In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and in particular
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
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
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
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 In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The corres ...
proved by
Donaldson
Donaldson is a Scottish and Irish patronymic surname meaning "son of Donald". It is a simpler Anglicized variant for the name MacDonald. Notable people with the surname include:
__NOTOC__
A
* Alastair Donaldson (1955–2013), Scottish musician ...
,
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
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
on a vector bundle over a Kähler manifold, which imply the
Yang-Mills equations. Let
be a
Hermitian connection on a Hermitian vector bundle
over a Kähler manifold
of dimension
. Then the Hermitian Yang-Mills equations are
:
for some constant
. Here we have
:
Notice that since
is assumed to be a Hermitian connection, the curvature
is
skew-Hermitian, and so
implies
.
When the underlying Kähler manifold
is compact,
may be computed using
Chern-Weil theory. Namely, we have
:
Since
and the identity endomorphism has trace given by the rank of
, we obtain
:
where
is the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the vector bundle
, given by
:
and the volume of
is taken with respect to the volume form
.
Due to the similarity of the second condition in the Hermitian Yang-Mills equations with the equations for an
Einstein metric
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian manifold, Riemannian or pseudo-Riemannian manifold, pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the Metric tensor, metric. T ...
, 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
, 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 In mathematics, a Hermitian connection \nabla is a connection on a Hermitian vector bundle E over a smooth manifold M which is compatible with the Hermitian metric
\langle \cdot, \cdot \rangle on E, meaning that
: v \langle s,t\rangle = \langle \na ...
. For the Chern connection, the condition that
is automatically satisfied. 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. From this perspective, the Hermitian Yang-Mills equations can be seen as a system of equations for the metric
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
is
, there is a splitting of the forms into self-dual and anti-self-dual forms. The complex structure interacts with this as follows:
:
When the degree of the vector bundle
vanishes, then the Hermitian Yang-Mills equations become
. By the above representation, this is precisely the condition that
. That is,
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
In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition i ...
*
Deformed Hermitian Yang–Mills equation
*
Gauge theory (mathematics)
References
*
*
{{DEFAULTSORT:Einstein-Hermitian vector bundle
Vector bundles
Albert Einstein
Differential geometry
Partial differential equations