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A [[Connection|connection]] in a [[Principal fibre bundle|principal fibre bundle]] over a (pseudo-)Riemannian manifold whose curvature satisfies the harmonicity condition (the Yang–Mills equation).

Yang–Mills fields, which are also called gauge fields, are used in modern physics to describe physical fields that play the role of carriers of an interaction. Thus, the electro-magnetic field in electro-dynamics, the field of vector  $  W $-bosons (carriers of the weak interaction in the Weinberg–Salam theory of electrically weak interactions), and finally, the gluon field (the carrier of the strong interaction) are described by Yang–Mills fields. The gravitational field can also be interpreted as a Yang–Mills field (see [[#References|[4]]]).

The idea of a connection as a field was first developed by H. Weyl (1917), who also attempted to describe the electro-magnetic field in terms of a connection. In 1954, C.N. Yang and R.L. Mills suggested that the space of intrinsic degrees of freedom of elementary particles (for example, the isotropic space describing the two degrees of freedom of a nucleon that correspond to its two pure states, proton and neutron) depends on the points of space-time, and the intrinsic spaces corresponding to different points are not canonically isomorphic. In geometrical terms, the suggestion of Yang and Mills was that the space of intrinsic degrees of freedom is a [[Vector bundle|vector bundle]] over [[Space-time|space-time]] that does not have a canonical trivialization, and physical fields are described by cross-sections of this bundle. To describe the differential evolution equation of a field one has to define a connection in the bundle, that is, a trivialization of the bundle along the curves in the base. Such a connection with a fixed holonomy group describes a physical field (usually called a Yang–Mills field). The equations for a free Yang–Mills field can be deduced from a variational principle; they are a natural non-linear generalization of the [[Maxwell equations|Maxwell equations]].

A more rigorous definition of a Yang–Mills field consists of the following. Let  $  \pi :  P \rightarrow M $
be a principal  $  G $-bundle over a Riemannian manifold  $  M $, 
and let  $  E ( M) = P \times _ {G} E \rightarrow M $
be the vector bundle associated with  $  \pi $
and a  $  G $-module  $  E $. 
A connection  $  \nabla $
of  $  \pi $
defines an operator  $  \nabla  ^ {E} :  \Gamma ( E) \rightarrow \Gamma ( T  ^ {*} ( M) \otimes E( M) ) $
acting on the space  $  \Gamma ( E) $
of cross-sections of  $  E ( M) $. 
It can be extended to an operator  $  d  ^  \nabla  :  \Gamma ( E _ {p} ) \rightarrow \Gamma ( E _ {p + 1 }  ) $, 
acting on the space  $  \Gamma ( E _ {p} ) $
of  $  E ( M) $-valued  $  p $-forms, by the formula  $  d  ^  \nabla  ( \omega \otimes e) = d \omega \otimes e + (- 1)  ^ {p} \omega \wedge \nabla  ^ {E} e $. 
The operator  $  \delta  ^  \nabla  $, 
on  $  p $-forms, formally conjugate to  $  d  ^  \nabla  $
is equal to  $  \delta  ^  \nabla  = (- 1) ^ {p + 1 } \star d  ^  \nabla  \star $, 
where  $  \star $
denotes the Hodge star operator.

A connection  $  \nabla $
in a principal  $  G $-bundle is called a Yang–Mills field if the curvature  $  R  ^  \nabla  $, 
considered as a  $  2 $-form with values in the vector bundle  $  \mathfrak g ( M) = P \times _ { \mathop{\rm Ad}   G } \mathfrak g $, 
where  $  \mathfrak g $
is the Lie algebra of the Lie group  $  G $, 
satisfies  $  \delta  ^  \nabla  R  ^  \nabla  = 0 $.

For a [[Riemannian connection|Riemannian connection]]  $  \nabla  ^ {g} $
of a Riemannian manifold  $  ( M, g) $, 
the Yang–Mills equation is equivalent to the symmetry condition

$$ 
( \nabla _ {X}  ^ {g}  \mathop{\rm Ric} ) ( Y, Z)  = \ 
( \nabla _ {Y}  ^ {g}  \mathop{\rm Ric} ) ( X, Z),\ \ 
X, Y, Z \in D ( M) = \Gamma ( TM) ,
$$

for the covariant derivative of the [[Ricci tensor|Ricci tensor]]  $   \mathop{\rm Ric} $. 
Thus, examples of Yang–Mills fields are Riemannian connections of Einstein spaces, and of direct products of such spaces. In particular, Riemannian connections of Kähler–Einstein spaces and quaternionic Riemannian spaces define Yang–Mills fields in the principal frame bundles with structure groups  $  U ( n/2) $
and  $   \mathop{\rm Sp} ( 1) \cdot  \mathop{\rm Sp} ( n/4) $. 
Examples of non-Einstein Riemannian connections satisfying the Yang–Mills equation are Riemannian connections of conformally-flat metrics with constant scalar curvature and non-constant sectional curvature. Examples of non-Riemannian connections satisfying the Yang–Mills equation are connections in the normal bundle of a totally-geodesic submanifold of a symmetric space, or of a quaternionic submanifold of a quaternionic space, induced by the Riemannian connections of these spaces.

