Higgs bundles for the non-compact dual of the special orthogonal group

Abstract

Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group $G$. In this paper we examine the case $G=\mathrm {SO}^*(2n)$. We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in $\mathrm {SO}^*(2n)$.

Appendix: \(G\)-Higgs bundles for other groups

We collect here some basic results about $G$-Higgs bundles for groups other than $\mathrm {SO}^*(2n)$ which play a role in our analysis of $\mathrm {SO}^*(2n)$-Higgs bundles. The groups include three complex reductive groups ($\mathrm {GL}(n,\mathbb {C})$, $\mathrm {SL}(n,\mathbb {C})$ and $\mathrm {SO}(n,\mathbb {C})$) and two non-compact real forms ($\mathrm {U}(p,q)$ and $\mathrm {U}^*(2n)$).

In all cases the basic definitions of stability properties follow from the general definition formulated for $G$-Higgs bundles in [15].

1.1 The groups $\mathrm {GL}(n,\mathbb {C}), \mathrm {SL}(n,\mathbb {C})$ and $\mathrm {SO}(n,\mathbb {C})$

We begin by recalling how the notion of $G$-Higgs bundle specializes when $G$ is a complex group. In this case, the complexified isotropy representation is just the adjoint representation of $G$ on $\mathfrak {g}$. Thus, a $G$-Higgs bundle for a complex group $G$ is a pair $(E,\varphi )$, where $E \rightarrow X$ is a holomorphic principal $G$-bundle and $\varphi \in H^0(X,{{\mathrm{Ad}}}E\otimes K)$; here ${{\mathrm{Ad}}}E = E\times _{{{\mathrm{Ad}}}}\mathfrak {g}$ is the adjoint bundle of $E$. We shall use this observation for all three groups considered in this section.

Consider first the case of $G=\mathrm {GL}(n,\mathbb {C})$. A $\mathrm {GL}(n,\mathbb {C})$-Higgs bundle may be viewed as a pair consisting of a rank $n$ holomorphic vector bundle $E$ over $X$ and a holomorphic section

$$\begin{aligned} \Phi \in H^0(X,K \otimes {{\mathrm{End}}}E). \end{aligned}$$

We refer the reader to [15] for the general statement of the stability conditions for $\mathrm {GL}(n,\mathbb {C})$-Higgs bundles. The notions of (semi-,poly-)stability in this case are equivalent to the original notions given by Hitchin in [23] (see [15]). Denote by $\mu (E) = \deg (E) / {{\mathrm{rk}}}(E)$ the slope of $E$.

Proposition 8.1

A $\mathrm {GL}(n,\mathbb {C})$-Higgs bundle $(E,\Phi )$ is semistable if and only if for any subbundle $E'\subset E$ such that $\Phi (E')\subset E' \otimes K$ we have $\mu (E')\le \mu (E)$. Furthermore, $(E,\Phi )$ is stable if for any nonzero and strict subbundle $E'\subset E$ such that $\Phi (E')\subset E' \otimes K$ we have $\mu (E')<\mu (E)$. Finally, $(E,\Phi )$ is polystable if it is semistable and for each subbundle $E'\subset E$ such that $\Phi (E')\subset E' \otimes K$ and $\mu (E')=\mu (E)$ there is another subbundle $E''\subset E$ satisfying $\Phi (E'')\subset E'' \otimes K$ and $E=E'\oplus E''$. As a consequence $(E,\Phi )=\oplus (E_i,\Phi _i)$ where $(E_i,\Phi _i)$ is a stable $\mathrm {GL}(n_i,\mathbb {C})$-Higgs bundle with $\mu (E_i)=\mu (E)$.

The group $\mathrm {SL}(n,\mathbb {C})$ is the subgroup of $\mathrm {GL}(n,\mathbb {C})$ defined by the usual condition on the determinant. A $\mathrm {SL}(n,\mathbb {C})$-Higgs bundle may thus be viewed as a $\mathrm {GL}(n,\mathbb {C})$-Higgs bundle $(E,\Phi )$ with the extra conditions that $E$ is endowed with a trivialization $\det E\simeq \mathcal {O}$ and $\Phi \in H^0(X,K\otimes {{\mathrm{End}}}_0 E)$ where ${{\mathrm{End}}}_0 E$ denotes the bundle of traceless endomorphisms of $E$. The (semi-,poly-)stability condition is the same as the one for $\mathrm {GL}(n,\mathbb {C})$-Higgs bundles given in Proposition 8.1.

