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Mirror symmetry for Nahm branes

Emilio Franco, Marcos Jardim

TL;DR

This work extends the Dirac–Higgs bundle and Nahm transform to the degree-zero, higher-rank setting by exploiting a flat unitary gerbe to replace a global universal bundle. It constructs a family of β_m-twisted BBB-branes on moduli spaces $ ext{M}_m^{ ext{st}}$ and analyzes their mirror partners via a β-twisted Fourier–Mukai transform on the smooth locus of the Hitchin fibration, showing the transforms live on Lagrangian multisections and thus encode partial $ ext{BAA}$-brane data. The tensorial spectral data are carefully tracked to support the higher-rank Nahm transform and its compatibility with Hitchin–BNR data. Overall, the paper provides a coherent algebraic framework for higher-rank Nahm branes and their mirror symmetry, contributing to the geometric Langlands program and SYZ-type dualities in Higgs bundle moduli.

Abstract

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.

Mirror symmetry for Nahm branes

TL;DR

This work extends the Dirac–Higgs bundle and Nahm transform to the degree-zero, higher-rank setting by exploiting a flat unitary gerbe to replace a global universal bundle. It constructs a family of β_m-twisted BBB-branes on moduli spaces and analyzes their mirror partners via a β-twisted Fourier–Mukai transform on the smooth locus of the Hitchin fibration, showing the transforms live on Lagrangian multisections and thus encode partial -brane data. The tensorial spectral data are carefully tracked to support the higher-rank Nahm transform and its compatibility with Hitchin–BNR data. Overall, the paper provides a coherent algebraic framework for higher-rank Nahm branes and their mirror symmetry, contributing to the geometric Langlands program and SYZ-type dualities in Higgs bundle moduli.

Abstract

The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than . This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.

Paper Structure

This paper contains 16 sections, 16 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Our constructions
  4. Organization of the paper
  5. Geometry of the Hitchin system
  6. Non-abelian Hodge theory
  7. The Hitchin fibration
  8. Fourier--Mukai transform and mirror symmetry for the Hitchin system
  9. Flat unitary gerbes and the universal bundle
  10. The Dirac--Higgs bundle
  11. Tensorization and spectral data
  12. Nahm transform of high rank
  13. Mirror branes
  14. A Fourier--Mukai transform for bundles twisted by gerbes
  15. The Dirac--Higgs bundle under Fourier--Mukai
  16. ...and 1 more sections

Key Result

Proposition 2.1

Consider the obvious projections $\pi_{\mathrm{M}} : X \times \mathrm{M}^\mathrm{st} \to \mathrm{M}^\mathrm{st}$ and $\pi_{X} : X \times \mathrm{M}^\mathrm{st} \to X$. The $\beta$-twisted Dirac--Higgs bundle is a $\beta$-twisted bundle over $\mathrm{M}^\mathrm{st}$ isomorphic to

Theorems & Definitions (34)

  • Proposition 2.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 24 more