Hermitian Yang–Mills connection

In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, and are often called instantons.

The Kobayashi–Hitchin correspondence proved by Donaldson, Uhlenbeck and Yau asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is slope polystable.

Hermitian Yang–Mills equations
Hermite-Einstein connections arise as solutions of the Hermitian Yang-Mills equations. These are a system of partial differential equations on a vector bundle over a Kähler manifold, which imply the Yang-Mills equations. Let $$A$$ be a Hermitian connection on a Hermitian vector bundle $$E$$ over a Kähler manifold $$X$$ of dimension $$n$$. Then the Hermitian Yang-Mills equations are


 * $$\begin{align}

&F_A^{0,2} = 0 \\ &F_A \cdot \omega = \lambda(E) \operatorname{Id}, \end{align}$$

for some constant $$\lambda(E)\in \mathbb{C}$$. Here we have


 * $$F_A \wedge \omega^{n-1} = (F_A \cdot \omega) \omega^n.$$

Notice that since $$A$$ is assumed to be a Hermitian connection, the curvature $$F_A$$ is skew-Hermitian, and so $$F_A^{0,2}=0$$ implies $$F_A^{2,0} = 0$$. When the underlying Kähler manifold $$X$$ is compact, $$\lambda(E)$$ may be computed using Chern-Weil theory. Namely, we have
 * $$\begin{align}

\deg(E) &:= \int_X c_1(E) \wedge \omega^{n-1}\\ &=\frac{i}{2\pi} \int_X \operatorname{Tr}(F_A) \wedge \omega^{n-1}\\ &=\frac{i}{2\pi} \int_X \operatorname{Tr}(F_A \cdot \omega) \omega^n. \end{align}$$

Since $$F_A \cdot \omega = \lambda(E) \operatorname{Id}_E$$ and the identity endomorphism has trace given by the rank of $$E$$, we obtain
 * $$\lambda(E) = -\frac{2\pi i}{n! \operatorname{Vol}(X)} \mu(E),$$

where $$\mu(E)$$ is the slope of the vector bundle $$E$$, given by


 * $$\mu(E) = \frac{\deg(E)}{\operatorname{rank}(E)},$$

and the volume of $$X$$ is taken with respect to the volume form $$\omega^n/n!$$.

Due to the similarity of the second condition in the Hermitian Yang-Mills equations with the equations for an Einstein metric, solutions of the Hermitian Yang-Mills equations are often called Hermite-Einstein connections, as well as Hermitian Yang-Mills connections.

Examples
The Levi-Civita connection of a Kähler–Einstein metric is Hermite-Einstein with respect to the Kähler-Einstein metric. (These examples are however dangerously misleading, because there are compact Einstein manifolds, such as the Page metric on $${\mathbb C P}^2 \# \overline{\mathbb C P}_2$$, that are Hermitian, but for which the Levi-Civita connection is not Hermite-Einstein.)

When the Hermitian vector bundle $$E$$ has a holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection. For the Chern connection, the condition that $$F_A^{0,2}=0$$ is automatically satisfied. The Hitchin-Kobayashi correspondence asserts that a holomorphic vector bundle $$E$$ admits a Hermitian metric $$h$$ such that the associated Chern connection satisfies the Hermitian Yang-Mills equations if and only if the vector bundle is polystable. From this perspective, the Hermitian Yang-Mills equations can be seen as a system of equations for the metric $$h$$ rather than the associated Chern connection, and such metrics solving the equations are called Hermite-Einstein metrics.

The Hermite-Einstein condition on Chern connections was first introduced by. These equation imply the Yang-Mills equations in any dimension, and in real dimension four are closely related to the self-dual Yang-Mills equations that define instantons. In particular, when the complex dimension of the Kähler manifold $$X$$ is $$2$$, there is a splitting of the forms into self-dual and anti-self-dual forms. The complex structure interacts with this as follows:


 * $$\Lambda_+^2 = \Lambda^{2,0} \oplus \Lambda^{0,2} \oplus \langle \omega \rangle,\qquad \Lambda_-^2 = \langle \omega \rangle^{\perp} \subset \Lambda^{1,1}$$

When the degree of the vector bundle $$E$$ vanishes, then the Hermitian Yang-Mills equations become $$F_A^{0,2} = F_A^{2,0} = F_A \cdot \omega=0$$. By the above representation, this is precisely the condition that $$F_A^+ = 0$$. That is, $$A$$ is an ASD instanton. Notice that when the degree does not vanish, solutions of the Hermitian Yang-Mills equations cannot be anti-self-dual, and in fact there are no solutions to the ASD equations in this case.