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Quantum difference equation for Nakajima varieties

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Abstract

For an arbitrary Nakajima quiver variety X, we construct an analog of the quantum dynamical Weyl group acting in its equivariant K-theory. The correct generalization of the Weyl group here is the fundamental groupoid of a certain periodic locally finite hyperplane arrangement in \({{\,\mathrm{Pic}\,}}(X)\otimes {\mathbb {C}}\). We identify the lattice part of this groupoid with the operators of quantum difference equation for X. The cases of quivers of finite and affine type are illustrated by explicit examples.

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Notes

  1. Here the threefold need not be Calabi–Yau, to point out a frequent misconception. For example, the equivariant Donaldson-Thomas theory of toric varieties is a very rich subject with many applications in mathematical physics.

  2. \({\mathsf {S}}(u,z)\) is denoted by \({\mathsf {S}}_{\sigma }(u,z)\) in [47] for a shift \(u\rightarrow u q^{\sigma }\) by a specific \({\mathsf {A}}\)-character \(\sigma \).

  3. See Theorem 9.3.1 in [37] for similar statement in the case of equivariant cohomology.

  4. Note, that we use modified quantum parameter z which differs by a sign:

    $$\begin{aligned} z^{{\mathsf {v}}} \mapsto (-1)^{\text {codim}/2} z^{{\mathsf {v}}}, \end{aligned}$$

    see Theorem 10.2.8 in [47]. Explicitly, this change of variables amounts to the following substitution of Kähler parameters:

    $$\begin{aligned} z_i \mapsto (-1)^{2 \kappa _i } z_i \end{aligned}$$

    for canonical vector (81). To get rid of the minus sign, we will use modified notations in this paper. :

  5. In this section we often switch from the variables z, denoting Kähler parameters, to their logarithms \(\lambda \) and back. The two are related via (72).

  6. Note the substitution \(\hbar \rightarrow \hbar ^{2}\) on the left side of this equality which was introduced at the begining of Sect. 6.1.10. We have \(\hbar ^{2 \Omega }={{\,\mathrm{diag}\,}}(1,\hbar ,\hbar ,1)\) and \(\hbar ^{-H \otimes H/2}={{\,\mathrm{diag}\,}}(\hbar ^{-1/2},\hbar ^{1/2},\hbar ^{1/2},\hbar ^{-1/2})=\hbar ^{-1/2}\hbar ^{2 \Omega }\).

  7. The factor 2 in \(\hbar ^{2\kappa }\) is from our conventions introduced at the beginning of Sect. 6.1.10.

  8. This expectation is in agreement with explicit computations of capped vertex functions [47] for the first values of k and n.

  9. Note that we need to substitute \(\hbar \rightarrow \hbar ^{2}\) in the geometric formulas to relate them to the action of \({{\mathscr {U}}_{\hbar }(\widehat{\mathfrak {gl}}_2)}\), as we explain in Sect. 6.1.10.

  10. This expectation is in agreement with explicit computations of the capped vertex functions [47] for the first several values of n and r.

  11. We use a Maple package, implemented by the second author, which computes the action of on the Fock space. The package is available from the author upon request.

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Acknowledgements

During our work on this project, we greatly benefited from interaction with M. Aganagic, R. Bezrukavnikov, H. Dinkins, S. Gautam, D. Maulik, S. Shakirov, V. Toledano Laredo and A. Oblomkov. We are particularly grateful to Pavel Etingof for his inspiration and guidance.

A.O. thanks the Simons foundation for being financially supported as a Simons investigator.

The work of A.S. was partially supported by NSF grant DMS-2054527 and by the RSF grant 19-11-00062.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    A. Okounkov

  2. Institute for Problems of Information Transmission, Bolshoy Karetny 19, Moscow, 127994, Russia

    A. Okounkov

  3. Laboratory of Representation Theory and Mathematical Physics, Higher School of Economics, Myasnitskaya 20, Moscow, 101000, Russia

    A. Okounkov

  4. Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599, USA

    A. Smirnov

  5. Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina 8, Moscow, 117966, Russia

    A. Smirnov

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Okounkov, A., Smirnov, A. Quantum difference equation for Nakajima varieties. Invent. math. 229, 1203–1299 (2022). https://doi.org/10.1007/s00222-022-01125-w

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