The Yang–Mills equation is the Euler–Lagrange variational equation for the functional  $  L ( \nabla ) $
on the space of connections of  $  \pi $, 
defined by

$$ 
L ( \nabla )  =  \int\limits _ { M } \langle  R  ^  \nabla  , R  ^  \nabla  \rangle.
$$

The Riemannian manifold  $  ( M, g) $
is assumed to be compact and oriented, and  $  \langle  \cdot , \cdot \rangle $
denotes the scalar product in the fibres of the vector bundle  $  \mathfrak g ( M) \otimes \Lambda  ^ {2} $
that is defined by the  $   \mathop{\rm Ad}  G $-invariant scalar product in the Lie algebra  $  \mathfrak g $
of  $  G $, 
and by the scalar product in the fibres of the bundle  $  \Lambda  ^ {2} $
of  $  2 $-forms on  $  M $
induced by the metric  $  g $. 
Thus, Yang–Mills fields are the critical points of  $  L ( \nabla ) $. 
A Yang–Mills field is called stable if the second differential of  $  L $
at  $  \nabla $
is positive definite (and, consequently,  $  \nabla $
is a local minimum of  $  L $), 
and weakly stable if the second differential is non-negative definite. It is known that there are no weakly-stable Yang–Mills fields in an arbitrary non-trivial principal bundle over the standard sphere  $  S  ^ {n} $
for  $  n \geq  5 $. 
On the other hand, for  $  n \geq  3 $
the Riemannian connection of the standard Riemannian metric of the quotient space  $  S  ^ {n} / \Gamma $
of the sphere with respect to a freely-acting non-trivial finite group  $  \Gamma $
of isometries is a stable Yang–Mills field [[#References|[5]]].

For physicists, the greatest interest is in Yang–Mills fields on four-dimensional Riemannian (and also Lorentz) manifolds. In this case the Hodge  $  \star $
operator maps the space of  $  2 $-forms on  $  M $ (with values in an arbitrary vector bundle) onto itself; moreover, it is an involution  $  ( \star  ^ {2} =  \mathop{\rm id} ) $, 
and depends only on the orientation and the conformal class of the metric  $  g $. 
A connection  $  \nabla $
in the principal bundle over  $  M  ^ {4} $
is called a self-dual connection or an instanton (respectively, anti-self-dual connection or anti-instanton) if the curvature  $  2 $-form  $  R  ^  \nabla  $
is an eigenvector of the Hodge operator with eigenvalue 1 (respectively,  $  - 1 $). 
By the Bianchi identity, instantons and anti-instantons are Yang–Mills fields. Moreover, they are points at which  $  L $
has an absolute minimum. In the case of a principal bundle over the standard sphere with structure group  $   \mathop{\rm SU} ( 2) $, 
$   \mathop{\rm SU} ( 3) $
or  $   \mathop{\rm SU} ( 4) $, 
the local minima of  $  L $
are exhausted by the instantons and anti-instantons (and so these are global minima). A Riemannian connection on a Riemannian manifold  $  M  ^ {4} $
is an instanton only for manifolds with holonomy group  $  G \subset   \mathop{\rm Sp} ( 1) $. 
All such compact manifolds are exhausted by the complex  $  K3 $-surfaces (cf. [[K3-surface| $  K3 $-surface]]).

The group  $  G ( \pi ) = \Gamma ( P _ { \mathop{\rm Ad}  }  ^ {*} G) $
of automorphisms of the bundle  $  \pi :  P \rightarrow M $
that are identities on the base is called the gauge group. It acts in a natural way on the set  $  C  ^ {+} ( \pi ) $
of instantons of  $  \pi $
with holonomy group  $  G $. 
The quotient space  $  M  ^ {+} ( \pi ) = C  ^ {+} ( \pi )/G ( \pi ) $
is called the moduli space of irreducible instantons of  $  \pi $. 
If the structure group  $  G $
of  $  \pi $
is compact and semi-simple and the base of  $  M  ^ {4} $
is a compact orientable Riemannian manifold with non-negative non-zero scalar curvature for which the Weyl conformal curvature tensor is self-dual, then the moduli space  $  M  ^ {+} ( \pi ) $
is either empty or a manifold of dimension

$$ 
 \mathop{\rm dim}  M  ^ {+} ( \pi )  = \ 
2p _ {1} ( \mathfrak g ( M)) -

\frac{ \mathop{\rm dim}  G }{2}
 ( \chi ( M  ^ {4} ) - \sigma ( M  ^ {4} ) ) ,
$$

where  $  p _ {1} ( \mathfrak g ( M)) $
is the first [[Pontryagin number|Pontryagin number]] of the bundle  $  \mathfrak g ( M) $
and  $  \chi ( M  ^ {4} ) $
and  $  \sigma ( M  ^ {4} ) $
are, respectively, the Euler–Poincaré characteristic (cf. [[Euler characteristic|Euler characteristic]]) and the [[Signature|signature]] of  $  M  ^ {4} $.

