[Paper Review] Quantum difference equation for Nakajima varieties

Andreĭ Okounkov, Andrey S. Smirnov|arXiv (Cornell University)|Feb 29, 2016

Algebraic structures and combinatorial models6 citations

TL;DR

This paper constructs a quantum difference equation for Nakajima quiver varieties by introducing a quantum dynamical Weyl group action in equivariant K-theory, realized through the fundamental groupoid of a periodic hyperplane arrangement in Pic(X)⊗ℂ. The key result identifies the lattice part of this groupoid with quantum difference operators, generalizing the quantum differential equation to K-theory and linking it to R-matrices and monodromy in Hilbert schemes.

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 $Pic(X)\otimes {\mathbb{C}}$. 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.

Motivation & Objective

• To generalize the quantum differential equation to equivariant K-theory of Nakajima quiver varieties.
• To construct a quantum dynamical Weyl group action in K-theory using a fundamental groupoid of a periodic hyperplane arrangement in Pic(X)⊗ℂ.
• To identify the lattice part of this groupoid with quantum difference operators acting on K-theory.
• To establish a connection between these operators and R-matrices of cyclic quiver varieties via stable bases.
• To show that the monodromy operators B_w(z) represent the fundamental group of the quantum differential equation's singular locus for Hilb^n(ℂ²).

Proposed method

• The authors define stable envelopes in equivariant K-theory to construct wall-crossing operators and R-matrices.
• They introduce the quantum loop algebra 𝒰_ℏ(ĝ_Q) and its wall subalgebras 𝒰_ℏ(𝔤_w) to model the quantum group action.
• The quantum difference operators M_L(z) are constructed as compositions of stable envelopes and R-matrices.
• The dynamical operators B_w(λ) are defined as limits of M_L(z) as z→∞, encoding monodromy data.
• Explicit matrices for B_w(z) are computed in the mixed stable basis, showing independence from equivariant parameters a=t₁/t₂ and dependence only on z and ℏ=t₁t₂.
• The operators B_w(z) are identified with K-theoretic R-matrices of cyclic quiver varieties, linking them to symplectic duality and Hilbert schemes.

Experimental results

Research questions

• RQ1 How can the quantum differential equation in cohomology be lifted to a quantum difference equation in equivariant K-theory for Nakajima quiver varieties?
• RQ2 What is the correct generalization of the Weyl group action in K-theory, and how is it related to hyperplane arrangements in Pic(X)⊗ℂ?
• RQ3 How do the quantum difference operators arise from the representation theory of quantum loop algebras and wall subalgebras?
• RQ4 What is the role of stable bases and R-matrices in realizing the quantum dynamical Weyl group?
• RQ5 How does the monodromy of the quantum differential equation for Hilb^n(ℂ²) relate to the operators B_w(z)?

Key findings

• The quantum difference equation for Nakajima varieties is constructed via a fundamental groupoid of a periodic hyperplane arrangement in Pic(X)⊗ℂ, generalizing the Weyl group.
• The lattice part of this groupoid corresponds exactly to the quantum difference operators acting on equivariant K-theory.
• The operators B_w(z) are shown to be independent of the equivariant parameter a=t₁/t₂ and depend only on z and ℏ=t₁t₂.
• The matrix of B_w(z) in the mixed stable basis coincides with the K-theoretic R-matrix of a cyclic quiver variety of d(w) vertices.
• The limit B_w = lim_{z→∞} B_w(z) realizes the monodromy of the quantum differential equation for Hilb^n(ℂ²) around its singularities.
• The construction provides a representation of the fundamental group π₁(ℙ¹∖{singularities}) in the Fock space via the operators B_w.