Finally we consider the case $G=\mathrm {SO}(n,\mathbb {C})$. A principal $\mathrm {SO}(n,\mathbb {C})$-bundle on $X$ corresponds to a rank $n$ holomorphic orthogonal vector bundle $(E,Q)$, where $E$ is a rank $n$ vector bundle and $Q$ is a holomorphic section of $S^2E^*$ whose restriction to each fibre of $E$ is non degenerate. The adjoint bundle can be identified with $\Lambda ^2_QE \subset {{\mathrm{End}}}(E)$, the subbundle of ${{\mathrm{End}}}(E)$ consisting of endomorphisms which are skew-symmetric with respect to $Q$. A $\mathrm {SO}(n,\mathbb {C})$-Higgs bundle is thus a pair consisting of a rank $n$ holomorphic orthogonal vector bundle $(E,Q)$ over $X$ and a section

$$\begin{aligned} \Phi \in H^0(X,\Lambda ^2_Q E\otimes K). \end{aligned}$$

The general notions of (semi-,poly-)stability specialize in the case of $\mathrm {SO}(n,\mathbb {C})$-Higgs bundles to the following (see [1, 2]).

Proposition 8.2

A $\mathrm {SO}(n,\mathbb {C})$-Higgs bundle $((E,Q),\Phi )$ is semistable if and only if for any isotropic subbundle $E'\subset E$ such that $\Phi (E')\subset K\otimes E'$ we have $\deg E'\le 0$. Furthermore, $((E,Q),\Phi )$ is stable if for any nonzero and strict isotropic subbundle $0\ne E'\subset E$ such that $\Phi (E')\subset K\otimes E'$ we have $\deg E'<0$. Finally, $((E,Q),\Phi )$ is polystable if it is semistable and for any nonzero and strict isotropic subbundle $E'\subset E$ such that $\Phi (E')\subset K\otimes E'$ and $\deg E'=0$ there is a coisotropic subbundle $E''\subset E$ such that $\Phi (E'')\subset K\otimes E''$ and $E=E'\oplus E''$.

Remark 8.3

Recall that if $(E,Q)$ is an orthogonal vector bundle, a subbundle $E'\subset E$ is said to be isotropic if the restriction of $Q$ to $E'$ is identically zero, and coisotropic if $E'^{\perp _Q}$ is isotropic.

Remark 8.4

For complex groups $G$, Definition 2.7 implies that a $G$-Higgs bundle $(E,\varphi )$ is simple if $\mathrm{Aut }(E,\varphi )=Z(H^{\mathbb {C}})$. For $G=\mathrm {GL}(n,\mathbb {C})$ or $\mathrm {SL}(n,\mathbb {C})$ it is well known that stability implies simplicity. This is not so for $\mathrm {SO}(n,\mathbb {C})$-Higgs bundles. For instance it is possible for a stable $\mathrm {SO}(n,\mathbb {C})$-Higgs bundle to decompose as sum of stable $\mathrm {SO}(n_i,\mathbb {C})$-Higgs bundles (with $\Sigma n_i=n$). In all cases though, the Higgs bundles which are stable and simple represent smooth points in their moduli spaces (see Proposition 2.14).

