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Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs

Hiraku Nakajima

TL;DR

The paper develops a geometric framework linking moduli spaces of classical-group instantons on ALE spaces with σ-quiver varieties and twisted Yangians. By leveraging stable envelopes, it constructs coideal subalgebras and computes explicit $K$-matrices in concrete examples, identifying them with Olshanski twisted Yangians and Molev–Ragoucy reflection algebras. It clarifies necessary conditions on polarization and shows that certain prior assumptions are unnecessary, broadening applicability to orthogonal/symplectic instantons. The results provide explicit algebraic realizations of twisted Yangian actions on equivariant (co)homology and give detailed case studies (instantons, partial flags) that illuminate the interaction between geometry and quantum symmetric pairs, with implications for representation theory and Coulomb-branch physics.

Abstract

Moduli spaces of instantons on ALE spaces for classical groups are examples of fixed point sets of involutions on quiver varieties, i.e., $σ$-quiver varieties. In 2018 Yiqiang Li considered their equivariant cohomology, and by stable envelope of Maulik-Okounkov, constructed representations of coideal subalgebras of Maulik-Okounkov Yangian, called twisted Yangian. We calculate $K$-matrices as matrices in examples, identified the twisted Yangians with ones studied in other literature, and clarify conditions which we should impose to make them well-defined.

Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum symmetric pairs

TL;DR

The paper develops a geometric framework linking moduli spaces of classical-group instantons on ALE spaces with σ-quiver varieties and twisted Yangians. By leveraging stable envelopes, it constructs coideal subalgebras and computes explicit -matrices in concrete examples, identifying them with Olshanski twisted Yangians and Molev–Ragoucy reflection algebras. It clarifies necessary conditions on polarization and shows that certain prior assumptions are unnecessary, broadening applicability to orthogonal/symplectic instantons. The results provide explicit algebraic realizations of twisted Yangian actions on equivariant (co)homology and give detailed case studies (instantons, partial flags) that illuminate the interaction between geometry and quantum symmetric pairs, with implications for representation theory and Coulomb-branch physics.

Abstract

Moduli spaces of instantons on ALE spaces for classical groups are examples of fixed point sets of involutions on quiver varieties, i.e., -quiver varieties. In 2018 Yiqiang Li considered their equivariant cohomology, and by stable envelope of Maulik-Okounkov, constructed representations of coideal subalgebras of Maulik-Okounkov Yangian, called twisted Yangian. We calculate -matrices as matrices in examples, identified the twisted Yangians with ones studied in other literature, and clarify conditions which we should impose to make them well-defined.
Paper Structure (65 sections, 20 theorems, 186 equations, 1 figure, 1 table)

This paper contains 65 sections, 20 theorems, 186 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Extended Yangian
  3. Yangian
  4. R-matrix
  5. Monodromy matrices
  6. $RTT$ relation
  7. Extended Yangian
  8. Hopf algebra structure
  9. Translation automorphism
  10. Homomorphism to the Yangian
  11. Extended Lie algebra g^ext
  12. Recovering the Yangian
  13. Unitarity
  14. Involutions on quiver varieties
  15. Quiver varieties
  16. ...and 50 more sections

Key Result

Theorem 2.18

(1) Let ${\mathsf{Y}}_R$ be the quotient ${\mathsf{X}}/(z_{F_i}(u) - 1)$. Here $(z_{F_i}(u) - 1)$ is the ideal generated by matrix entries of coefficients of $z_{F_i}(u) - 1$ for all $i\in I$. Then $\tilde{\Phi}$ induces an isomorphism (2) The center of ${\mathsf{X}}$ is isomorphic to the polynomial ring of generators $z_i^{(r)}$ for $i\in I$, $r\ge 2$, appeared in the expansion of $z_{F_i}(u)$.

Figures (1)

  • Figure 1: The involution induced by the longest element of the Weyl group

Theorems & Definitions (57)

  • Definition 2.11
  • Definition 2.14
  • Remark 2.15
  • Theorem 2.18: MR3849990
  • Theorem 2.19: MR3849990
  • Theorem 2.21: MR3849990
  • Lemma 2.22
  • proof
  • Remark 2.23
  • Remark 3.2: convention on ${\mathbb C}_{\hbar}^\times$-action
  • ...and 47 more