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PreprintCC BY 4.0arXiv

Zero velocity Lagrangians in the Hitchin system

  • Peter Gothen
  • André Oliveira
  • Rodrigo Pereira
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Abstract

Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the ℂ*-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.

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Cite

BibTeXRIS
BibTeX
@misc{gothen2026zero,
  title = {{Zero velocity Lagrangians in the Hitchin system}},
  author = {Peter Gothen and André Oliveira and Rodrigo Pereira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38548},
  eprint = {2609.38548},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38548},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Zero velocity Lagrangians in the Hitchin system
AU  - Peter Gothen
AU  - André Oliveira
AU  - Rodrigo Pereira
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.38548
UR  - https://arxiv.org/abs/2609.38548
AB  - Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the ℂ^*-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.
ER  -