1.2 The groups $\mathrm {U}(p,q)$ and $\mathrm {U}^*(2n)$

1.2.1 $\mathrm {U}(p,q)$-Higgs bundles

The maximal compact subgroups of $\mathrm {U}(p,q)$ are isomorphic to $H=\mathrm {U}(p)\times \mathrm {U}(q)$ and hence $H^{\mathbb {C}}=\mathrm {GL}(p,\mathbb {C})\times \mathrm {GL}(q,\mathbb {C})$. The complexified isotropy representation space is $\mathfrak {m}^{\mathbb {C}}={{\mathrm{Hom}}}(\mathbb {C}^q,\mathbb {C}^p)\oplus {{\mathrm{Hom}}}(\mathbb {C}^p,\mathbb {C}^q)$. A $\mathrm {U}(p,q)$-Higgs bundle may thus be described by the data $(V, W,\varphi =\beta +\gamma )$, where $V$ and $W$ are vector bundles of rank $p$ and $q$, respectively, $\beta \in H^0(X,{{\mathrm{Hom}}}(W,V)\otimes K)$ and $\gamma \in H^0(X,{{\mathrm{Hom}}}(V,W)\otimes K)$.

The following proposition gives the simplified stability conditions for $\mathrm {U}(p,q)$-Higgs bundles. It can be proved using arguments similar to the ones for other real groups (cf. Sect. 3.2 and [15, Section 4]).

Proposition 8.5

A $\mathrm {U}(p,q)$-Higgs bundle $(V,W,\varphi =\beta +\gamma )$ is semistable if

$$\begin{aligned} \mu (V'\oplus W') \le \mu (V\oplus W), \end{aligned}$$

is satisfied for all $\varphi$-invariant pairs of subbundles $V'\subset V$ and $W'\subset W$, i.e. for pairs such that

$$\begin{aligned} \beta&:W'\longrightarrow V'\otimes K\nonumber \\ \gamma&:V'\longrightarrow W'\otimes K. \end{aligned}$$

A $\mathrm {U}(p,q)$-Higgs bundle $(V,W,\varphi )$ is stable if the slope inequality is strict whenever $V'\oplus W'$ is a proper non-zero $\varphi$-invariant subbundle of $V\oplus W$.

A $\mathrm {U}(p,q)$-Higgs bundle $(V,W,\varphi )$ is polystable if it is semistable and for any $\varphi$-invariant pair of subbundles $V'\subset V$ and $W'\subset W$ satisfying $\mu (V'\oplus W')=\mu (V\oplus W)$ there is another $\varphi$-invariant pair of subbundles $V''\subset V$ and $W''\subset W$ such that $V=V'\oplus V''\ \mathrm {and}\ W=W'\oplus W''$. As a consequence there is a decomposition

$$\begin{aligned} (V,W,\beta ,\gamma )=\bigoplus (V_i,W_i,\beta _i,\gamma _i), \end{aligned}$$

where $V=\bigoplus V_i$, $W=\bigoplus W_i, \beta =\Sigma \beta _i ,\gamma =\Sigma \gamma _i$ and $(V_i,W_i,\beta _i,\gamma _i)$ is a stable $\mathrm {U}(p_i,q_i)$-Higgs bundle with $\mu (V_i\oplus W_i)=\mu (V\oplus W)$.

Remark 8.6

In the case $q=0$, the group is $\mathrm {U}(p)$ and hence $\varphi = 0$. Thus a $\mathrm {U}(p)$-Higgs bundle is an ordinary vector bundle. Proposition 8.5 shows that in this case the $\mathrm {U}(p,q)$-Higgs bundles stability condition coincides with the usual one for vector bundles.

1.2.2 $\mathrm {U}^*(2n)$-Higgs bundles

The group $\mathrm {U}^*(2n)$ is a non-compact real form of $\mathrm {GL}(2n,\mathbb {C})$ consisting of matrices $M$ verifying that $\bar{M}J_n=J_nM$ where $J_n=\left( \begin{matrix} 0 &{} I_n \\ -I_n &{} 0 \end{matrix}\right) .$ A maximal compact subgroup of $\mathrm {U}^*(2n)$ is the compact symplectic group $\mathrm {Sp}(2n)$ (or, equivalently, the group of $n\times n$ quaternionic unitary matrices), whose complexification is $\mathrm {Sp}(2n,\mathbb {C})$, the complex symplectic group. The group $\mathrm {U}^*(2n)$ is the non-compact dual of $\mathrm {U}(2n)$, in the sense that the non-compact symmetric space $\mathrm {U}^*(2n)/\mathrm {Sp}(2n)$ is the dual of the compact symmetric space $\mathrm {U}(2n)/\mathrm {Sp}(2n)$ in Cartan’s classification of symmetric spaces (cf. [22]).

