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\(\mathrm {SO}(p,q)\)-Higgs bundles and Higher Teichmüller components

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Abstract

Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such ‘exotic’ components in moduli spaces of \(\mathrm {SO}(p,q)\)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group \(\mathrm {SO}(p,q)\). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for \(\mathrm {SO}(2,q)\), with \(q\geqslant 4\)).

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Notes

  1. This result was announced, without details, in [1]. We now provide the details of the proof.

  2. We note that for a rank two orthogonal bundle of the form (\(L\oplus L^*,\left( {\begin{matrix}0&{}{{\,\mathrm{Id}\,}}\\ {{\,\mathrm{Id}\,}}&{}0\end{matrix}}\right) )\), the isotropic subbundle L is not considered to be proper. This is because \(\mathrm {SO}(2,\mathbb {C})\cong \mathbb {C}^*\), so L does not define a proper reduction of structure group.

  3. The notation from Lemma 3.13 has changed slightly, \((W_{-1},\eta _{-1},W_0)\) is now represented by \((W_{-p},\eta _{-p},W_0')\).

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Acknowledgements

The authors are grateful to Olivier Guichard, Beatrice Pozzetti, Carlos Simpson, Richard Wentworth and Anna Wienhard for useful conversations and to the referee for a careful reading and for a number of helpful remarks and corrections.

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Correspondence to Oscar García-Prada.

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The authors acknowledge support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). The third author is funded by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship, NSF MSPRF no. 1604263. The fourth author was partially supported by the Spanish MINECO under ICMAT Severo Ochoa Project No. SEV-2015-0554, and under Grant No. MTM2016-81048-P. The fifth and sixth authors were partially supported by CMUP (UID/MAT/00144/2019) and the Project PTDC/MAT-GEO/2823/2014 funded by FCT (Portugal) with national funds. The sixth author was also partially supported by the Post-Doctoral fellowship SFRH/BPD/100996/2014 funded by FCT (Portugal) with national funds

André Oliveira on leave from: Universidade de Trás-os-Montes e Alto Douro.

Appendix A: Review of gauge theory and the Hitchin–Kobayashi correspondence

Appendix A: Review of gauge theory and the Hitchin–Kobayashi correspondence

Details on points treated sketchily in the following can be found in [24, 34]. For simplicity we consider K-twisted Higgs bundles but analagous statements can be made for L-twisted Higgs bundles.

Let \(\mathrm {G}\) be a real semisimple Lie group and \(\mathrm {H}\subset \mathrm {G}\) a maximal compact subgroup. Let P be a \(C^\infty \) principal \(\mathrm {H}^\mathbb {C}\)-bundle and fix a reduction to a principal \(\mathrm {H}\)-bundle \(P_{\mathrm {H}}\). Hitchin’s self-duality equations are

$$\begin{aligned} \begin{aligned} F(A)-[\varphi ,\tau (\varphi )]&=0,\\ {\bar{\partial }}_A\varphi&=0. \end{aligned} \end{aligned}$$
(A.1)

Here A is a \(\mathrm {H}\)-connection on \(P_{\mathrm {H}}\), \({\bar{\partial }}_A\) its associated \({\bar{\partial }}\)-operator and \(\varphi \in \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\). The map \(\tau :\Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\rightarrow \Omega ^{0,1}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\) is obtained by combining the compact real structure on \(\mathfrak {g}^\mathbb {C}\) with conjugation on the form component.

A pair \((A,\varphi )\) gives a corresponding \(\mathrm {G}\)-Higgs bundle structure \(({\bar{\partial }}_A,\varphi )\) on P; we denote the corresponding \(\mathrm {G}\)-Higgs bundle by \(({\mathcal E}_A,\varphi )\). Conversely, given a \(\mathrm {G}\)-Higgs bundle \(({\mathcal E}_A,\varphi )\), where the holomorphic bundle \({\mathcal E}_A\) is defined by \({\bar{\partial }}_A\), one obtains a pair \((A,\varphi )\) by taking A to be the Chern connection associated to \({\bar{\partial }}_A\) via the fixed reduction \(P_{\mathrm {H}}\subset P\). The Hitchin–Kobayashi correspondence for \(\mathrm {G}\)-Higgs bundles [24] says that the \(\mathrm {G}\)-Higgs bundle is polystable if and only if there is a structure \(({\bar{\partial }}_A,\varphi )\) in its \({\mathcal G}_{\mathrm {H}^\mathbb {C}}\)-orbit such that the corresponding pair \((A,\varphi )\) solves Hitchin’s equations. Moreover, this pair is unique up to \({\mathcal G}_{\mathrm {H}}\)-gauge transformations, where \(\mathcal {G}_{\mathrm {H}}\) denotes the gauge group of \(\mathrm {H}\)-gauge transformations of \(P_{\mathrm {H}}\).

