Contents

Idea

A Higgs bundle is a holomorphic? vector bundle $E$ together with a 1-form $\Phi$ with values in the endomorphisms of (the fibers of) $E$, such that $\Phi \wedge \Phi =0$.

The term was introduced by Nigel Hitchin as a reference to roughly analogous structures in the standard model of particle physics related to the Higgs field.

Higgs bundles play a central role in nonabelian Hodge theory.

Definition

Let $ℰ$ be a sheaf of sections of a holomorphic bundle $E$ on complex manifold $M$ with structure sheaf ${𝒪}_{M}$ and module of Kähler differentials ${\Omega }_{M}^{1}$.

A Higgs field on $ℰ$ is an ${𝒪}_{M}$-linear map

$\Phi :ℰ\to {\Omega }_{M}^{1}{\otimes }_{{𝒪}_{M}}ℰ$\Phi : \mathcal{E}\to \Omega^1_M\otimes_{\mathcal{O}_M}\mathcal{E}

satisfying the integrability condition $\Phi \wedge \Phi =0$. The pair of data $\left(E,\Phi \right)$ is then called a Higgs bundle.

Higgs bundle can be considered as a limiting case of a flat connection in the limit in which its exterior differential tends to zero, be obtained by rescaling. So the equation $du/\mathrm{dz}=A\left(z\right)u$ where $A\left(z\right)$ is a matrix of connection can be rescaled by putting a small parameter in front of $du/\mathrm{dz}$.

Properties

Stability

For a Higgs bundle to admit a harmonic metric (…) it needs to be stable (…).

In nonabelian Hodge theory

In nonabelian Hodge theory the moduli space of stable Higgs bundles overa Riemann surface $X$ is identified with that of special linear group $\mathrm{SL}\left(n,ℂ\right)$ irreducible representations of its fundamental group ${\pi }_{1}\left(X\right)$.

Examples

Rank 1

In the special case that $E$ has rank 1, hence is a line bundle, the form $\Phi$ is simply any holomorphic 1-form. This case is also called that of an abelian Higgs bundle.

References

The moduli space of Higgs bundles over an algebraic curve is one of the principal topics in works of Nigel Hitchin and Carlos Simpson in late 1980-s and 1990-s (and later Ron Donagi, Tony Pantev…).

Around lemma 6.4.1 in

• Kevin Costello, Notes on supersymmetric and holomorphic field theories in dimension 2 and 4 (pdf)