The corresponding Cartan decomposition of the complex Lie algebra is

$$\begin{aligned} \mathfrak {gl}(2n,\mathbb {C})=\mathfrak {sp}(2n,\mathbb {C})\oplus \mathfrak {m}^{\mathbb {C}}, \end{aligned}$$

where $\mathfrak {m}^{\mathbb {C}}=\{A\in \mathfrak {gl}(2n,\mathbb {C})\;|\;A^tJ_n=J_nA\}$. Hence a $\mathrm {U}^*(2n)$-Higgs bundle over $X$ is a pair $(E,\varphi )$, where $E$ is a holomorphic $\mathrm {Sp}(2n,\mathbb {C})$-principal bundle and the Higgs field $\varphi$ is a global holomorphic section of $E\times _{\mathrm {Sp}(2n,\mathbb {C})}\mathfrak {m}^{\mathbb {C}}\otimes K$.

Given a symplectic vector bundle $(W,\Omega )$, denote by $S^{2}_{\Omega }W$ the bundle of endomorphisms $\xi$ of $W$ which are symmetric with respect to $\Omega$ i.e. such that $\Omega (\xi \,\cdot ,\cdot )=\Omega (\cdot ,\xi \,\cdot )$. In terms of vector bundles, we have that a $\mathrm {U}^*(2n)$-Higgs bundle over $X$ is a triple $(W,\Omega ,\varphi )$, where $W$ is a holomorphic vector bundle of rank $2n$, $\Omega \in H^0(X,\Lambda ^2W^*)$ is a symplectic form on $W$, and the Higgs field $\varphi \in H^0(X,S_{\Omega }^2 W\otimes K)$ is a $K$-twisted endomorphism $W\rightarrow W \otimes K$, symmetric with respect to $\Omega$.

Given the symplectic form $\Omega$, we have the usual skew-symmetric isomorphism

$$\begin{aligned} \omega :W\mathop {\longrightarrow }\limits ^{\simeq }W^* \end{aligned}$$

given by

$$\begin{aligned} \omega (v)=\Omega (v,-). \end{aligned}$$

The map $f\mapsto f\omega ^{-1}$ defines an isomorphism between $S^2_\Omega W$ and $\Lambda ^2W$. Hence we can think of a $\mathrm {U}^*(2n)$-Higgs bundle as a triple $(W,\Omega ,\varphi )$ with $\varphi \in H^0(X,S_{\Omega }^2W\otimes K)$ or as a triple $(W,\Omega ,\tilde{\varphi })$ with $\tilde{\varphi }\in H^0(X,\Lambda ^2W\otimes K)$ given by

$$\begin{aligned} \tilde{\varphi }=\varphi \omega ^{-1}. \end{aligned}$$

(8.1)

The general (semi-,poly-)stability conditions for $\mathrm {U}^*(2n)$-Higgs bundles are studied in [16], where simplified conditions (similarly to the case of other groups) are given. We have the following ([16, Proposition 3.6]).

Proposition 8.7

A $\mathrm {U}^*(2n)$-Higgs bundle $(W,\Omega ,\varphi )$ semistable if and only if $\deg W'\le 0$ for any isotropic and $\varphi$-invariant subbundle $W'\subset W$.

A $\mathrm {U}^*(2n)$-Higgs bundle $(W,\Omega ,\varphi )$ is stable if and only if it is semistable and $\deg W'<0$ for any isotropic and $\varphi$-invariant strict subbundle $0\ne W' \subset W$.

The $\mathrm {U}^*(2n)$-Higgs bundle $(W,\Omega ,\varphi )$ is polystable if and only if it is semistable and, for any isotropic (respectively coisotropic) and $\varphi$-invariant strict subbundle $0\ne W'\subset W$ such that $\deg W'=0$, there is another coisotropic (respectively isotropic) and $\varphi$-invariant subbundle $0\ne W''\subset W$ such that $W\simeq W'\oplus W''$.