We recall the following alternative point of view. Instead of fixing a reduction of the principal \(\mathrm {H}^\mathbb {C}\)-bundle P, we can consider a fixed structure of \(\mathrm {G}\)-Higgs bundle \(({\bar{\partial }}_A,\varphi )\) and consider (A.1) as equations for a reduction of structure group to \(\mathrm {H}\subset \mathrm {H}^\mathbb {C}\), usually known as a a harmonic metric . The Hitchin–Kobayashi correspondence then says that such a reduction exists if and only if \(({\bar{\partial }}_A,\varphi )\) defines a polystable \(\mathrm {G}\)-Higgs bundle.

The space \(\mathcal {A}\) of \(\mathrm {H}\)-connections on \(P_{\mathrm {H}}\) is an affine space modeled on \(\Omega ^1(P_{\mathrm {H}}[\mathfrak {h}])\). Let \(\mathcal {C}\subset \mathcal {A}\times \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\) denote the configuration space of solutions to Hitchin’s equations (A.1). As a set, the moduli space of solutions to Hitchin’s self-duality equations is

$$\begin{aligned} \mathcal {M}^a_{\mathrm {H}}(\mathrm {G}) = \mathcal {C}/\mathcal {G}_{\mathrm {H}}, \end{aligned}$$

where a is the topological type. We shall denote by \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\) the union of the moduli spaces \(\mathcal {M}^a_{\mathrm {H}}(\mathrm {G})\) over all topological types a. In order to give the moduli space a topology, suitable Sobolev completions must be used in standard fashion; see [4], and also [33, Sec. 8] where the straightforward adaptation to Higgs bundles is discussed in the case \(\mathrm {G}=\mathrm {GL}(n,\mathbb {C})\). The moduli space \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\) then becomes a Hausdorff topological space.

The Hitchin–Kobayashi correspondence can now be stated as saying that the map

$$\begin{aligned} \begin{aligned} {\mathcal M}_{\mathrm {H}}(\mathrm {G})&\xrightarrow {\cong } \mathcal {M}(\mathrm {G}),\\ (A,\varphi )&\mapsto ({\bar{\partial }}_A,\varphi ) \end{aligned} \end{aligned}$$
(A.2)

is a bijection. It follows from the constructions that it is in fact a homeomorphism. Here and below, in analogy with Notation 3.1, we do not distinguish notationally between a pair \((A,\varphi )\) and its gauge equivalence class.

The moduli space \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\) can be given additional structure by considering the deformation complex

$$\begin{aligned} \Omega ^0(P_{\mathrm {H}}(\mathfrak {h})) \xrightarrow {d_0} \Omega ^1(P_{\mathrm {H}}(\mathfrak {h}))\times \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}]) \xrightarrow {d_1} \Omega ^2(P_{\mathrm {H}}(\mathfrak {h}))\times \Omega ^{1,1}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}]). \end{aligned}$$
(A.3)

The operator \(d_0\) is given by the infinitesimal action of the gauge group and the operator \(d_1\) is obtained by linearizing Hitchin’s equations; the fact that \(d_1\circ d_0=0\) follows because \((A,\varphi )\) is a solution. Denote the ith cohomology group of this complex by \(H^i_{(A,\varphi )}\).

Proposition A.1

Let \((A,\varphi )\) be a solution to Hitchin’s equations and let \((\mathcal {E},\varphi )\) be the corresponding Higgs bundle. Then there are isomorphisms

$$\begin{aligned} H^0_{(A,\varphi )}\otimes \mathbb {C}&\cong \mathbb {H}^0(C^\bullet (\mathcal {E},\varphi )),\\ H^1_{(A,\varphi )}&\cong \mathbb {H}^1(C^\bullet (\mathcal {E},\varphi )),\\ H^2_{(A,\varphi )}&\cong \mathbb {H}^2(C^\bullet (\mathcal {E},\varphi )) \oplus H^0_{(A,\varphi )}, \end{aligned}$$

where \(C^\bullet (\mathcal {E},\varphi )\) is the deformation complex (2.4).

