35 |
| url | https://arxiv.org/abs/2106.06281 |