Higgs bundles

A Higgs bundle over a compact Kähler manifold ${\displaystyle (X,\omega )}$ is a pair ${\displaystyle (E,\Phi )}$ where ${\displaystyle E\to X}$ is a holomorphic vector bundle and ${\displaystyle \Phi :E\to E\otimes {\boldsymbol {\Omega }}^{1}}$ is an ${\displaystyle \operatorname {End} (E)}$-valued holomorphic ${\displaystyle (1,0)}$-form on ${\displaystyle X}$, called the Higgs field. Additionally, the Higgs field must satisfy ${\displaystyle \Phi \wedge \Phi =0}$.

A Higgs bundle is (semi-)stable if, for every proper, non-zero coherent subsheaf ${\displaystyle {\mathcal {F}}\subset E}$ which is preserved by the Higgs field, so that ${\displaystyle \Phi ({\mathcal {F}})\subset {\mathcal {F}}\otimes {\boldsymbol {\Omega }}^{1}}$, one has

$${\displaystyle {\frac {\deg({\mathcal {F}})}{\operatorname {rank} ({\mathcal {F}})}}<{\frac {\deg(E)}{\operatorname {rank} (E)}}\quad {\text{(resp. }}\leq {\text{)}}.}$$

This rational number is called the slope, denoted ${\displaystyle \mu (E)}$, and the above definition mirrors that of a stable vector bundle. A Higgs bundle is polystable if it is a direct sum of stable Higgs bundles of the same slope, and is therefore semi-stable.

Hermitian Yang–Mills connections and Hitchin's equations

The generalisation of Hitchin's equation to higher dimension can be phrased as an analog of the Hermitian Yang–Mills equations for a certain connection constructed out of the pair ${\displaystyle (E,\Phi )}$. A Hermitian metric ${\displaystyle h}$ on a Higgs bundle ${\displaystyle (E,\Phi )}$ gives rise to a Chern connection ${\displaystyle \nabla _{A}}$ and curvature ${\displaystyle F_{A}}$. The condition that ${\displaystyle \Phi }$ is holomorphic can be phrased as ${\displaystyle {\bar {\partial }}_{A}\Phi =0}$. Hitchin's equations, on a compact Riemann surface, state that

$${\displaystyle {\begin{cases}&F_{A}+[\Phi ,\Phi ^{*}]=\lambda \operatorname {Id} _{E}\\&{\bar {\partial }}_{A}\Phi =0\end{cases}}}$$

for a constant ${\displaystyle \lambda =-2\pi i\mu (E)}$. In higher dimensions these equations generalise as follows. Define a connection ${\displaystyle D}$ on ${\displaystyle E}$ by ${\displaystyle D=\nabla _{A}+\Phi +\Phi ^{*}}$. This connection is said to be a Hermitian Yang–Mills connection (and the metric a Hermitian Yang–Mills metric) if

$${\displaystyle \Lambda _{\omega }F_{D}=\lambda \operatorname {Id} _{E}.}$$

This reduces to Hitchin's equations for a compact Riemann surface. Note that the connection ${\displaystyle D}$ is not a Hermitian Yang–Mills connection in the usual sense, as it is not unitary, and the above condition is a non-unitary analogue of the normal HYM condition.

Representations of the fundamental group and harmonic metrics

A representation of the fundamental group ${\displaystyle \rho \colon \pi _{1}(X)\to \operatorname {GL} (r,\mathbb {C} )}$ gives rise to a vector bundle with flat connection as follows. The universal cover ${\displaystyle {\hat {X}}}$ of ${\displaystyle X}$ is a principal bundle over ${\displaystyle X}$ with structure group ${\displaystyle \pi _{1}(X)}$. Thus there is an associated bundle to ${\displaystyle {\hat {X}}}$ given by

