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.
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.
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.
is assumed to be a Hermitian connection, the curvature
When the underlying Kähler manifold
and the identity endomorphism has trace given by the rank of
is the slope of the vector bundle
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.
(These examples are however dangerously misleading, because there are compact Einstein manifolds, such as the Page metric on
, that are Hermitian, but for which the Levi-Civita connection is not Hermite–Einstein.)
When the Hermitian vector bundle
has a holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection.
For the Chern connection, the condition that
The Hitchin–Kobayashi correspondence asserts that a holomorphic vector bundle
admits a Hermitian metric
such that the associated Chern connection satisfies the Hermitian Yang–Mills equations if and only if the vector bundle is polystable.
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
The complex structure interacts with this as follows: When the degree of the vector bundle
vanishes, then the Hermitian Yang–Mills equations become
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.