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Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

Ana Peón-Nieto

TL;DR

This work provides a birational description of the fixed-point loci and nilpotent-cone components for Higgs-bundle moduli with nilpotent order two, establishing Drinfeld’s conjecture for the $(n_0,n_1)$ sublocus and linking wobbliness to a refined real-form criterion via $ ext{U}(n_0,n_1)$-Higgs bundles. It introduces the notion of $ ext{U}(n_0,n_1)$-wobbliness and proves its equivalence with classical wobbliness for length-two fixed-point components, yielding a computable criterion for wobbliness based on Brill–Noether data and Higgs-field rank. The paper also computes virtual equivariant multiplicities and a symmetric Euler pairing for downward-flow branes, showing that these invariants often fail to detect all wobbliness, especially outside the small-rank or maximal-Toledo regimes, and highlights a tight Brill–Noether–geometric interaction governing the fixed-point and nilpotent-cone geometry. Overall, the results advance understanding of the nilpotent cone, flow dynamics in Higgs moduli, and the interplay between Higgs geometry and real forms, with implications for Brill–Noether theory and mirror symmetry in Hitchin systems.

Abstract

We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the $\CC^\times$-fixed point locus of length two in the moduli space of Higgs bundles. Using these results, we prove Drinfeld's conjecture for the sublocus of type $(n_0,n_1)$ fixed points. We introduce the notion of $\U(n_0,n_1)$-wobbliness (stronger than the one of wobbliness) and show that fixed point components of type $(n_0,n_1)$ are wobbly in rank higher than three, if and only if they are also $\U(n_0,n_1)$-wobbly. This yields a computable criterion to check wobbliness of fixed point components, that simplifies the existing ones. We analyse the virtual equivariant multiplicities of fixed points of type $(n_0,n_1)$ and their Euler pairings with downward flows for type $(1,\dots, 1)$ fixed points. We find that both invariants fail to fully detect all wobbly components for ordered partitions other than $(2,1)$.

Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

TL;DR

This work provides a birational description of the fixed-point loci and nilpotent-cone components for Higgs-bundle moduli with nilpotent order two, establishing Drinfeld’s conjecture for the sublocus and linking wobbliness to a refined real-form criterion via -Higgs bundles. It introduces the notion of -wobbliness and proves its equivalence with classical wobbliness for length-two fixed-point components, yielding a computable criterion for wobbliness based on Brill–Noether data and Higgs-field rank. The paper also computes virtual equivariant multiplicities and a symmetric Euler pairing for downward-flow branes, showing that these invariants often fail to detect all wobbliness, especially outside the small-rank or maximal-Toledo regimes, and highlights a tight Brill–Noether–geometric interaction governing the fixed-point and nilpotent-cone geometry. Overall, the results advance understanding of the nilpotent cone, flow dynamics in Higgs moduli, and the interplay between Higgs geometry and real forms, with implications for Brill–Noether theory and mirror symmetry in Hitchin systems.

Abstract

We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the -fixed point locus of length two in the moduli space of Higgs bundles. Using these results, we prove Drinfeld's conjecture for the sublocus of type fixed points. We introduce the notion of -wobbliness (stronger than the one of wobbliness) and show that fixed point components of type are wobbly in rank higher than three, if and only if they are also -wobbly. This yields a computable criterion to check wobbliness of fixed point components, that simplifies the existing ones. We analyse the virtual equivariant multiplicities of fixed points of type and their Euler pairings with downward flows for type fixed points. We find that both invariants fail to fully detect all wobbly components for ordered partitions other than .
Paper Structure (13 sections, 32 theorems, 192 equations, 1 table)

This paper contains 13 sections, 32 theorems, 192 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Higgs bundles, the Hitchin map, and fixed points
  4. Wobbly and very stable points and components
  5. $\mathrm{U}(p,q)$-Higgs bundles
  6. Fixed points of length two and $\mathrm{U}(n_0,n_1)$-Higgs bundles
  7. Wobbly divisors of type $(n_0,n_1)$
  8. Criteria for wobbliness
  9. Wobbliness of length two components
  10. $\mathrm{GL}_n(\mathbb{C})$-wobbliness of length two components
  11. $\mathrm{U}(n_0,n_1)$-wobbliness of length two components.
  12. Virtual equivariant multiplicities of wobbly components
  13. Euler pairings

Key Result

Lemma 2.6

Let $\mathcal{E}\in\mathrm{M}_X(n,d)^{\mathbb{C}^\times}$ be stable. Then, $\mathcal{E}$ is very stable if and only if

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • proof
  • Definition 2.8
  • ...and 81 more