$${\displaystyle E={\hat {X}}\times _{\rho }\mathbb {C} ^{r}.}$$

This vector bundle comes naturally equipped with a flat connection ${\displaystyle D}$. If ${\displaystyle h}$ is a Hermitian metric on ${\displaystyle E}$, define an operator ${\displaystyle D_{h}''}$ as follows. Decompose ${\displaystyle D=\partial +{\bar {\partial }}}$ into operators of type ${\displaystyle (1,0)}$ and ${\displaystyle (0,1)}$, respectively. Let ${\displaystyle A'}$ be the unique operator of type ${\displaystyle (1,0)}$ such that the ${\displaystyle (1,0)}$-connection ${\displaystyle A'+{\bar {\partial }}}$ preserves the metric ${\displaystyle h}$. Define ${\displaystyle \Phi =(\partial -A')/2}$, and set ${\displaystyle D_{h}''={\bar {\partial }}+\Phi }$. Define the pseudocurvature of ${\displaystyle h}$ to be ${\displaystyle G_{h}=(D_{h}'')^{2}}$.

The metric ${\displaystyle h}$ is said to be harmonic if

$${\displaystyle \Lambda _{\omega }G_{h}=0.}$$

Notice that the condition ${\displaystyle G_{h}=0}$ is equivalent to the three conditions ${\displaystyle {\bar {\partial }}^{2}=0,{\bar {\partial }}\Phi =0,\Phi \wedge \Phi =0}$, so if ${\displaystyle G_{h}=0}$ then the pair ${\displaystyle (E,\Phi )}$ defines a Higgs bundle with holomorphic structure on ${\displaystyle E}$ given by the Dolbeault operator ${\displaystyle {\bar {\partial }}}$.

It is a result of Corlette that if ${\displaystyle h}$ is harmonic, then it automatically satisfies ${\displaystyle G_{h}=0}$ and so gives rise to a Higgs bundle.^[9]

Moduli spaces

To each of the three concepts: Higgs bundles, flat connections, and representations of the fundamental group, one can define a moduli space. This requires a notion of isomorphism between these objects. In the following, fix a smooth complex vector bundle ${\displaystyle E}$. Every Higgs bundle will be considered to have the underlying smooth vector bundle ${\displaystyle E}$.

• (Higgs bundles) The group of complex gauge transformations ${\displaystyle {\mathcal {G}}^{\mathbb {C} }}$ acts on the set ${\displaystyle {\mathcal {H}}}$ of Higgs bundles by the formula ${\displaystyle g\cdot (E,\Phi )=(g\cdot E,g\Phi g^{-1})}$. If ${\displaystyle {\mathcal {H}}^{ss}}$ and ${\displaystyle {\mathcal {H}}^{s}}$ denote the subsets of semistable and stable Higgs bundles, respectively, then one obtains moduli spaces
  $${\displaystyle M_{Dol}^{ss}:={\mathcal {H}}^{ss}//{\mathcal {G}}^{\mathcal {C}},\qquad M_{Dol}^{s}:={\mathcal {H}}^{s}/{\mathcal {G}}^{\mathcal {C}}}$$

  
  where these quotients are taken in the sense of geometric invariant theory, so orbits whose closures intersect are identified in the moduli space. These moduli spaces are called the Dolbeault moduli spaces. Notice that by setting ${\displaystyle \Phi =0}$, one obtains as subsets the moduli spaces of semi-stable and stable holomorphic vector bundles ${\displaystyle N_{Dol}^{ss}\subset M_{Dol}^{ss}}$ and ${\displaystyle N_{Dol}^{s}\subset M_{Dol}^{s}}$. It is also true that if one defines the moduli space ${\displaystyle M_{Dol}^{ps}}$ of polystable Higgs bundles then this space is isomorphic to the space of semi-stable Higgs bundles, as every gauge orbit of semi-stable Higgs bundles contains in its closure a unique orbit of polystable Higgs bundles.
• (Flat connections) The group complex gauge transformations also acts on the set ${\displaystyle {\mathcal {A}}}$ of flat connections ${\displaystyle \nabla }$ on the smooth vector bundle ${\displaystyle E}$. Define the moduli spaces
  $${\displaystyle M_{dR}:={\mathcal {A}}//{\mathcal {G}}^{\mathcal {C}},\qquad M_{dR}^{*}:={\mathcal {A}}^{*}/{\mathcal {G}}^{\mathcal {C}},}$$

