[Paper Review] Instantons on ALE spaces for classical groups

Hiraku Nakajima|arXiv (Cornell University)|Jan 19, 2018

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper extends the ADHM construction of instantons on ALE spaces from unitary groups to classical groups (SO(n) and Sp(n/2)) by introducing an involution combining the diagram automorphism $*$, the reflection functor $\mathcal{F}$ for the longest Weyl group element, and a transpose operation $t$. The key result is that the moduli space of SO(n)/Sp(n/2) framed instantons is isomorphic to the fixed point locus of this combined involution on the moduli space of U(n) instantons.

ABSTRACT

We give an ADHM type description of instantons on ALE spaces for classical groups as an extension of the description in [KN90] for unitary groups.

Motivation & Objective

• To extend the ADHM description of instantons on ALE spaces from unitary gauge groups to classical groups, specifically SO(n) and Sp(n/2).
• To resolve the gap in the literature where SO/Sp instantons on ALE spaces were not explicitly described in ADHM form despite the availability of related tools.
• To provide a complete and explicit construction of SO(n)/Sp(n/2) instanton moduli spaces using the reflection functor $\mathcal{F}$, diagram automorphism $*$, and transpose $t$, building on [KN90] and [Nak03].
• To establish that the fixed point locus of the composition $t \circ * \circ \mathcal{F}$ on the U(n) instanton moduli space realizes the moduli space for SO(n)/Sp(n/2) instantons.
• To extend the construction to the Uhlenbeck partial compactification of the moduli space, showing the involution extends as a homeomorphism and the fixed locus remains well-defined.

Proposed method

• Use the McKay correspondence to associate irreducible representations $\rho_i$ of $\Gamma \subset \mathrm{SU}(2)$ to vertices of the affine Dynkin diagram.
• Define an involution $*$ on the Dynkin diagram via $\rho_i^* \cong \rho_{i^*}$, induced by the longest Weyl group element $w_0$, which exchanges vertices according to $i^* = \ell - i + 1$ for $A_\ell$, and similar rules for $D_\ell$, $E_6$.
• Apply the reflection functor $\mathcal{F}$ associated with $w_0$ to relate the ADHM data for $\zeta$ and $-\zeta$, enabling the description of dual instantons.
• Introduce a transpose operation $t$ on the framing spaces $W_i$, defined via a bilinear form that respects the symplectic or orthogonal structure depending on the group.
• Construct the composite involution $t \circ * \circ \mathcal{F}$ on the moduli space $\mathfrak{M}_\zeta^{\mathrm{reg}}(\mathbf{v},\mathbf{w})$ of U(n) instantons.
• Show that the fixed point locus of this involution gives the moduli space of SO(n)/Sp(n/2) framed instantons, using the hyperkähler moment map and quiver representation structure.

Experimental results

Research questions

• RQ1 How can the ADHM construction for U(n) instantons on ALE spaces be extended to describe SO(n) and Sp(n/2) instantons?
• RQ2 What is the role of the diagram automorphism $*$ induced by the longest Weyl group element in relating dual instanton moduli spaces?
• RQ3 How does the reflection functor $\mathcal{F}$ for $w_0$ facilitate the transition between ADHM data for $\zeta$ and $-\zeta$?
• RQ4 What is the precise geometric and algebraic structure of the fixed point locus of $t \circ * \circ \mathcal{F}$ on the U(n) instanton moduli space?
• RQ5 Can the Uhlenbeck partial compactification of the moduli space be equipped with an extended involution, and does the fixed locus remain well-defined?

Key findings

• The moduli space of SO(n)/Sp(n/2) framed instantons on an ALE space $X_\zeta$ is isomorphic to the fixed point locus of the involution $t \circ * \circ \mathcal{F}$ on the moduli space of U(n) instantons.
• The involution $t \circ * \circ \mathcal{F}$ is well-defined on the Uhlenbeck partial compactification $\mathfrak{M}_\zeta(\mathbf{v},\mathbf{w})$, extending the isomorphism to the compactified setting.
• The fixed point locus inherits the structure of a hyperkähler manifold and, in the algebro-geometric setting, becomes a quasiprojective variety.
• The construction relies on the interplay between the diagram automorphism $*$, the reflection functor $\mathcal{F}$, and the transpose $t$, with the sign of the bilinear form determining the gauge group (SO or Sp).
• For $D_\ell$ with odd $\ell$, the involution $*$ exchanges the two 'tails' of the Dynkin diagram, and the resulting representation structure determines the orthogonal/symplectic nature of $\rho_i \cong \rho_i^*$.
• The method does not require new input beyond [KN90] and [Nak03], and the result is stated without proof, relying on established tools in hyperkähler geometry and quiver varieties.