## PD Dr. Jan Swoboda

### Project leader

Ruprecht-Karls-Universität Heidelberg

E-mail: swoboda(at)mathi.uni-heidelberg.de

Telephone: +49-6221-54-14224

Homepage: https://www.mathi.uni-heidelberg.de/~swo…

## Project

**32**Asymptotic geometry of the Higgs bundle moduli space

## Publications within SPP2026

This paper relates different approaches to the asymptotic geometry of the

Hitchin moduli space of SL(2,C) Higgs bundles on a closed Riemann surface and,

via the nonabelian Hodge theorem, the character variety of SL(2,C)

representations of a surface group. Specifically, we find an asymptotic

correspondence between the analytically defined limiting configuration of a

sequence of solutions to the self-duality equations constructed by

Mazzeo-Swoboda-Weiss-Witt, and the geometric topological shear-bend parameters

of equivariant pleated surfaces due to Bonahon and Thurston. The geometric link

comes from a study of high energy harmonic maps. As a consequence we prove: (1)

the local invariance of the partial compactification of the moduli space by

limiting configurations; (2) a refinement of the harmonic maps characterization

of the Morgan-Shalen compactification of the character variety; and (3) a

comparison between the family of complex projective structures defined by a

quadratic differential and the realizations of the corresponding flat

connections as Higgs bundles, as well as a determination of the asymptotic

shear-bend cocycle of Thurston's pleated surface.

**Related project(s):****27**Invariants and boundaries of spaces**32**Asymptotic geometry of the Higgs bundle moduli space

Given a generic stable strongly parabolic $SL(2,\mathbb{C})$-Higgs bundle

$(\mathcal{E}, \varphi)$, we describe the family of harmonic metrics $h_t$ for

the ray of Higgs bundles $(\mathcal{E}, t \varphi)$ for $t\gg0$ by perturbing

from an explicitly constructed family of approximate solutions

$h_t^{\mathrm{app}}$. We then describe the natural hyperK\"ahler metric on

$\mathcal{M}$ by comparing it to a simpler "semi-flat" hyperK\"ahler metric. We

prove that $g_{L^2} - g_{\mathrm{sf}} = O(\mathrm{e}^{-\gamma t})$ along a

generic ray, proving a version of Gaiotto-Moore-Neitzke's conjecture.

Our results extend to weakly parabolic $SL(2,\mathbb{C})$-Higgs bundles as

well.

In the case of the four-puncture sphere, we describe the moduli space and

metric more explicitly. In this case, we prove that the hyperk\"ahler metric is

ALG and show that the rate of exponential decay is the conjectured optimal one,

$\gamma=4L$, where $L$ is the length of the shortest geodesic on the base curve

measured in the singular flat metric $|\mathrm{det}\, \varphi|$.

Pages | 73 pages |

Link to preprint version |

**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

We study the asymptotics of the natural \(L^2\) metric on the Hitchin moduli space with group \(G=SU(2)\). Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

Journal | Comm. Math. Phys. |

Publisher | Springer |

Volume | 367, no. 1 |

Pages | 151-191 |

Link to preprint version | |

Link to published version |

**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

We prove a gluing theorem for solutions \((A_0, \Phi_0)\) of Hitchin's self-duality equations with logarithmic singularities on a rank-\(2\) vector bundle over a noded Riemann surface \( \Sigma\) representing a boundary point of Teichmüller moduli space. We show that every nearby smooth Riemann surface \( \Sigma_1\) carries a smooth solution \((A_1, \Phi_1)\) of the self-duality equations, which may be viewed as a desingularization of \((A_0, \Phi_0)\).

Journal | Adv. Math. |

Publisher | Elsevier |

Volume | 322 |

Pages | 637-681 |

Link to preprint version | |

Link to published version |

**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

In this note we study some analytic properties of the linearized self-duality equations on a family of smooth Riemann surfaces \( \Sigma_R\) converging for \(R\searrow 0\) to a surface \( \Sigma_0\) with a finite number of nodes. It is shown that the linearization along the fibres of the Hitchin fibration \(\mathcal M_d \to \Sigma_R\) gives rise to a graph-continuous Fredholm family, the index of it being stable when passing to the limit. We also report on similarities and differences between properties of the Hitchin fibration in this degeneration and in the limit of large Higgs fields as studied in Mazzeo et al. (Duke Math. J. 165(12):2227–2271, 2016).

Journal | Abh. Math. Semin. Univ. Hambg. |

Publisher | Springer Berlin Heidelberg |

Volume | 86 |

Pages | 189--201 |

Link to preprint version | |

Link to published version |

**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space

We associate to each stable Higgs pair \((A_0,\Phi_0)\) on a compact Riemann surface *X* a singular limiting configuration \((A_\infty,\Phi_\infty)\), assuming that \(\det\Phi\) has only simple zeroes. We then prove a desingularization theorem by constructing a family of solutions \((A_t,\Phi_t) \) to Hitchin's equations which converge to this limiting configuration as \(t\to\infty\). This provides a new proof, via gluing methods, for elements in the ends of the Higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space.

Journal | Duke Math. J. |

Publisher | Duke University Press |

Volume | 165 |

Pages | 2227-2271 |

Link to preprint version | |

Link to published version |

**Related project(s):****32**Asymptotic geometry of the Higgs bundle moduli space