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Higgs bundle

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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.

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 ϕ\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 ϕ\phi is a one-form on CC (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=4N = 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 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 completion 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.

Bundles of holomorphic forms

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

This is discussed in (Seaman 98)

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)

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

Revised on October 19, 2014 21:14:42 by Urs Schreiber (185.26.182.34)