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

{\displaystyle A}

be a Hermitian connection on a Hermitian vector bundle

E

{\displaystyle E}

over a Kähler manifold

X

{\displaystyle X}

of dimension

n

{\displaystyle n}

. Then the Hermitian Yang–Mills equations are:

F

A

0

,

2

=

0

F

A

⋅

ω

=

λ

(

E

)

Id

,

{\displaystyle {\begin{aligned}&F_{A}^{0,2}=0\\&F_{A}\cdot \omega =\lambda (E)\operatorname {Id} ,\end{aligned}}}

for some constant

λ

(

E

)

∈

C

{\displaystyle \lambda (E)\in \mathbb {C} }

. Here we have:

F

A

∧

ω

n

−

1

=

(

F

A

⋅

ω

)

ω

n

=

λ

(

E

)

ω

n

Id

.

{\displaystyle F_{A}\wedge \omega ^{n-1}=(F_{A}\cdot \omega )\omega ^{n}=\lambda (E)\omega ^{n}\operatorname {Id} .}

Notice that since

A

{\displaystyle A}

is assumed to be a Hermitian connection, the curvature

F

A

{\displaystyle F_{A}}

is skew-Hermitian, and so

F

A

0

,

2

=

0

{\displaystyle F_{A}^{0,2}=0}

implies

F

A

2

,

0

=

0

{\displaystyle F_{A}^{2,0}=0}

.

When the underlying Kähler manifold

X

{\displaystyle X}

is compact,

λ

(

E

)

{\displaystyle \lambda (E)}

may be computed using Chern–Weil theory.