nLab
Higgs bundle

Contents

Idea

A Higgs bundle is a holomorphic vector bundle EE together with a 1-form Φ\Phi with values in the endomorphisms of (the fibers of) EE, such that ΦΦ=0\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 EE on complex manifold MM with structure sheaf 𝒪 M\mathcal{O}_M and module of Kähler differentials Ω M 1\Omega^1_M.

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

Φ:Ω M 1 𝒪 M \Phi : \mathcal{E}\to \Omega^1_M\otimes_{\mathcal{O}_M}\mathcal{E}

satisfying the integrability condition ΦΦ=0\Phi\wedge\Phi = 0. The pair of data (E,Φ)(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 Φ\Phi is just \mathbb{C}-linear and the integrability condiiton is dϕ+ΦΦ=0\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 du/dz=A(z)ud u/dz = A(z)u where A(z)A(z) is a matrix of connection can be rescaled by putting a small parameter in front of du/dzd u/dz.

Formulation in D-geometry

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

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

H(X dR,BG)H(X Dol,BG) ss,0, \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 XX is identified with that of special linear group SL(n,)SL(n,\mathbb{C}) irreducible representations of its fundamental group π 1(X)\pi_1(X).

Examples

Rank 1

In the special case that EE has rank 1, hence is a line bundle, the form Φ\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)

See also

Discussion in terms of X DolX_{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)

Revised on January 28, 2014 09:19:26 by Urs Schreiber (82.113.121.62)