View source for Hermitian Yang–Mills connection

{{More citations needed|date=October 2024}}
In [[mathematics]], and in particular [[gauge theory (mathematics)|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 connection]]s, and are often called '''instantons'''.

The [[Kobayashi–Hitchin correspondence]] proved by [[Simon Donaldson|Donaldson]], [[Karen Uhlenbeck|Uhlenbeck]] and [[Shing-Tung Yau|Yau]] asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is [[Stable_vector_bundle|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>A</math> be a [[Hermitian connection]] on a Hermitian vector bundle <math>E</math> over a Kähler manifold <math>X</math> of dimension <math>n</math>. Then the '''Hermitian Yang–Mills equations''' are:

:<math>\begin{align}
&F_A^{0,2} = 0 \\
&F_A \cdot \omega = \lambda(E) \operatorname{Id},
\end{align}</math>

for some constant <math>\lambda(E)\in \mathbb{C}</math>. Here we have:

:<math>F_A \wedge \omega^{n-1} = (F_A \cdot \omega) \omega^n.</math>
Notice that since <math>A</math> is assumed to be a Hermitian connection, the curvature <math>F_A</math> is [[skew-Hermitian]], and so <math>F_A^{0,2}=0</math> implies <math>F_A^{2,0} = 0</math>. 
When the underlying Kähler manifold <math>X</math> is compact, <math>\lambda(E)</math> may be computed using [[Chern–Weil theory]]. Namely, we have
:<math>\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>F_A \cdot \omega = \lambda(E) \operatorname{Id}_E</math> and the identity endomorphism has trace given by the rank of <math>E</math>, we obtain
:<math>\lambda(E) = -\frac{2\pi i}{n! \operatorname{Vol}(X)} \mu(E),</math>

where <math>\mu(E)</math> is the [[Stable_vector_bundle#Slope_stability|slope]] of the vector bundle <math>E</math>, given by

:<math>\mu(E) = \frac{\deg(E)}{\operatorname{rank}(E)},</math>

and the volume of <math>X</math> is taken with respect to the volume form <math>\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 manifold|Einstein manifolds]], such as the Page metric on <math>{\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>E</math> has a [[holomorphic vector bundle|holomorphic structure]], there is a natural choice of Hermitian connection, the [[Chern connection]]. For the Chern connection, the condition that <math>F_A^{0,2}=0</math> is automatically satisfied. The [[Hitchin–Kobayashi correspondence]] asserts that a holomorphic vector bundle <math>E</math> admits a Hermitian metric <math>h</math> such that the associated Chern connection satisfies the Hermitian Yang–Mills equations if and only if the vector bundle is [[stable vector bundle|polystable]]. From this perspective, the Hermitian Yang–Mills equations can be seen as a system of equations for the metric <math>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 {{harvs|txt|last=Kobayashi|authorlink=Shoshichi Kobayashi|year=1980|loc=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>X</math> is <math>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>\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>E</math> vanishes, then the Hermitian Yang–Mills equations become <math>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>F_A^+ = 0</math>. That is, <math>A</math> is an [[Yang–Mills equations#Anti-self-duality equations|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.<ref>Donaldson, S. K., Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.</ref>

==See also==
* [[Einstein manifold]]
* [[Deformed Hermitian Yang–Mills equation]]
* [[Gauge theory (mathematics)]]

==References==
*{{Citation | last1=Kobayashi | first1=Shoshichi | title=First Chern class and holomorphic tensor fields | url=http://projecteuclid.org/getRecord?id=euclid.nmj/1118786013 |mr=556302 | year=1980 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=77 | pages=5–11| doi=10.1017/S0027763000018602 | s2cid=118228189 | doi-access=free }}
*{{Citation | last1=Kobayashi | first1=Shoshichi | title=Differential geometry of complex vector bundles | publisher=[[Princeton University Press]] | series=Publications of the Mathematical Society of Japan | isbn=978-0-691-08467-1 |mr=909698 | year=1987 | volume=15}}

{{Reflist}}

{{DEFAULTSORT:Einstein-Hermitian vector bundle}}
[[Category:Vector bundles]]
[[Category:Albert Einstein]]
[[Category:Differential geometry]]
[[Category:Partial differential equations]]