# 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

### In components

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

A Higgs field on $\mathcal{E}$ is an $\mathcal{O}_M$-linear map

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

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

(Notice that this is similar to but crucially different the definition of a flat connection on a vector bundle. For that the map $\Phi$ is just $\mathbb{C}$-linear and the integrability condiiton is $\mathbf{d}\phi + \Phi\wedge\Phi = 0$.)

Higgs bundles 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 $d u/dz = A(z)u$ where $A(z)$ is a matrix of connection can be rescaled by putting a small parameter in front of $d u/dz$.

### Formulation in D-geometry

Analogous to how the de Rham stack $\int_{inf} X = X_{dR}$ of $X$ is the (homotopy) quotient of $X$ by the first order infinitesimal neighbourhood of the diagonal in $X \times X$, so there is a space (stack) $X_{Dol}$ which is the formal competion of the 0-section of the tangent bundle of $X$ (Simpson 96).

Now a flat vector bundle on $X$ is essentially just a vector bundle on the de Rham stack $X_{dR}$, and a Higgs bundle is essentially just a vector bundle on $X_{Dol}$. Therefore in this language the nonabelian Hodge theorem reads (for $G$ a linear algebraic group over $\mathbb{C}$)

$\mathbf{H}(X_{dR}, \mathbf{B}G) \simeq \mathbf{H}(X_{Dol}, \mathbf{B}G)^{ss,0} \,,$

where the superscript on the right denotes restriction to semistable Higgs bundles with vanishing first Chern class (see Raboso 14, theorem 4.2).

## 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 $SL(n,\mathbb{C})$ irreducible representations of its fundamental group $\pi_1(X)$.

## 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)

Discussion in terms of $X_{Dol}$ is in