# nLab Higgs bundle

Contents

### Context

#### Bundles

bundles

fiber bundles in physics

## Constructions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $\Phi \wedge \Phi = 0$.

Higgs bundles play a central role in nonabelian Hodge theory.

## Terminology

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.

(Witten 08, remark 2.1): As an aside, one may ask how closely related $\phi$, known in the present context as the Higgs field, is to the Higgs fields of particle physics. Thus, to what extent is the terminology that was introduced in Hitchin (1987a) actually justified? The main difference is that Higgs fields in particle physics are scalar fields, while $\phi$ is a one-form on $C$ (valued in each case in some representation of the gauge group). However, although Hitchin’s equations were first written down and studied directly, they can be obtained from N = 4 supersymmetric gauge theory via a sort of twisting procedure (similar to the procedure that leads from N = 2 supersymmetric gauge theory to Donaldson theory). In this twisting procedure, some of the Higgs-like scalar fields of $N = 4$ super Yang-Mills theory are indeed converted into the Higgs field that enters in Hitchin’s equations. $[$ Kapustin-Witten 06 $]$ This gives a reasonable justification for the terminology.

## Definition

### In components

Let $\mathcal{E}$ be a sheaf of sections of a holomorphic vector 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 completion 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 (…).

Stability is defined similarly to stability for holomorphic vector bundles except that instead of quantifying over all proper non-zero sub-bundles, one only considers proper $\Phi$ invariant sub-bundles. So in particular for $\Phi=0$, this is stability of the underlying vector bundle.

### In nonabelian Hodge theory

In nonabelian Hodge theory the moduli space of stable Higgs bundles over a 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.

### Bundles of holomorphic forms

Let $X$ be a complex manifold and $\omega \in \Omega^{k,0}(X)$ for odd $k$. Then $\Omega^{\bullet,0}(X)$ becomes a Higgs bundle when equipped with the endomorphis-valued 1-form which sends a holomorphic vector $v$ to the wedge product operation with the contraction of $\omega$ with $v$.

This is discussed in (Seaman 98)

### General

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, etc….).

Around lemma 6.4.1 in

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

Discussion of the topology of the moduli space of Higgs bundles is in

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

• Carlos Simpson, The Hodge filtration on nonabelian cohomology, Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 217{281. MR

1492538 (99g:14028) (arXiv:9604005)

• Alberto García Raboso, A twisted nonabelian Hodge correspondence, PhD thesis 2014 (pdf slides)

Discussion of the example of homolorphic forms is in

• Walter Seaman, Higgs Bundles and Holomorphic Forms (arXiv:9811097)

Discussion in the context of geometric Langlands duality includes