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Moduli space of homologically trivial parabolic (Higgs) bundles on the projective line and applications

Xueqing Wen

TL;DR

This work establishes a precise isomorphism between the moduli space $oldsymbol{M}_P$ of semistable homologically trivial parabolic bundles on $oldsymbol{P}^1$ and the quiver variety $oldsymbol{R}_{oldsymbol{ ikhchi}}(oldsymbol{v})$, and between the moduli space of parabolic Higgs bundles $ ext{Higgs}^{ullet}_P$ and the quiver variety $oldsymbol{M}_{oldsymbol{ ikhchi}}(oldsymbol{v})$ for a star‑shaped quiver, thereby encoding parabolic data into quiver data. This bridge enables two major results: a closed formula for Littlewood–Richardson coefficients via Verlinde theory, expressed as a detailed sum over a spectral parameter ${oldsymbol{v}}$ with sine factors and Schur polynomials, and a geometric treatment of the nilpotent additive Deligne–Simpson problem, yielding irreducible solutions under explicit rank and gcd conditions through spectral curves and line bundles on their normalization. The approach unifies moduli of parabolic (Higgs) bundles, quiver varieties, and representation‑theoretic/combinatorial problems, providing concrete computational tools for LR coefficients and explicit constructions for Deligne–Simpson problems.

Abstract

We establish an isomorphism between the moduli space of homologically trivial parabolic (Higgs) bundles on $\mathbb{P}^1$ and the quiver variety associated to a star-shaped quiver. As applications, we deduce a closed formula for the Littlewood-Richardson coefficients from the Verlinde formula, and solve the nilpotent case of the Deligne-Simpson problem via geometric methods.

Moduli space of homologically trivial parabolic (Higgs) bundles on the projective line and applications

TL;DR

This work establishes a precise isomorphism between the moduli space of semistable homologically trivial parabolic bundles on and the quiver variety , and between the moduli space of parabolic Higgs bundles and the quiver variety for a star‑shaped quiver, thereby encoding parabolic data into quiver data. This bridge enables two major results: a closed formula for Littlewood–Richardson coefficients via Verlinde theory, expressed as a detailed sum over a spectral parameter with sine factors and Schur polynomials, and a geometric treatment of the nilpotent additive Deligne–Simpson problem, yielding irreducible solutions under explicit rank and gcd conditions through spectral curves and line bundles on their normalization. The approach unifies moduli of parabolic (Higgs) bundles, quiver varieties, and representation‑theoretic/combinatorial problems, providing concrete computational tools for LR coefficients and explicit constructions for Deligne–Simpson problems.

Abstract

We establish an isomorphism between the moduli space of homologically trivial parabolic (Higgs) bundles on and the quiver variety associated to a star-shaped quiver. As applications, we deduce a closed formula for the Littlewood-Richardson coefficients from the Verlinde formula, and solve the nilpotent case of the Deligne-Simpson problem via geometric methods.

Paper Structure

This paper contains 10 sections, 23 theorems, 99 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A closed formula for the Littlewood–Richardson coefficients
  3. The nilpotent case of the additive Deligne--Simpson problem
  4. Moduli space of parabolic Higgs bundles over projective line
  5. Parabolic bundles on $\mathbb{P}^1$
  6. Parabolic Higgs bundles on $\mathbb{P}^1$
  7. Parabolic Hitchin map
  8. Quiver varieties and the isomorphism theorem
  9. A closed formula for the Littlewood–Richardson coefficients
  10. The nilpotent case of the additive Deligne--Simpson problem

Key Result

Theorem 1.1

There exist isomorphisms $\mathbf{M}_P \cong \mathpzc{R}_{\chi}(\mathbf{v})$ and $\mathbf{Higgs}^{\circ}_P \cong \mathfrak{M}_{\chi}(\mathbf{v})$, where the quiver $Q$, the character $\chi$, and the dimension vector $\mathbf{v}$ are determined by the parabolic data of the parabolic (Higgs) bundles.

Figures (2)

  • Figure 1: The star‑shaped quiver $Q$
  • Figure 2: The doubled quiver $\overline{Q}$

Theorems & Definitions (49)

  • Theorem 1.1: Theorem \ref{['main1']}, Theorem \ref{['main2']}
  • Theorem 1.2: Theorem \ref{['thm closed formula']}
  • Theorem 1.3: Theorem \ref{['DSP']}
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Example 2.4
  • ...and 39 more