The most complete results have been obtained in the physically important case of bundles over the standard sphere  $  S  ^ {4} $
with classical compact structure groups  $  G $. 
In particular, all instantons in such bundles can be described in purely algebraic terms (for example, in terms of certain modules over a Grassmann algebra, or in terms of the solutions to certain quaternion matrix equations (see [[#References|[1]]])). For the case  $  G =  \mathop{\rm Sp} ( 1) $, 
all instantons can be described explicitly. For example, for an  $   \mathop{\rm Sp} ( 1) $-bundle  $  \pi _ {1} :  P _ {1} \rightarrow S  ^ {4} $
with Pontryagin number 1, the moduli space  $  M  ^ {+} ( \pi _ {1} ) \approx \mathbf R  ^ {+} \times \mathbf H $, 
where  $  \mathbf R  ^ {+} $
is the set of positive numbers and  $  \mathbf H $
is the set of quaternions. To the pair  $  ( \lambda , a) \in \mathbf R  ^ {+} \times \mathbf H $
there corresponds the instanton defined by the  $  \mathfrak g $-valued  $  1 $-form of the connection

$$ 
A ( x)  =   \mathop{\rm Im}  
\frac{\overline{ {u ( x) }}\; \cdot du ( x) }{1 + | u ( x) |  ^ {2} }
 ,
$$

where  $  u ( x) = \lambda ( \overline{ {a - x }}\; )  ^ {- 1} $, 
$  x \in \mathbf H $. 
Quaternions in  $  \mathbf H \approx \mathbf R  ^ {4} $
are identified with points of  $  S  ^ {4} $
by using the stereographic projection, and the Lie algebra  $  \mathfrak g =  \mathop{\rm Sp} ( 1) $
is regarded as the Lie algebra  $   \mathop{\rm Im}  \mathbf H $
of purely-imaginary quaternions.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.I. Manin,   "Gauge fields and holomorphic geometry"  ''J. Soviet Math.'' , '''21''' :  4  (1983)  pp. 465–507  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''17'''  (1981)  pp. 3–55</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Shvarts,   "Elliptic operators in quantum field theory"  ''J. Soviet Math.'' , '''21''' :  4  (1983)  pp. 551–601  ''Itogi Nauk. i Tekhn. Sovr. Probl. Mat.'' , '''17'''  (1981)  pp. 113–173</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  M.F. Atiyah,   N.J. Hitchin,   I.M. Singer,   "Self-duality in four-dimensional Riemannian geometry"  ''Proc. Roy. Soc. London A'' , '''362'''  (1978)  pp. 425–461</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  D.A. Popov,   L.I. Daikhin,   "Einstein spaces, and Yang–Mills fields"  ''Dokl. Akad. Nauk SSSR'' , '''225''' :  4  (1975)  pp. 790–793  (In Russian)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  J. Bourguignon,   ''Proc. Nat. Acad. Sci. USA'' , '''76''' :  4  (1979)  pp. 1550–1553</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  N.P. Konopleva,   V.N. Popov,   "Gauge fields" , Horwood  (1981)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top">  C.N. Yang,   R.L. Mills,   "Conservation of isotopic spin and isotopic gauge invariance"  ''Phys. Rev.'' , '''96''' :  1  (1954)  pp. 191–195</TD></TR>
<TR><TD valign="top">[8]</TD> <TD valign="top"> , ''Geometrical ideas in physics'' , Moscow  (1983)  (In Russian; translated from English)</TD></TR>
<TR><TD valign="top">[9]</TD> <TD valign="top">  D.S. Freed,   K.K. Uhlenbeck,   "Instantons and four manifolds" , Springer  (1984)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  Yu.I. Manin,   "Gauge field theory and complex geometry" , Springer  (1988)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Albeverio,   S. Paycha,   S. Scarlatti,   "A short overview of mathematical approaches to functional integration"  Z. Haba (ed.)  J. Sobczyk (ed.) , ''Functional Integration, Geometry and Strings'' , Birkhäuser  (1989)  pp. 230–276</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  "Dynamical theory of groups and fields"  C.M. DeWitt (ed.)  B.S. DeWitt (ed.) , ''Relativity, groups and topology'' , Gordon &amp; Breach  (1964)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  T.-P. Cheng,   L.-F. Li,   "Gauge theory of elementary particle physics" , Clarendon Press  (1984)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Nélipa,   "Physique des particules élémentaires" , MIR</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  M.F. Atiyah,   "Geometry of Yang–Mills fields" , Scuola Norm. Sup. Pisa  (1979)</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  B. [B. Dubrovin] Doubrovine,   S. Novikov,   A. Fomenko,   "Contemporary geometry" , Springer  (1990)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  B. Lawson,   M.-L. Michelsohn,   "Spin geometry" , Princeton Univ. Press  (1989)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Bleecker,   "Gauge theory and variational principles" , Addison-Wesley  (1981)</TD></TR>
</table>