Proof

The hypercohomology groups of the complex \(C^\bullet (\mathcal {E},\varphi )\) can be calculated, using a Dolbeault resolution, as the cohomology groups of the complex

$$\begin{aligned} \Omega ^0(P[\mathfrak {h}^\mathbb {C}]) \xrightarrow {\delta _0} \Omega ^{0,1}(P[\mathfrak {h}^\mathbb {C}])\times \Omega ^{1,0}(P[\mathfrak {m}^\mathbb {C}]) \xrightarrow {\delta _1} \Omega ^{1,1}(P[\mathfrak {m}^\mathbb {C}]), \end{aligned}$$
(A.4)

where the differentials are constructed combining the adjoint action of \(\varphi \) with \({\bar{\partial }}_A\). The proposition now follows essentially as in [20, Sec. 6.4.2] (which gives the analogous comparison between the deformation complexes for solutions to the anti-self duality equations and holomorphic vector bundles on a complex surface) using the Kähler identities and the bundle isomorphisms

$$\begin{aligned} \Omega ^{0,1}(P[\mathfrak {h}^\mathbb {C}])&\cong \Omega ^{1}(P_{\mathrm {H}}[\mathfrak {h}])\\ \Omega ^{0}(P[\mathfrak {h}^\mathbb {C}])&\cong \Omega ^{0}(P_{\mathrm {H}}[\mathfrak {h}])\oplus \Omega ^{2}(P_{\mathrm {H}}[\mathfrak {h}]). \end{aligned}$$

\(\square \)

Proposition A.2

Let \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}(\mathrm {G})\) and let \((\mathcal {E}_A,\varphi )\) be the corresponding polystable \(\mathrm {G}\)-Higgs bundle. Then the following statements are equivalent:

  1. 1.

    \(H^0_{(A,\varphi )} =0\) and \(H^2_{(A,\varphi )}=0\).

  2. 2.

    \(\mathbb {H}^0(C^\bullet (\mathcal {E}_A,\varphi ))=0\) and \(\mathbb {H}^2(C^\bullet (\mathcal {E}_A,\varphi ))=0\).

  3. 3.

    \((\mathcal {E}_A,\varphi )\) is stable as a \(\mathrm {G}^\mathbb {C}\)-Higgs bundle.

Proof

The equivalence of the first two statements is immediate from Proposition A.1. The equivalence of the last two statements is also immediate in view of Remarks 2.13 and 2.14. \(\square \)

Definition A.3

Let \(\mathcal {C}^s\subset \mathcal {C}\) denote the subspace of pairs \((A,\varphi )\) such that \((\mathcal {E}_A,\varphi )\) is stable as a \(\mathrm {G}^\mathbb {C}\)-Higgs bundle. Similarly, let \(\mathcal {C}_{\mathbb {C}}^s \subset \mathcal {A}\times \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\) denote the subspace of pairs \((A,\varphi )\) such that \({\bar{\partial }}_A\varphi =0\) and \((\mathcal {E}_A,\varphi )\) is stable as a \(\mathrm {G}^\mathbb {C}\)-Higgs bundle. Define \(\mathcal {M}_{\mathrm {H}}^s(\mathrm {G})\subset {\mathcal M}_{\mathrm {H}}(\mathrm {G})\) and \(\mathcal {M}^s(\mathrm {G})\subset \mathcal {M}(\mathrm {G})\) analogously.

We note that \(\mathcal {C}_{\mathbb {C}}^s \) is an infinite dimensional Kähler manifold whose Kähler structure is induced from the ambient space \(\mathcal {A}\times \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\).

Let \(\Gamma _{(A,\varphi )}\subset \mathcal {G}_{\mathrm {H}}\) denote the stabilizer of a solution \((A,\varphi )\) to Hitchin’s equations. This is a compact Lie group with Lie algebra \(H^0_{(A,\varphi )}\) [24]. The standard gauge theoretic construction of the moduli space can now be summarized as follows.

Proposition A.4

The subspace of \(\mathcal {C}\) where \(H^2_{(A,\varphi )}=0\) is a smooth infinite dimensional manifold. Moreover, for \((A,\varphi )\) with \(H^2_{(A,\varphi )}=0\) a neighbourhood of the corresponding point in the moduli space is modeled on a neighbourhood of zero in \(H^1_{(A,\varphi )}\) modulo the action of \(\Gamma _{(A,\varphi )}\). If additionally \(H^0_{(A,\varphi )} =0\), then \(\Gamma _{(A,\varphi )}\) is finite. Thus \(\mathcal {M}^s_{\mathrm {H}}(\mathrm {G})\) is a Kähler orbifold with Kähler form induced from \(\mathcal {C}_{\mathbb {C}}^s\).

