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

:$\begin &F_A^ = 0 \\ &F_A \cdot \omega = \lambda(E) \operatorname, \end$

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

:$F_A \wedge \omega^ = (F_A \cdot \omega) \omega^n.$
Notice that since $A$ is assumed to be a Hermitian connection, the curvature $F_A$ is skew-Hermitian, and so $F_A^=0$ implies $F_A^ = 0$.
When the underlying Kähler manifold $X$ is compact, $\lambda(E)$ may be computed using Chern-Weil theory. Namely, we have
:$\begin \deg(E) &:= \int_X c_1(E) \wedge \omega^\\ &=\frac \int_X \operatorname(F_A) \wedge \omega^\\ &=\frac \int_X \operatorname(F_A \cdot \omega) \omega^n. \end$

Since $F_A \cdot \omega = \lambda(E) \operatorname_E$ and the identity endomorphism has trace given by the rank of $E$, we obtain
:$\lambda(E) = -\frac \mu(E),$

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

:$\mu(E) = \frac,$

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 $^2 \# \overline_2$, 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 $F_A^=0$ 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:

:$\Lambda_+^2 = \Lambda^ \oplus \Lambda^ \oplus \langle \omega \rangle,\qquad \Lambda_-^2 = \langle \omega \rangle^ \subset \Lambda^$
When the degree of the vector bundle $E$ vanishes, then the Hermitian Yang-Mills equations become $F_A^ = F_A^ = F_A \cdot \omega=0$. 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.Donaldson, S. K., Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.

See also

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

References

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{{DEFAULTSORT:Einstein-Hermitian vector bundle
Vector bundles
Albert Einstein
Differential geometry
Partial differential equations