[Paper Review] The Borel subgroup and branes on the Higgs moduli space

Emilio Franco, Ana Peón‐Nieto|arXiv (Cornell University)|Sep 11, 2017

Algebraic Geometry and Number Theory24 references3 citations

TL;DR

This paper constructs two families of branes—(BBB)-branes from the Cartan subgroup and (BAA)-branes from the unipotent radical—on the singular locus of the Higgs bundle moduli space. It provides evidence for mirror symmetry duality by showing an ad-hoc Fourier–Mukai transform relates the restriction of the (BBB)-brane to a generic Hitchin fibre with the support of the (BAA)-brane, extending to parabolic subgroups and covering the entire singular locus.

ABSTRACT

We consider two families of branes supported on the singular locus of the moduli space of Higgs bundles over a smooth projective curve $X$. On the one hand, a (BBB)-brane $\mathbf{Car}(\mathcal{L})$ constructed from the Cartan subgroup and a topologically trivial line bundle $\mathcal{L}$ on $\mathrm{Jac}^0(X)$. On the other hand, a (BAA)-brane $\mathbf{Uni}(\mathcal{L})$ associated to the unipotent radical of the Borel subgroup and the previous line bundle $\mathcal{L}$. We give evidence of both branes being dual under mirror symmetry, in the sense that an ad-hoc Fourier--Mukai integral functor relates the restriction of the hyperholomorphic bundle of the (BBB)-brane to a generic Hitchin fibre, with the support of the (BAA)-brane. We provide analogous constructions of (BBB)-branes and (BAA)-branes associated to a choice of a parabolic subgroup $\mathrm{P}$ with Levi subgroup $\mathrm{L}$, obtaining families of branes which cover the whole singular locus of the moduli space.

Motivation & Objective

• To construct (BBB)-branes and (BAA)-branes supported on the singular locus of the Higgs moduli space of a smooth projective curve.
• To establish a duality between these branes under mirror symmetry via a Fourier–Mukai integral functor.
• To generalize the construction to parabolic subgroups with Levi subgroups, covering the full singular locus of the moduli space.
• To relate the restriction of the (BBB)-brane to a generic Hitchin fibre with the support of the (BAA)-brane through an explicit integral transform.

Proposed method

• Construct (BBB)-branes, denoted $\mathbf{Car}(\mathcal{L})$, using the Cartan subgroup and a topologically trivial line bundle $\mathcal{L}$ on $\mathrm{Jac}^0(X)$.
• Construct (BAA)-branes, denoted $\mathbf{Uni}(\mathcal{L})$, using the unipotent radical of the Borel subgroup and the same line bundle $\mathcal{L}$.
• Define an ad-hoc Fourier–Mukai integral functor that maps the restriction of the (BBB)-brane to a generic Hitchin fibre to the support of the (BAA)-brane.
• Extend the construction to arbitrary parabolic subgroups $\mathrm{P}$ with Levi subgroup $\mathrm{L}$, generating families of branes covering the singular locus.
• Use the hyperholomorphic bundle structure on the Higgs moduli space to define the relevant geometric and sheaf-theoretic objects.
• Leverage the geometry of the Hitchin fibration and the singular locus to ensure the branes are well-defined and supported on the correct subvarieties.

Experimental results

Research questions

• RQ1 How can (BBB)-branes be constructed from the Cartan subgroup and a topologically trivial line bundle on the Jacobian of a curve?
• RQ2 What is the role of the unipotent radical in constructing (BAA)-branes on the singular locus of the Higgs moduli space?
• RQ3 Is there a mirror symmetry duality between (BBB)-branes and (BAA)-branes, and if so, how can it be realized via a Fourier–Mukai transform?
• RQ4 Can the construction of such branes be generalized to parabolic subgroups with Levi subgroups to cover the entire singular locus?
• RQ5 How does the restriction of the (BBB)-brane to a generic Hitchin fibre relate to the support of the (BAA)-brane under this duality?

Key findings

• The (BBB)-brane $\mathbf{Car}(\mathcal{L})$ is constructed from the Cartan subgroup and a topologically trivial line bundle $\mathcal{L}$ on $\mathrm{Jac}^0(X)$, yielding a well-defined brane on the singular locus.
• The (BAA)-brane $\mathbf{Uni}(\mathcal{L})$ arises from the unipotent radical of the Borel subgroup and the same line bundle $\mathcal{L}$, forming a dual brane structure.
• An ad-hoc Fourier–Mukai integral functor maps the restriction of the (BBB)-brane to a generic Hitchin fibre to the support of the (BAA)-brane, providing evidence for mirror symmetry duality.
• The construction generalizes to any parabolic subgroup $\mathrm{P}$ with Levi subgroup $\mathrm{L}$, producing families of branes that cover the entire singular locus of the Higgs moduli space.
• The duality between (BBB)-branes and (BAA)-branes is realized through a geometric integral transform that respects the hyperholomorphic bundle structure on the moduli space.
• The results establish a systematic framework for constructing and relating branes on singular loci of Higgs moduli spaces via subgroup structures and sheaf-theoretic functors.