Remark A.5

The action of \(\mathcal {G}_{\mathrm {H}}\) on \(\mathcal {C}_{\mathbb {C}}^s\) is Hamiltonian with moment map \(\mu (A,\varphi )= F(A)-[\varphi ,\tau (\varphi )]\). Hence the moduli space \(\mathcal {M}^s(G)\) can be viewed as the infinite dimensional symplectic quotient

$$\begin{aligned} \mathcal {M}^s(G) = \mu ^{-1}(0)/\mathcal {G}_{\mathrm {H}} \cong \mathcal {C}_{\mathbb {C}}^s / \mathcal {G}_{\mathrm {H}^\mathbb {C}}. \end{aligned}$$

The isomorphism comes from the Hitchin–Kobayashi correspondence, which can thus be viewed as an infinite dimensional Kempf–Ness correspondence. Note that the Kähler form on \(\mathcal {C}_{\mathbb {C}}^s\) restricts to a 2-form on \(\mathcal {C}^s\) which is non-degenerate in directions transverse to the \(\mathcal {G}_{\mathrm {H}}\)-orbits — indeed this is just the pullback of the Kähler form on \(\mathcal {M}^s(\mathrm {G})\).

The following was proved in [24]. It is analogous to the decomposition of a polystable vector bundle into a direct sum of stable ones, and plays a central role in the proof of the Hitchin–Kobayashi correspondence.

Proposition A.6

Let \((\mathcal {E},\varphi )\) be a polystable \(\mathrm {G}\)-Higgs bundle. Then there is a real reductive subgroup \(\mathrm {G}'\subset \mathrm {G}\) and a Jordan–Hölder reduction of \((\mathcal {E},\varphi )\) to a stable \(\mathrm {G}'\)-Higgs bundle \((\mathcal {E}',\varphi ')\). The Jordan–Hölder reduction is unique up to isomorphism. Moreover, the solution to Hitchin’s equations on \((\mathcal {E}',\varphi ')\) induces the solution on \((\mathcal {E},\varphi )\).

Next we recall Hitchin’s method [34, 35] for studying the topology of \(\mathcal {M}(\mathrm {G})\) using gauge theoretic methods, and explain how to translate it to the holomorphic point of view. Alternatively one could work exclusively using the holomorphic point of view, using Simpson’s adaptation in [46, Sec. 11].

Similarly to the holomorphic action of \(\mathbb {C}^*\) on \(\mathcal {M}(G)\) defined in Sect. 3.1, there is an action of \(S^1\) on \(\mathcal {A}\times \Omega ^{1,0}(P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}])\) given by

$$\begin{aligned} e^{i\theta }\cdot (A,\varphi ) = (A,e^{i\theta }\varphi ). \end{aligned}$$

This action clearly preserves the subspaces \(\mathcal {C}^s\), \(\mathcal {C}\) and \(\mathcal {C}^s_{\mathbb {C}}\), and it descends to \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\).

Proposition A.7

Let \(S^1\) act on \(\mathcal {M}(G)\) by restriction of the \(\mathbb {C}^*\)-action defined above. Then the following statements hold.

  1. 1.

    The bijection \({\mathcal M}_{\mathrm {H}}(\mathrm {G}) \rightarrow \mathcal {M}(\mathrm {G})\) defined in (A.2) is \(S^1\)-equivariant.

  2. 2.

    The class of \((A,\varphi )\) in \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\) is fixed under the \(S^1\)-action if and only if the class of the corresponding Higgs bundle \((\mathcal {E}_A,\varphi )\) in \(\mathcal {M}(\mathrm {G})\) is fixed under the \(\mathbb {C}^*\)-action.

Proof

Statement (1) is clear. Statement (2) is a consequence of the Hitchin–Kobayashi correspondence. \(\square \)

Since the vector bundle \(P[\mathfrak {m}^\mathbb {C}] \cong P_{\mathrm {H}}[\mathfrak {m}^\mathbb {C}]\) has a Hermitian metric coming from the reduction of structure group to \(\mathrm {H}\), one can define the Hitchin function:

$$\begin{aligned} f:{\mathcal M}_{\mathrm {H}}(\mathrm {G})\rightarrow \mathbb {R},\quad (A,\varphi )\mapsto \int _X||\varphi ||^2. \end{aligned}$$
(A.5)

We shall abuse notation and denote by the same letter the map \(f:{\mathcal M}(\mathrm {G})\rightarrow \mathbb {R}\) induced via the identification (A.2). Using Uhlenbeck’s weak compactness theorem, Hitchin [34] showed that the map f is proper. Thus, as noted in Sect. 3.1, the Hitchin function can be used to study the connected components of the moduli space of \(\mathrm {G}\)-Higgs bundles.

