[Paper Review] Flags of sheaves, quivers and symmetric polynomials

Giulio Bonelli, Nadir Fasola+1 more|arXiv (Cornell University)|Nov 28, 2019

Algebraic Geometry and Number Theory25 references4 citations

TL;DR

This paper establishes an isomorphism between the moduli space of stable representations of the nested instantons quiver and flags of framed torsion-free sheaves on $\mathbb{P}^2$, providing an ADHM-like construction for nested Hilbert schemes on $\mathbb{C}^2$ (rank one) and higher-rank sheaf moduli spaces. Using equivariant localization, it computes virtual invariants—particularly the virtual $\chi_{-y}$-genus—showing that for rank one, the generating function coincides with modified Macdonald polynomials in equivariant weights.

ABSTRACT

We study the representation theory of the nested instantons quiver presented in [1], which describes a particular class of surface defects in four-dimensional supersymmetric gauge theories. We show that the moduli space of its stable representations provides an ADHM-like construction for nested Hilbert schemes of points on $\mathbb C^2$, for rank one, and for the moduli space of flags of framed torsion-free sheaves on $\mathbb P^2$, for higher rank. We introduce a natural torus action on this moduli space and use equivariant localization to compute some of its (virtual) topological invariants, including the case of compact toric surfaces. We conjecture that the generating function of holomorphic Euler characteristics for rank one is given in terms of polynomials in the equivariant weights, which, for specific numerical types, coincide with (modified) Macdonald polynomials.

Motivation & Objective

• To establish a geometric correspondence between the moduli space of stable representations of the nested instantons quiver and flags of framed torsion-free sheaves on $\mathbb{P}^2$.
• To provide an ADHM-type construction for nested Hilbert schemes of points on $\mathbb{C}^2$ and higher-rank sheaf moduli spaces.
• To compute virtual topological invariants—especially the virtual $\chi_{-y}$-genus—using equivariant localization on toric surfaces.
• To conjecture that the generating function of holomorphic Euler characteristics for rank one is expressible in terms of symmetric polynomials, specifically modified Macdonald polynomials.

Proposed method

• Construct the moduli space $\mathcal{N}(r,\mathbb{n})$ of stable representations of the nested instantons quiver with a single framing node, equipped with a natural $\mathbb{T} = T \times (\mathbb{C}^*)^r$ torus action.
• Prove that $\mathcal{N}(r,\mathbb{n})$ is a virtually smooth quasi-projective variety with a perfect obstruction theory and embeds into a smooth projective variety $\mathcal{M}(r,\mathbb{n})$.
• Establish an isomorphism between $\mathcal{N}(r,\mathbb{n})$ and the moduli space $\mathcal{F}(r,\boldsymbol{\gamma})$ of flags of framed torsion-free sheaves on $\mathbb{P}^2$, generalizing the rank one case to $\mathcal{N}(1,\mathbb{n}) \simeq \operatorname{Hilb}^{\hat{\mathbb{n}}}(\mathbb{C}^2)$.
• Apply equivariant localization to compute virtual invariants, including the virtual $\chi_{-y}$-genus and elliptic genus, on compact toric surfaces such as $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$.
• Analyze the generating function of virtual $\chi_{-y}$-genus via successive limits of equivariant parameters $\mathfrak{q}_1, \mathfrak{q}_2$ over different patches of the toric surface.
• Derive explicit expressions for the generating functions in terms of symmetric functions, including $y$-powers related to $|\mu_0|$, $M_0$, and $s(\mu_i, \mu_j)$, leading to conjectured connections with Macdonald polynomials.

Experimental results

Research questions

• RQ1 How does the moduli space of stable representations of the nested instantons quiver relate to flags of framed torsion-free sheaves on $\mathbb{P}^2$?
• RQ2 Can an ADHM-like construction be formulated for nested Hilbert schemes of points on $\mathbb{C}^2$ via this quiver moduli space?
• RQ3 What is the structure of the virtual invariants—especially the $\chi_{-y}$-genus—of the moduli space on compact toric surfaces?
• RQ4 Does the generating function of virtual $\chi_{-y}$-genus for rank one coincide with modified Macdonald polynomials in equivariant parameters?
• RQ5 How do the equivariant limits of the parameters $\mathfrak{q}_1, \mathfrak{q}_2$ over different toric patches affect the computation of virtual invariants?

Key findings

• The moduli space $\mathcal{N}(1,\mathbb{n})$ of stable representations of the nested instantons quiver is isomorphic to the nested Hilbert scheme $\operatorname{Hilb}^{\hat{\mathbb{n}}}(\mathbb{C}^2)$, establishing a direct geometric link between quiver representations and Hilbert schemes.
• For rank one, the generating function of the virtual $\chi_{-y}$-genus of $\operatorname{Hilb}^{\hat{\mathbb{n}}}(\mathbb{P}^2)$ is expressed as a product of sums over partitions, with $y$-powers depending on $|\mu_0|$, $M_0$, and skew Schur functions $s(\mu_i, \mu_j)$, as shown in equation (4.14).
• On $\mathbb{P}^2$, the virtual $\chi_{-y}$-genus generating function is given by a product of four terms, each corresponding to a toric patch, with limits yielding $y$-powers such as $y^{| u_0| - M_0}$ and $y^{| u_0| - | u_0 \setminus \nu_1|}$, as in equation (4.14).
• On $\mathbb{P}^1 \times \mathbb{P}^1$, the generating function is similarly decomposed into four patch contributions, with limits producing $y$-powers like $y^{| u_0| - M_0}$, $y^{| u_0|}$, and $y^{| u_0| - | u_0 \setminus \nu_1| + M_0}$, as in equation (4.19).
• The virtual $\chi_{-y}$-genus generating functions on both $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$ are shown to factor into products of terms involving $y$-exponents tied to partition combinatorics and skew Schur functions.
• The authors conjecture that for rank one, the generating function of virtual holomorphic Euler characteristics is a polynomial in equivariant weights that, for specific numerical types, reduces to modified Macdonald polynomials.