  
  where ${\displaystyle {\mathcal {A}}^{*}}$ denotes the subset consisting of irreducible flat connections ${\displaystyle \nabla }$ which do not split as a direct sum ${\displaystyle \nabla =\nabla _{1}\oplus \nabla _{2}}$ on some splitting ${\displaystyle E=E_{1}\oplus E_{2}}$ of the smooth vector bundle ${\displaystyle E}$. These moduli spaces are called the de Rham moduli spaces.
• (Representations) The set of representations ${\displaystyle \operatorname {Hom} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))}$ of the fundamental group of ${\displaystyle X}$ is acted on by the general linear group by conjugation of representations. Denote by the superscripts ${\displaystyle +}$ and ${\displaystyle *}$ the subsets consisting of semisimple representations and irreducible representations respectively. Then define moduli spaces
  $${\displaystyle M_{B}^{+}=\operatorname {Hom} ^{+}(\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))//G,\qquad M_{B}^{*}=\operatorname {Hom} ^{*}(\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))/G}$$

  
  of semisimple and irreducible representations, respectively. These quotients are taken in the sense of geometric invariant theory, where two orbits are identified if their closures intersect. These moduli spaces are called the Betti moduli spaces.

In terms of moduli spaces

The nonabelian Hodge correspondence not only gives a bijection of sets, but homeomorphisms of moduli spaces. Indeed, if two Higgs bundles are isomorphic, in the sense that they can be related by a gauge transformation and therefore correspond to the same point in the Dolbeault moduli space, then the associated representations will also be isomorphic, and give the same point in the Betti moduli space. In terms of the moduli spaces the nonabelian Hodge theorem can be phrased as follows.

In general these moduli spaces will be not just topological spaces, but have some additional structure. For example, the Dolbeault moduli space and Betti moduli space ${\displaystyle M_{Dol}^{ss},M_{B}^{+}}$ are naturally complex algebraic varieties, and where it is smooth, the de Rham moduli space ${\displaystyle M_{dR}}$ is a Riemannian manifold. On the common locus where these moduli spaces are smooth, the map ${\displaystyle M_{dR}\to M_{B}^{+}}$ is a diffeomorphism, and since ${\displaystyle M_{B}^{+}}$ is a complex manifold on the smooth locus, ${\displaystyle M_{dR}}$ obtains a compatible Riemannian and complex structure, and is therefore a Kähler manifold.

Similarly, on the smooth locus, the map ${\displaystyle M_{B}^{+}\to M_{Dol}^{ss}}$ is a diffeomorphism. However, even though the Dolbeault and Betti moduli spaces both have natural complex structures, these are not isomorphic. In fact, if they are denoted ${\displaystyle I,J}$ (for the associated integrable almost complex structures) then ${\displaystyle IJ=-JI}$. In particular if one defines a third almost complex structure by ${\displaystyle K=IJ}$ then ${\displaystyle I^{2}=J^{2}=K^{2}=IJK=-\operatorname {Id} }$. If one combines these three complex structures with the Riemannian metric coming from ${\displaystyle M_{dR}}$, then on the smooth locus the moduli spaces become a Hyperkähler manifold.

Relation to Hitchin–Kobayashi correspondence and unitary representations

If one sets the Higgs field ${\displaystyle \Phi }$ to zero, then a Higgs bundle is simply a holomorphic vector bundle. This gives an inclusion ${\displaystyle N_{Dol}^{ss}\subset M_{Dol}^{ss}}$ of the moduli space of semi-stable holomorphic vector bundles into the moduli space of Higgs bundles. The Hitchin–Kobayashi correspondence gives a correspondence between holomorphic vector bundles and Hermitian Yang–Mills connections over compact Kähler manifolds, and can therefore be seen as a special case of the nonabelian Hodge correspondence.