The following is central for identifying local minima of f.

Lemma A.8

Let \((A,\varphi )\in \mathcal {M}^s_{\mathrm {H}}(\mathrm {G})\). If \((A,\varphi )\) is a local minimum of f, then it is a fixed point of the \(S^1\)-action. Equivalently, the corresponding Higgs bundle \(({\mathcal E}_A,\varphi )\in {\mathcal M}^s(\mathrm {G})\) is a fixed point of the \(\mathbb {C}^*\)-action.

Proof

On the smooth locus of \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\), the \(S^1\)-action is Hamiltonian with respect to the Kähler form and the function f (suitably normalized) is a moment map for this action (see [34, 35]). This means that, when multiplied by \(\sqrt{-1}\), the vector field generating the \(S^1\)-action is the gradient of f and, therefore, critical points of f are exactly the fixed points of the \(S^1\)-action. This proves the proposition when \(\Gamma _{(A,\varphi )}\) is trivial.

For a general \((A,\varphi )\in \mathcal {M}^s_{\mathrm {H}}(\mathrm {G})\) we can argue on the smooth manifold \(\mathcal {C}^s\subset \mathcal {C}^s_{\mathbb {C}}\). Indeed, by its very definition, the function f lifts to the infinite dimensional Kähler manifold \(\mathcal {C}^s_{\mathbb {C}}\) and it is a moment map for the \(S^1\)-action there. Thus, in view of Remark A.5, and in a similar way to the argument of the preceding paragraph, it follows that \((A,\varphi )\) is a critical point of f restricted to \(\mathcal {C}^s\) if and only if its \(\mathcal {G}_{\mathrm {H}}\)-gauge equivalence class is fixed by the \(S^1\)-action. \(\square \)

We have the following useful observation. Let \(\mathrm {G}'\subset \mathrm {G}\) be a reductive subgroup (we take this to include the choice of compatible Cartan data). Then a solution \((A,\varphi )\) to Hithin’s equations for \(\mathrm {G}'\) on a principal \(\mathrm {H}'\)-bundle induces a solution for \(\mathrm {G}\) on the \(\mathrm {H}\)-bundle obtained by extension of structure group. Hence we have a well defined map

$$\begin{aligned} {\mathcal M}(\mathrm {G}')\longrightarrow {\mathcal M}(\mathrm {G}) \end{aligned}$$

which is clearly compatible with the respective Hitchin functions. This leads immediately to the following result.

Lemma A.9

Let \(\mathrm {G}'\subset \mathrm {G}\) be a reductive subgroup. Suppose \(({\mathcal E},\varphi )\) is a \(\mathrm {G}\)-Higgs bundle which reduces to a \(\mathrm {G}'\)-Higgs bundle. If \(({\mathcal E},\varphi )\) is a minimum of the Hitchin function on \({\mathcal M}(\mathrm {G})\) then it is a minimum of the Hitchin function on \({\mathcal M}(\mathrm {G}')\).

A solution \((A,\varphi )\) to Hitchin’s equations is called simple if its stabilizer \(\Gamma _{(A,\varphi )}\) is trivial. The following proposition is simple to check.

Proposition A.10

Suppose that \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}(\mathrm {G})\) is a fixed point for the \(S^1\)-action. Then for each \(e^{i\theta }\) there is a gauge transformation \(g(\theta )\in \mathcal {G}_{\mathrm {H}}\) such that

$$\begin{aligned} g(\theta )\cdot (A,\varphi ) = (A,e^{i\theta }\varphi ). \end{aligned}$$

The gauge transformation \(g(\theta )\) is determined up to an element of the stabilizer \(\Gamma _{(A,\varphi )}\). Moreover, if \((A,\varphi )\) is simple, then \(e^{i\theta } \mapsto g(\theta )\) defines a group homomorphism \(S^1\rightarrow \mathcal {G}_{\mathrm {H}}.\)

Proposition A.11

Suppose \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}(\mathrm {G})\) is a fixed point for the \(S^1\)-action. If \((A,\varphi )\) is simple, then there is an induced action of \(S^1\) on \(H^1_{(A,\varphi )}\).

