Hermitian Yang–Mills connection

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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 [math]\displaystyle{ A }[/math] be a Hermitian connection on a Hermitian vector bundle [math]\displaystyle{ E }[/math] over a Kähler manifold [math]\displaystyle{ X }[/math] of dimension [math]\displaystyle{ n }[/math]. Then the Hermitian Yang-Mills equations are

[math]\displaystyle{ \begin{align} &F_A^{0,2} = 0 \\ &F_A \cdot \omega = \lambda(E) \operatorname{Id}, \end{align} }[/math]

for some constant [math]\displaystyle{ \lambda(E)\in \mathbb{C} }[/math]. Here we have

[math]\displaystyle{ F_A \wedge \omega^{n-1} = (F_A \cdot \omega) \omega^n. }[/math]

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

[math]\displaystyle{ \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} }[/math]

Since [math]\displaystyle{ F_A \cdot \omega = \lambda(E) \operatorname{Id}_E }[/math] and the identity endomorphism has trace given by the rank of [math]\displaystyle{ E }[/math], we obtain

[math]\displaystyle{ \lambda(E) = -\frac{2\pi i}{n! \operatorname{Vol}(X)} \mu(E), }[/math]

where [math]\displaystyle{ \mu(E) }[/math] is the slope of the vector bundle [math]\displaystyle{ E }[/math], given by

[math]\displaystyle{ \mu(E) = \frac{\deg(E)}{\operatorname{rank}(E)}, }[/math]

and the volume of [math]\displaystyle{ X }[/math] is taken with respect to the volume form [math]\displaystyle{ \omega^n/n! }[/math].

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 [math]\displaystyle{ {\mathbb C P}^2 \# \overline{\mathbb C P}_2 }[/math], that are Hermitian, but for which the Levi-Civita connection is not Hermite-Einstein.)

When the Hermitian vector bundle [math]\displaystyle{ E }[/math] has a holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection. For the Chern connection, the condition that [math]\displaystyle{ F_A^{0,2}=0 }[/math] is automatically satisfied. The Hitchin-Kobayashi correspondence asserts that a holomorphic vector bundle [math]\displaystyle{ E }[/math] admits a Hermitian metric [math]\displaystyle{ h }[/math] 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 [math]\displaystyle{ h }[/math] 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 [math]\displaystyle{ X }[/math] is [math]\displaystyle{ 2 }[/math], there is a splitting of the forms into self-dual and anti-self-dual forms. The complex structure interacts with this as follows:

[math]\displaystyle{ \Lambda_+^2 = \Lambda^{2,0} \oplus \Lambda^{0,2} \oplus \langle \omega \rangle,\qquad \Lambda_-^2 = \langle \omega \rangle^{\perp} \subset \Lambda^{1,1} }[/math]

When the degree of the vector bundle [math]\displaystyle{ E }[/math] vanishes, then the Hermitian Yang-Mills equations become [math]\displaystyle{ F_A^{0,2} = F_A^{2,0} = F_A \cdot \omega=0 }[/math]. By the above representation, this is precisely the condition that [math]\displaystyle{ F_A^+ = 0 }[/math]. That is, [math]\displaystyle{ A }[/math] 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.[1]

See also

References

  1. ↑ Donaldson, S. K., Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.



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