Difference between Higgs bundle and vector bundle with connection

Question

• Let the pair $(E,A)$ be a $G$ equivariant vector bundle with connection over an algebraic variety $X$ and let $G$ be a classical Lie group. $A$ is the connection 1-form, i.e. $A \in \Omega^1(X)$. Of course, by considering a specific associated bundle, i.e. the adjoint bundle the induced connection is a Lie algebra valued 1-form. In a local coordinate system we write $A = A(x)dx$ with $A(x)$ a matrix.
• Let the pair $(E', \phi)$ be a $G$ equivariant vector bundle with a Higgs field $\phi \in End(E')\otimes K_X$ and if $X$ is a complex curve (which is what I am interested for) then $\phi \in End(E')\otimes \Omega^1 (X)$. Locally we write $\phi = \phi(x) dx$ where $\phi(x)$ is a matrix again. The Higgs field satisfies $\phi \wedge \phi = 0$.

My understanding is that for a $G$ equivariant vector bundle $F$ with Lie algebra $\mathfrak{g}$ we can consider $End(F) \cong \mathfrak{g}$. How true is this?

If this is the case (at least more or less) then what is the difference between a vector bundle with connection and a Higgs bundle?

If I have stated something not quite precisely can you please help me make it precise and if not answer at least provide some references that target this question?

Answer 1

The "connection form" $A$ is not really a 1-form. It is a 1-form locally i.e. if we have affine open cover $X=\cup_i U_i$ then on $U_i$ object $A$ is a 1-form $A_i$ with values in $\operatorname{End}(E)$, but on an intersections $U_{ij}=U_i \cap U_j$ we have $$ A_i=gA_j g^{-1} +g^{-1}dg $$ where $g: U_{ij} \to \operatorname{End}(E)$ is the transition function for $E$. It is invertible.

Because of this additional term $g^{-1}dg$ local objects $A_i$ don't glue to a global 1-form but to an object called connection form.

We can think about connection on $E$ an $\mathbb{C}$-linear map $$ \nabla: E \to E \otimes \Omega^1_X, $$ satisfying Leibniz rule and Higgs field on $E$ is an $\mathcal{O}_X$-linear map $$ \phi: E \to E \otimes \Omega_X^1, $$ satisfying $\phi \wedge \phi=0$. This condition $\phi \wedge \phi=0$ is similar to condition $\nabla^2=0$ for connection $\nabla$, such connections are called flat.

Other then this formal analogy connections and Higgs fields are different additional structures on $E$, not immediately related.

Finally, you don't need condition that $E$ is a $G-$equivariant bundle here, you use $G$ as a structure group of the bundle. It means that transition functions are functions with values in $G$ i.e. $g: U_{ij} \to G$ and they act on fibers of $E$ by some representation of $G$. In this case connection form takes values in Lie algebra of $G$ and additional term $g^{-1}dg$ is the Maurer-Cartan form on $G$.

To say that $E$ is a $G-$equivariant bundle we need $G$ to act on $X$ and then put some additional structure on $E$, but we don't need this to work with connections/Higgs fields.

Answer 2

Essentially my answer is similar to Alex's, but I would like to clarify some ideas and analogies and give a more conceptual prospect. Most of the things below are basic, though I needed to write them down for making analogies more explicit.

Personally, I also like to think about connections as choices of vertical subbundles in the tangent bundle to the total space (which are known as Ehresmann connections). However, sometimes it is usefull to think a bit more algebraically and it seems that this is the case.

1.Flat $d$-connections aka flat connections.

What is a connection on bundle $E$, after all? This is an operator $$\nabla \colon \Gamma(E) \to \Gamma(E \otimes \Omega^1(X)),$$ which satisfies the Leibniz rule: for any section $s \in \Gamma(E)$ and function $f$ one has $$ \nabla(fs) = f\nabla s + df \otimes s. $$ (This doesn't really matter, but notice that from this approach one can define the notion of connection for arbitrary modules over dg-algebras).