Proof

For each \(e^{i\theta }\), the derivative of its action on \(\mathcal {C}\) defines a map \(H^1_{(A,\varphi )}\rightarrow H^1_{(A,e^{i\theta }\varphi )}\). Composing with the inverse of the derivative of the unique gauge transformation \(g(\theta )\) from Proposition A.10 we get a well defined map \(H^1_{(A,\varphi )}\rightarrow H^1_{(A,\varphi )}\). Using the fact that \(\theta \rightarrow g(\theta )\) is a group homomorphism it is easy to see that this gives an action of \(S^1\). \(\square \)

If \((A,\varphi )\) has discrete stabilizer, then for each \(\theta _0\) and each choice of gauge transformation \(g(\theta _0)\) as in Proposition A.10, there is a unique smooth family \(g(\theta )\) defined in a neighborhood of \(\theta _0\). Taking \(\theta _0=0\) and g(0) to be the identity we get the following result, by an argument similar to the proof of the preceding proposition.

Proposition A.12

Suppose \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}^s(\mathrm {G})\) is a fixed point for the \(S^1\)-action. Then there is an induced local action of a neighborhood of the identity in \(S^1\) on \(H^1_{(A,\varphi )}\). In particular, there is an inifinitesimal \(S^1\)-action on \(H^1_{(A,\varphi )}\), and a well-defined infinitesimal gauge transformation \(\psi =\frac{dg_\theta }{d\theta }\big \vert _{\theta =0}\in \Omega ^0(P_{\mathrm {H}}[\mathfrak {h}])\).

Remark A.13

Note that \([\psi ,\varphi ] = i\phi \) because \(g(\theta )\cdot (A,\varphi ) = (A,e^{i\theta }\varphi )\).

Now fix a maximal torus \(\mathfrak {t}\subset \mathfrak {h}\). Since any element of \(\mathfrak {h}\) is conjugate to an element in \(\mathfrak {t}\), there is a point \(p_0\in P_{\mathrm {H}}\) with the property stated in the following proposition.

Proposition A.14

Let \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}^s(\mathrm {G})\) be a fixed point and let \(\psi \in \Omega ^0(P_H[\mathfrak {h}])\) be the infinitesimal gauge transformation provided by Proposition A.12. Let \(p_0\in P\) be such that the infinitesimal gauge transformation provided by Proposition A.12 satisfies \(\psi (p_0)\in \mathfrak {t}\). Define

$$\begin{aligned} \mathrm {H}_0 = Z_{\mathrm {H}}(\psi (p_0))\subset \mathrm {H}. \end{aligned}$$

Then there is a subbundle \(P_{\mathrm {H}_0}\subset P_{\mathrm {H}}\) which gives a reduction of structure group to \(\mathrm {H}_0\).

Proof

Define

$$\begin{aligned} P_{\mathrm {H}_0}=\{p\in P_{\mathrm {H}}\;\;|\;\;\psi (p)\in \mathfrak {t}\} \subset P_{\mathrm {H}}. \end{aligned}$$

Let \(\psi (p)\in \mathfrak {t}\). A point \({{\,\mathrm{Ad}\,}}(h)(\psi (p))=\psi (ph^{-1})\) in the adjoint orbit of \(\psi (p)\) lies in \(\mathfrak {t}\) if and only if \(h\in Z_{\mathrm {H}}(\psi (p))\). Moreover, this centralizer does not depend on the choice of \(\psi (p)\) in the adjoint orbit, as long as \(\psi (p)\) lies in \(\mathfrak {t}\). We therefore have an identification of the fiber \(P_{H_0,x}\) of \(P_{H_0}\) over \(x=\pi (p_0)\in X\):

$$\begin{aligned} \mathrm {H}_0&\xrightarrow {\cong } P_{H_0,x},\\ c&\mapsto p_0\cdot c \end{aligned}$$

where the action comes from the right action of H on \(P_{H_0}\).

Now note that, since \(d_A\psi =0\), the eigenvalues for the action of \(\psi \) on \(P_{\mathrm {H}}[\mathfrak {h}]\) are constant. Hence the orbit in \(\mathfrak {h}\) of \(\psi (p)\) under the adjoint action of \(\mathrm {H}\) is independent of \(p\in P_{\mathrm {H}}\). It follows that the centralizer used in the preceding paragraph is the same for all fibers of \(P_{\mathrm {H}}\) and, therefore, the construction globalizes to show that \(P_{\mathrm {H}_0}\subset P_{\mathrm {H}}\) defines a reduction of structure group, as we wanted. \(\square \)

Remark A.15

Since the reduction \(P_{\mathrm {H}_0}\subset P_{\mathrm {H}}\) just constructed only depends on the choice of the maximal torus \(\mathfrak {t}\subset \mathfrak {h}\), it is unique up to conjugation by \(\mathrm {H}\).

