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

$\begin{array}{rl}& F_A^{0,2}=0\\ & F_A\cdot \omega =\lambda (E)\mathrm{Id},\end{array}$

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

$\begin{array}{rl}\mathrm{deg}(E)& :=\underset{X}{\int }{c}_1(E)\wedge {\omega }^{n-1}\\ & =\frac{i}{2\pi }\underset{X}{\int }\mathrm{Tr}(F_A)\wedge {\omega }^{n-1}\\ & =\frac{i}{2\pi }\underset{X}{\int }\mathrm{Tr}(F_A\cdot \omega ){\omega }^n.\end{array}$

Since $F_A\cdot \omega =\lambda (E){\mathrm{Id}}_E$ and the identity endomorphism has trace given by the rank of $E$, we obtain

$\lambda (E)=-\frac{2\pi i}{n!\mathrm{Vol}(X)}\mu (E),$

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

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

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 ${\mathbb{ℂ}P}^2\mathrm{\#}{\overline{\mathbb{ℂ}P}}_2$, 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 Kobayashi (1980, section 6). 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:

${\mathrm{\Lambda }}_{+}^2={\mathrm{\Lambda }}^{2,0}\oplus {\mathrm{\Lambda }}^{0,2}\oplus ⟨\omega ⟩,{\mathrm{\Lambda }}_{-}^2=⟨\omega {⟩}^{\perp }\subset {\mathrm{\Lambda }}^{1,1}$

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.

See also

• Einstein manifold
• Deformed Hermitian Yang–Mills equation
• Gauge theory (mathematics)

References

• Kobayashi, Shoshichi (1980), "First Chern class and holomorphic tensor fields", Nagoya Mathematical Journal 77: 5–11, doi:10.1017/S0027763000018602, ISSN 0027-7630, http://projecteuclid.org/getRecord?id=euclid.nmj/1118786013 
• Kobayashi, Shoshichi (1987), Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, ISBN 978-0-691-08467-1