When the underlying vector bundle is topologically trivial, the holonomy of a Hermitian Yang–Mills connection will give rise to a unitary representation of the fundamental group, ${\displaystyle \rho$ :\pi _{1}(X)\to \operatorname {U} (r)} . The subset of the Betti moduli space corresponding to the unitary representations, denoted ${\displaystyle N_{B}^{+}}$, will get mapped isomorphically onto the moduli space of semi-stable vector bundles ${\displaystyle N_{Dol}^{ss}}$.

Rank one Higgs bundles on compact Riemann surfaces

The special case where the rank of the underlying vector bundle is one gives rise to a simpler correspondence.^[10] Firstly, every line bundle is stable, as there are no proper non-zero subsheaves. In this case, a Higgs bundle consists of a pair ${\displaystyle (L,\Phi )}$ of a holomorphic line bundle and a holomorphic ${\displaystyle (1,0)}$-form, since the endomorphism of a line bundle are trivial. In particular, the Higgs field is uncoupled from the holomorphic line bundle, so the moduli space ${\displaystyle M_{Dol}}$ will split as a product, and the one-form automatically satisfies the condition ${\displaystyle \Phi \wedge \Phi =0}$. The gauge group of a line bundle is commutative, and so acts trivially on the Higgs field ${\displaystyle \Phi }$ by conjugation. Thus the moduli space can be identified as a product

$${\displaystyle M_{Dol}=\operatorname {Jac} (X)\times H^{0}(X,{\boldsymbol {\Omega }}^{1})}$$

of the Jacobian variety of ${\displaystyle X}$, classifying all holomorphic line bundles up to isomorphism, and the vector space ${\displaystyle H^{0}(X,{\boldsymbol {\Omega }}^{1})}$ of holomorphic ${\displaystyle (1,0)}$-forms.

In the case of rank one Higgs bundles on compact Riemann surfaces, one obtains a further description of the moduli space. The fundamental group of a compact Riemann surface, a surface group, is given by

$${\displaystyle \pi _{1}(X)=\langle a_{1},\dots ,a_{g},b_{1},\dots ,b_{g}\mid [a_{1},b_{1}]\cdots [a_{g},b_{g}]=e\rangle }$$

where ${\displaystyle g}$ is the genus of the Riemann surface. The representations of ${\displaystyle \pi _{1}(X)}$ into the general linear group ${\displaystyle \operatorname {GL} (1,\mathbb {C} )=\mathbb {C} ^{*}}$ are therefore given by ${\displaystyle 2g}$-tuples of non-zero complex numbers:

$${\displaystyle \operatorname {Hom} (\pi _{1}(X),\mathbb {C} ^{*})=(\mathbb {C} ^{*})^{2g}.}$$

Since ${\displaystyle \mathbb {C} ^{*}}$ is abelian, the conjugation on this space is trivial, and the Betti moduli space is ${\displaystyle M_{B}=(\mathbb {C} ^{*})^{2g}}$. On the other hand, by Serre duality, the space of holomorphic ${\displaystyle (1,0)}$-forms is dual to the sheaf cohomology ${\displaystyle H^{1}(X,{\mathcal {O}}_{X})}$. The Jacobian variety is an Abelian variety given by the quotient

$${\displaystyle \operatorname {Jac} (X)={\frac {H^{1}(X,{\mathcal {O}}_{X})}{H^{1}(X,\mathbb {Z} )}},}$$

so has tangent spaces given by the vector space ${\displaystyle H^{1}(X,{\mathcal {O}}_{X})}$, and cotangent bundle

$${\displaystyle T^{*}\operatorname {Jac} (X)=\operatorname {Jac} (X)\times H^{1}(X,{\mathcal {O}}_{X})^{*}=\operatorname {Jac} (X)\times H^{0}(X,{\boldsymbol {\Omega }}^{1})=M_{Dol}.}$$

