Hermitian Yang–Mills connection explained

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 ^{0 , 2} _{A} = 0 \\ &F _{A} ⋅ \omega = λ (E) \operatorname{Id}, \end{align}

for some constant

λ (E)\in C . Here we haveF _{A} \wedge \omega ^{n-1} = (F _{A} ⋅ \omega) \omega ^{n.} Notice that since A is assumed to be a Hermitian connection, the curvature F _{A} is skew-Hermitian, and so F ^{0 , 2} _{A} =0 implies F ^{2 , 0} _{A} = 0 . When the underlying Kähler manifold X is compact, λ (E) may be computed using Chern-Weil theory. Namely, we have\begin{align} \deg(E) &: = \int _{X} c _{1(E)} \wedge \omega ^{n-1} \\ & = (i)/(2 \pi) \int _{X} \operatorname{Tr}(F _{A)} \wedge \omega ^{n-1} \\ & = (i)/(2 \pi) \int _{X} \operatorname{Tr}(F _{A} ⋅ \omega) \omega ^{n. \end{align}}

Since

F _{A} ⋅ \omega = λ (E) \operatorname{Id} _{E} and the identity endomorphism has trace given by the rank of E , we obtainλ (E) =- (2 \pi i)/(n! \operatorname{Vol) (X)} \mu(E),

where

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

{ C P} ^{2} \# \overline{ 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 ^{0 , 2} _{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:Λ ^{2} _{+} = Λ ^{2,0} ⊕ Λ ^{0,2} ⊕ \langle \omega \rangle, Λ ^{2} _{-} = \langle \omega \rangle ^{\perp} \subset Λ ^{1,1} When the degree of the vector bundle E vanishes, then the Hermitian Yang-Mills equations become F ^{0 , 2} _{A} = F ^{2 , 0} _{A} = F _{A} ⋅ \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.^[1]

See also

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

Notes and References

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