K-theoretic Gromov–Witten invariants of line degrees on flag varieties
Creators
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1.
Rutgers, The State University of New Jersey
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2.
Swarthmore College
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3.
California Institute of Technology
Abstract
A homology class d ∈ H₂(X,Z) of a complex flag variety X = G ∕ P is called a line degree if the moduli space ℳ 0,0(X,d) of 0-pointed stable maps to X of degree d is also a flag variety G ∕ P'. We prove a quantum equals classical formula stating that any n-pointed (equivariant, K-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety G ∕ P’. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags G ∕ B. Our formulas make it straightforward to compute the big quantum K-theory ring QK^(big)(X) modulo the ideal ⟨Q^d⟩ generated by degrees d larger than line degrees.
Copyright and License
© 2024 World Scientific Publishing Company.
Acknowledgement
This project was initiated at the ICERM Women in Algebraic Geometry workshop in July 2020 and the ICERM Combinatorial Algebraic Geometry program in Spring 2021. We thank A. Gibney, L. Heller, E. Kalashnikov, H. Larson as well as P. E. Chaput, L. C. Mihalcea and N. Perrin for inspiring collaborations on related projects. We also thank an anonymous referee for a careful reading of our paper and for several helpful suggestions. A.B. was partially supported by NSF Grants DMS-2152316 as well as DMS-1929284 while in residence at ICERM during the Spring of 2021. L.C. was partially supported by NSF Grant DMS-2101861.
Funding
A.B. was partially supported by NSF Grants DMS-2152316 as well as DMS-1929284 while in residence at ICERM during the Spring of 2021. L.C. was partially supported by NSF Grant DMS-2101861.
Additional details
Related works
- Is new version of
- Discussion Paper: arXiv:2409.09580 (arXiv)
Funding
- National Science Foundation
- DMS-2152316
- National Science Foundation
- DMS-1929284
- National Science Foundation
- DMS-2101861
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- Publication Status
- Published