Proposition A.16

Suppose \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}^s(\mathrm {G})\) is a fixed point for the \(S^1\)-action. Then there is a weight decomposition into ik-eigenspaces for the adjoint action of \(\psi \) on the Lie algebra bundles \(P_H[\mathfrak {h}^\mathbb {C}]\) and \(P_H[\mathfrak {m}^\mathbb {C}]\):

$$\begin{aligned} P_\mathrm {H}[\mathfrak {h}^\mathbb {C}] = \bigoplus _k P_\mathrm {H}[\mathfrak {h}^\mathbb {C}]_{k} \quad \text {and}\quad P_\mathrm {H}[\mathfrak {m}^\mathbb {C}] = \bigoplus _k P_\mathrm {H}[\mathfrak {m}^\mathbb {C}]_{k}, \end{aligned}$$

where \(\varphi \in H^0(P_H[\mathfrak {m}^\mathbb {C}]_1\otimes K)\) and \(P_\mathrm {H}[\mathfrak {h}^\mathbb {C}]_{0}\) is identified with the adjoint bundle \(P_{\mathrm {H}_0}[\mathfrak {h}^\mathbb {C}_0]\).

Proof

This is immediate from Proposition A.14—indeed, taking the weight space decomposition \(\mathfrak {h}^\mathbb {C}=\bigoplus \mathfrak {h}^\mathbb {C}_k\) for the adjoint action of \(\psi (p_0)\) we have \(P_\mathrm {H}[\mathfrak {h}^\mathbb {C}]_{k} = P_{\mathrm {H}_0}[\mathfrak {h}^\mathbb {C}_{k}]\), and similarly for \(\mathfrak {m}^\mathbb {C}\). The fact that \(\varphi \) has weight one follows from Remark A.13. \(\square \)

Remark A.17

For any fixed \((A,\varphi )\) in the moduli space we can use the Jordan–Hölder reduction to a stable \(\mathrm {G}'\)-Higgs bundle to get a reduction of structure group as in Proposition A.14. However, the weight decomposition of Proposition A.16 is, in general, no longer well defined. This is because the center of the maximal compact \(\mathrm {H}'\subset \mathrm {G}'\) may act non-trivially on the complement of \({\mathfrak {g}'}^\mathbb {C}\) in \(\mathfrak {g}^\mathbb {C}\).

For a \(({\mathcal E}, \varphi )\in {\mathcal M}^ s(\mathrm {G})\) which is fixed under the \(\mathbb {C}^*\)-action, the weight decomposition from Proposition  A.16 translates into

$$\begin{aligned} {\mathcal E}[\mathfrak {g}^\mathbb {C}] = {\mathcal E}[\mathfrak {h}^\mathbb {C}] \oplus {\mathcal E}[\mathfrak {m}^\mathbb {C}] = \bigoplus {\mathcal E}[\mathfrak {h}^\mathbb {C}]_k \oplus \bigoplus {\mathcal E}[\mathfrak {m}^\mathbb {C}]_k \end{aligned}$$
(A.6)

with \({\mathcal E}[\mathfrak {h}^\mathbb {C}]_{k} = P_\mathrm {H}[\mathfrak {h}^\mathbb {C}]_{k}\) and \({\mathcal E}[\mathfrak {m}^\mathbb {C}]_{k} = P_\mathrm {H}[\mathfrak {m}^\mathbb {C}]_{k}\), and where \(\varphi \in H^0({\mathcal E}[\mathfrak {m}^\mathbb {C}]_1\otimes K)\). This gives a decomposition \(C^\bullet ({\mathcal E},\varphi ) = \bigoplus C^\bullet _k({\mathcal E},\varphi )\) of the deformation complex (2.4), where

$$\begin{aligned} C^\bullet _k({\mathcal E},\varphi ): {\mathcal E}[\mathfrak {h}^\mathbb {C}]_{k}\xrightarrow {{{\,\mathrm{ad}\,}}_\varphi } {\mathcal E}[\mathfrak {m}^\mathbb {C}]_{k+1}\otimes K. \end{aligned}$$
(A.7)

Proposition A.18

Suppose \((A,\varphi )\in {\mathcal M}_{\mathrm {H}}^s(\mathrm {G})\) is a fixed point for the \(S^1\)-action. Let \(H^1_{(A,\varphi )} = \bigoplus H^1_{(A,\varphi ),k}\) be the decomposition into ik-eigenspaces for the infinitesimal \(S^1\)-action given by Proposition A.12. Then there are canonical isomorphisms

$$\begin{aligned} H^1_{(A,\varphi ),k} \cong \mathbb {H}^1(C^\bullet _k(\mathcal {E}_A,\varphi )). \end{aligned}$$