Using the Leibniz rule one extends $\nabla$ to an operator $\nabla \colon \Gamma(E \otimes \Omega^pX) \to \Gamma(E \otimes \Omega^{p+1}X)$ and a flat connection is the one that satisfies $\nabla\circ \nabla = \nabla^2 = 0$.

If the bundle $E$ is trivial then the de Rham differential itself is a flat connection on $E$. For any other flat connection $\nabla$ the difference $\alpha = \nabla - d$ is clearly $C^{\infty}$-linear, that is $$\alpha(fs) = f\alpha(s).$$ That is why it is not only a map between the spaces of sections but a map between vector bundles themselves and hence $\alpha \in \Gamma(\operatorname{Hom}(E,E \otimes \Omega^1X)) = \Gamma(\operatorname{End}(E) \otimes \Omega^1X)$. In this case $\alpha$ is precisely the connection form. The flatness condition leads to the Maurer-Cartan equation $$2d\alpha + [\alpha \wedge \alpha] = 0.$$

2. Flat $\overline{\partial}$-connections aka Higgs bundles.

Now assume that the base of our bundle is a complex manifold $X$. Thus we have the Dolbeault operator $\overline{\partial} \colon \Gamma(C^{\infty}(X)) \to \Gamma(\Omega^{0,1}(X)) \subset \Gamma(\Omega^1(X))$. Let us say that an operator $D \colon \Gamma(E) \to \Gamma(E \otimes \Omega^1_X)$ is a $\overline{\partial}$-connection if it satisfies the $\overline{\partial}$-Leibniz rule $$ D(fs)= fD(s) + \overline{\partial}f \otimes s. $$ Let us also say that $\overline{\partial}$-connection is integrable (or flat), if $D^2 = 0$ (as a map $\Gamma(E) \to \Gamma(E \otimes \Omega^2(X))$).

Take any integrable $\overline{\partial}$-connection $D$ and decompose it as $D = D^{1,0} + D^{0,1}$. Denote then $\overline{\partial}_E:=D^{0,1}$ and $\phi:= D^{1,0}$. A simple computation shows that $D^2=0$ implies $\overline{\partial}_E$ is an operator of holomorphic structure, i.e. an operator $\overline{\partial}_E \colon \Gamma(E) \to \Gamma(E \otimes \Omega^{0,1}(X))$ which satisfies $\overline{\partial_E}^2 = 0$.Such an operator is the same as choice of holomorphic structure on $E$.

At the same time its $(1,0)$-part is linear and holomorphic (w.r.to $\overline{\partial_E}$), so it is a holomorphic section $\phi$ of $\operatorname{End}(E) \otimes \Omega^{1,0}(X)$. Moreover it satisfies the Higgs equation $$[\phi \wedge \phi] = 0.$$ If you compare this to the Maurer-Cartan equation, you'll notice that the first term $2\overline{\partial}\phi$ disappeared. Indeed in the $\overline{\partial}$-world its vanishing is independent to the vanishing of the second term (since they are of different Hodge type) and is equivalent to holomorphicity of $\phi$.

2+1. All together.

So, answering your question, one can say that the connection form is a difference of a flat connection (or $d$-connection) and $d$, while the Higgs field is the difference of a flat $\overline{\partial}$-connection and the Dolbeault differential $\overline{\partial}_E$. In very vague terms a structure of Higgs bundle is kind of "holomorphic shadow of a flat connection".

The last statement can be might rigorous, but this is a very non-trivial theorem (or rather a collection of hard theorems by Simpson, Hitchin, Corlette, Uhlenbeck and Yau), also known as non-abelian Hodge-de Rham theorem. It states that if $X$ is Kähler, then (after adding some stability conditions) one has a bijection between flat connections and Higgs bundles (and, moreover, equivalence of categories and diffeomorphisms between moduli spaces).

In fact, this result is a kind of magic. The flat connections are the same as the representations of $\pi_1(X)$, so these are objects of topological origin, while Higgs bundles are objects of holomorphic origin, and, even more, can be defined in positive characteristic!