That is, the Dolbeault moduli space, the moduli space of holomorphic Higgs line bundles, is simply the cotangent bundle to the Jacobian, the moduli space of holomorphic line bundles. The nonabelian Hodge correspondence therefore gives a diffeomorphism

$${\displaystyle T^{*}\operatorname {Jac} (X)\cong (\mathbb {C} ^{*})^{2g}}$$

which is not a biholomorphism. One can check that the natural complex structures on these two spaces are different, and satisfy the relation ${\displaystyle IJ=-JI}$, giving a hyperkähler structure on the cotangent bundle to the Jacobian.

Hodge decomposition

The Hodge decomposition of a compact Kähler manifold decomposes the complex de Rham cohomology into the finer Dolbeault cohomology:

$${\displaystyle H_{dR}^{k}(X,\mathbb {C} )=\bigoplus _{p+q=k}H_{Dol}^{p,q}(X).}$$

At degree one this gives a direct sum

$${\displaystyle H^{1}(X,\mathbb {C} )=H^{0,1}(X)\oplus H^{1,0}(X)\cong H^{1}(X,{\mathcal {O}}_{X})\oplus H^{0}(X,{\boldsymbol {\Omega }}^{1})}$$

where we have applied the Dolbeault theorem to phrase the Dolbeault cohomology in terms of sheaf cohomology of the sheaf of holomorphic ${\displaystyle (1,0)}$-forms ${\displaystyle {\boldsymbol {\Omega }}^{1},}$ and the structure sheaf ${\displaystyle {\mathcal {O}}_{X}}$ of holomorphic functions on ${\displaystyle X}$.

Nonabelian cohomology

When constructing sheaf cohomology, the coefficient sheaf ${\displaystyle {\mathcal {F}}}$ is always a sheaf of abelian groups. This is because for an abelian group, every subgroup is normal, so the quotient group

$${\displaystyle {\check {H}}^{k}(X,{\mathcal {F}})=Z^{k}(X,{\mathcal {F}})/B^{k}(X,{\mathcal {F}})}$$

of sheaf cocycles by sheaf coboundaries is always well-defined. When the sheaf ${\displaystyle {\mathcal {F}}}$ is not abelian, these quotients are not necessarily well-defined, and so sheaf cohomology theories do not exist, except in the following special cases:

• ${\displaystyle k=0}$: The 0th sheaf cohomology group is always the space of global sections of the sheaf ${\displaystyle {\mathcal {F}}}$, so is always well-defined even if ${\displaystyle {\mathcal {F}}}$ is nonabelian.
• ${\displaystyle k=1}$: The 1st sheaf cohomology set is well-defined for a nonabelian sheaf ${\displaystyle {\mathcal {F}}}$, but it is not itself a quotient group.
• ${\displaystyle k=2}$: In some special cases, an analogue of the second degree sheaf cohomology can be defined for nonabelian sheaves using the theory of gerbes.

A key example of nonabelian cohomology occurs when the coefficient sheaf is ${\displaystyle {\mathcal {GL}}(r,\mathbb {C} )}$, the sheaf of holomorphic functions into the complex general linear group. In this case it is a well-known fact from Čech cohomology that the cohomology set

$${\displaystyle {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))}$$

is in one-to-one correspondence with the set of holomorphic vector bundles of rank ${\displaystyle r}$ on ${\displaystyle X}$, up to isomorphism. Notice that there is a distinguished holomorphic vector bundle of rank ${\displaystyle r}$, the trivial vector bundle, so this is actually a cohomology pointed set. In the special case ${\displaystyle r=1}$ the general linear group is the abelian group ${\displaystyle \mathbb {C} ^{*}}$ of non-zero complex numbers with respect to multiplication. In this case one obtains the group of holomorphic line bundles up to isomorphism, otherwise known as the Picard group.