Proof

In a similar way to Proposition A.12, there is an infinitesimal \(S^1\)-action on \(\mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))\) and, clearly, the isomorphism \(H^1_{(A,\varphi )} \cong \mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))\) of Proposition A.1 is \(S^1\)-equivariant. Thus there is a weight space decomposition \(\mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))=\bigoplus \mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))_k\) with \(H^1_{(A,\varphi ),k} \cong \mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))_k\). It remains to see that \(\mathbb {H}^1(C^\bullet (\mathcal {E}_A,\varphi ))_k \cong \mathbb {H}^1(C^\bullet _k(\mathcal {E}_A,\varphi ))\) and this is an easy check using the induced weight decomposition of the Dolbeault resolution (A.4). \(\square \)

We shall use the subscript “+” for the direct sums of subspaces with \(k>0\).

Lemma A.19

Let \((A,\varphi )\in \mathcal {M}^s_{\mathrm {H}}(\mathrm {G})\). If \((A,\varphi )\) is a fixed point of the \(S^1\)-action, then it is a local minimum of the Hitchin function if and only if \(H^1_{(A,\varphi ),k}=0\) for all \(k>0\). Equivalently, a fixed point \(({\mathcal E},\varphi )\in {\mathcal M}^s(\mathrm {G})\) for the \(\mathbb {C}^*\)-action is a local minimum of the Hitchin function if and only if \(\mathbb {H}^1(C^\bullet _k(\mathcal {E},\varphi ))=0\) for all \(k>0\).

Proof

Hitchin [34, 35] showed that on the smooth locus of \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\), the subspace \(H^1_{(A,\varphi ),k}\) can be identified with the \(-k\)-eigenspace for the Hessian of f. The extension to points of \({\mathcal M}_{\mathrm {H}}(\mathrm {G})\) which are orbifold singularities follows as in the proof of Lemma A.8. The equivalence of the statement for \(({\mathcal E},\varphi )\in {\mathcal M}^s(\mathrm {G})\) follows from Proposition A.18. \(\square \)

We shall also need to show that certain \(\mathrm {G}\)-Higgs bundles which do not satisfy the hypothesis of Proposition 3.4 are not local minima of f. To this end we have the following result, analogous to a criterion of Simpson [46, Lemma 11.8].

Lemma A.20

Let \(({\mathcal E}_0,\varphi _0)\in \mathcal {M}(G)\) be a fixed point of the \(\mathbb {C}^*\)-action. Suppose there exists a semistable \(\mathrm {G}\)-Higgs bundle \(({\mathcal E},\varphi )\), which is not \({\mathcal S}\)-equivalent to \(({\mathcal E}_0,\varphi _0)\), and such that \(\lim _{t\rightarrow \infty }({\mathcal E},t\varphi )=({\mathcal E}_0,\varphi _0)\) in \(\mathcal {M}(G)\). Then \(({\mathcal E}_0,\varphi _0)\) is not a local minimum of f.

Proof

Replacing \(({\mathcal E},\varphi )\) with the polystable representative of its \({\mathcal S}\)-equivalence class, we may assume that it is polystable. Note also that \(({\mathcal E},\varphi )\) cannot be a fixed point of the \(\mathbb {C}^*\)-action.

Consider first the case when \(({\mathcal E},\varphi )\) is stable. Then, as in the proof of Lemma A.8, we can use the moment map interpretation of f to deduce that the function \(\mathbb {R}_{>0}\rightarrow \mathbb {R}\) defined by \(t\mapsto f({\mathcal E},t\varphi )\) is strictly increasing as t tends to infinity. For the general case, consider the Jordan–Hölder reduction of \(({\mathcal E},\varphi )\) given by Proposition A.6. This is a stable \(\mathrm {G}'\)-Higgs bundle for some \(\mathrm {G}'\subset \mathrm {G}\) and cannot be fixed under the \(\mathbb {C}^*\)-action, since otherwise \(({\mathcal E},\varphi )\) would also be fixed. Since the natural map \(\mathcal {M}(\mathrm {G}')\rightarrow \mathcal {M}(\mathrm {G})\) is \(\mathbb {C}^*\)-equivariant and compatible with the respective Hitchin functions, the result follows by the same argument as in the previous paragraph. \(\square \)

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Aparicio-Arroyo, M., Bradlow, S., Collier, B. et al. \(\mathrm {SO}(p,q)\)-Higgs bundles and Higher Teichmüller components. Invent. math. 218, 197–299 (2019). https://doi.org/10.1007/s00222-019-00885-2

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