nLab Higgs bundle

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 condition is $\mathbf{d}\phi + \Phi\wedge\Phi = 0$.)

Higgs bundles can be considered as a limiting case of a (holomorphic) flat connection in the limit in which its exterior differential $d\Phi$ is small in comparison to the exterior square $\Phi\wedge\Phi$, what can be obtained by rescaling the connection form. 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 $d u/dz$, or equivalently a large parameter in front of $A(z)$. Then the Maurer-Cartan equation has a term quadratic in the large parameter and the exterior derivative term is linear in large parameter and can be neglected, so the flatness reduces to the Higgs bundle condition $\Phi\wedge\Phi = 0$.

In fact, there is a notion interpolating between flat connections and Higgs bundles due Deligne, the Deligne’s $\lambda$-connection.

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)

Related concepts

• Hitchin system

• nonabelian Hodge theory

• topological recursion

References

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….), hence also known as the Hitchin moduli space.

• N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. , 54 (1987) pp. 91–114 MR89a:32021 doi euclid; The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, MR887284 doi; Flat connections and geometric quantization, Comm. Math. Phys. 131, n 2 (1990), 347-380, euclid

• Carlos Simpson, Higgs bundles and local systems, Publ. Mathématiques de l’IHÉS 75 (1992), p. 5-95, numdam, MR94d:32027

• Ludmil Katzarkov, Dmitri Orlov, Tony Pantev, Notes on Higgs bundles and D-branes, (delivered as a lecture by Tony Pantev at winter school at Guanajuato 2013 link) draft pdf

• S. B. Bradlow, O. García-Prada, P. B. Gothen, WHAT IS…a Higgs Bundle?, Notices AMS, pdf

• Michael Murray, Danny Stevenson, Higgs fields, bundle gerbes and string structures, arxiv:math.DG/0106179](https://arxiv.org/abs/math.DG/0106179)

• David Baraglia, Cyclic Higgs bundles and the affine Toda equations, arxiv:1011.6421

• Kevin Costello, around lemma 6.4.1 in: Notes on supersymmetric and holomorphic field theories in dimension 2 and 4, talk at Geometric and Algebraic Structures in Mathematics, Stony Brook (2011) [arXiv:1111.4234]

Review:

• Jan Swoboda, Moduli Spaces of Higgs Bundles – Old and New, Dtsch. Math. Ver. 123 (2021) 65–130 [doi:10.1365/s13291-021-00229-1]

See also:

• Donu Arapura, MO comment on Higgs bundles

• math.stackexchange:difference-between-higgs-bundle-and-vector-bundle-with-connection

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

• Steven Rayan, Aspects of the topology and combinatorics of Higgs bundle moduli spaces, SIGMA 14 (2018) 129, 18 pages (arXiv:1809.05732)

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

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Communications in Number Theory and Physics Volume 1 (2007) Number 1 (arXiv:hep-th/0604151)

• Edward Witten, Mirror Symmetry, Hitchin’s Equations, And Langlands Duality (arXiv:0802.0999)

The following paper starts with a survey of nonabelian Hodge correspondence,

• Peter B. Gothen, Higgs bundles and the real symplectic group, arXiv:1102.4175

Relation to $Spin(7)$-manifolds

Relating M-theory on Spin(7)-manifolds with F-theory on Spin(7)-manifolds via Higgs bundles:

• Mirjam Cvetic, Jonathan Heckman, Thomas B. Rochais, Ethan Torres, Gianluca Zoccarato, Geometric Unification of Higgs Bundle Vacua (arXiv:2003.13682)

Mirror symmetry

• Victoria Hoskins, Simon Pepin Lehalleur. Motivic mirror symmetry for Higgs bundles (2022). (arXiv:2205.15393).