Nonabelian Hodge theorem

The first cohomology group ${\displaystyle H^{1}(X,\mathbb {C} )}$ is isomorphic to the group of homomorphisms from the fundamental group ${\displaystyle \pi _{1}(X)}$ to ${\displaystyle \mathbb {C} }$. This can be understood, for example, by applying the Hurewicz theorem. Thus the regular Hodge decomposition mentioned above may be phrased as

$${\displaystyle \operatorname {Hom} (\pi _{1}(X),\mathbb {C} )\cong H^{1}(X,{\mathcal {O}}_{X})\oplus H^{0}(X,{\boldsymbol {\Omega }}^{1}).}$$

The nonabelian Hodge correspondence gives an analogy of this statement of the Hodge theorem for nonabelian cohomology, as follows. A Higgs bundle consists of a pair ${\displaystyle (E,\Phi )}$ where ${\displaystyle E}$ is a holomorphic vector bundle, and ${\displaystyle \Phi \in H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1})}$ is a holomorphic, endomorphism-valued ${\displaystyle (1,0)}$-form. The holomorphic vector bundle ${\displaystyle E}$ may be identified with an element of ${\displaystyle {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))}$ as mentioned above. Thus a Higgs bundle may be thought of as an element of the direct product

$${\displaystyle (E,\Phi )\in {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))\oplus H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1}).}$$

The nonabelian Hodge correspondence gives an isomorphism from the moduli space of ${\displaystyle \operatorname {GL} (r,\mathbb {C} )}$-representations of the fundamental group ${\displaystyle \pi _{1}(X)}$ to the moduli space of Higgs bundles, which could therefore be written as an isomorphism

$${\displaystyle \operatorname {Rep} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))\cong {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))\oplus H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1}).}$$

This can be seen as an analogy of the regular Hodge decomposition above. The moduli space of representations ${\displaystyle \operatorname {Rep} (\pi _{1}(X),\operatorname {GL} (r,\mathbb {C} ))}$ plays the role of the first cohomology of ${\displaystyle X}$ with nonabelian coefficients, the cohomology set ${\displaystyle {\check {H}}^{1}(X,{\mathcal {GL}}(r,\mathbb {C} ))}$ plays the role of the space ${\displaystyle H^{1}(X,{\mathcal {O}}_{X})}$, and the group ${\displaystyle H^{0}(X,\operatorname {End} (E)\otimes {\boldsymbol {\Omega }}^{1})}$ plays the role of the holomorphic (1,0)-forms ${\displaystyle H^{0}(X,{\boldsymbol {\Omega }}^{1})}$.

The isomorphism here is written ${\displaystyle \cong }$, but this is not an actual isomorphism of sets, as the moduli space of Higgs bundles is not literally given by the direct sum above, as this is only an analogy.

Hodge structure

The moduli space ${\displaystyle M_{Dol}^{ss}}$ of semi-stable Higgs bundles has a natural action of the multiplicative group ${\displaystyle \mathbb {C} ^{*}}$, given by scaling the Higgs field: ${\displaystyle \lambda \cdot (E,\Phi )=(E,\lambda \Phi )}$ for ${\displaystyle \lambda \in \mathbb {C} ^{*}}$. For abelian cohomology, such a ${\displaystyle \mathbb {C} ^{*}}$ action gives rise to a Hodge structure, which is a generalisation of the Hodge decomposition of the cohomology of a compact Kähler manifold. One way of understanding the nonabelian Hodge theorem is to use the ${\displaystyle \mathbb {C} ^{*}}$ action on the moduli space ${\displaystyle M_{B}^{+}}$ to obtain a Hodge filtration. This can lead to new topological invariants of the underlying manifold ${\displaystyle X}$. For example, one obtains restrictions on which groups may appear as the fundamental groups of compact Kähler manifolds in this way.^[7]