Affine nil-Hecke algebras and quantum cohomology

Gonzalez, Eduardo, Mak, Cheuk Yu and Pomerleano, Dan (2023) Affine nil-Hecke algebras and quantum cohomology. Advances in Mathematics, 415, [108867]. (doi:10.1016/j.aim.2023.108867).

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Abstract

Let G be a compact, connected Lie group and T⊂G a maximal torus. Let (M, ω) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G. We construct a module action of the affine nil-Hecke algebra H^S1×T∗(LG/T) on the S1×T-equivariant quantum cohomology of M, QH∗S1×T(M). Our construction generalizes the theory of shift operators for Hamiltonian torus actions [46,40]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that the G-equivariant quantum cohomology QH∗G(M)defines a canonical holomorphic Lagrangian subvariety LG(M) to BFM(G∨C) in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [51].

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Accepted/In Press date: 2 January 2023

e-pub ahead of print date: 26 January 2023

Published date: 15 February 2023

Additional Information: Funding Information: C.M. would like to thank Pavel Safronov for helpful communications. D.P. would like to thank Constantin Teleman for his generous and patient explanations of [51,52] . He would also like to thank Victor Ginzburg for a helpful email exchange. C.M. was supported by the Simons Collaboration on Homological Mirror Symmetry , Award # 652236 while working on this project. D.P. was partly supported by the Simons Collaboration in Homological Mirror Symmetry , Award # 652299 while working on this project. Publisher Copyright: © 2023 Elsevier Inc.

Keywords: G-equivariant Quantum cohomology, Loop group, affine nil-Hecke algebra, shift operator, Equivariant quantum cohomology, Integrable systems