In differential geometry, the
Simple important cases of
-Yang–Mills connections include exponential Yang–Mills connections using the exponential function for
as exponent of a potence of the norm of the curvature form similar to the
Also often considered are Yang–Mills–Born–Infeld connections (or YMBI connections) with positive or negative sign in a function
involving the square root.
This makes the Yang–Mills–Born–Infeld equation similar to the minimal surface equation.
be a strictly increasing
function, one can also consider the following constant:[2] Let
be a compact Lie group with Lie algebra
-bundle with an orientable Riemannian manifold
and a volume form
be its adjoint bundle.
is the space of connections,[3] which are either under the adjoint representation
invariant Lie algebra–valued or vector bundle–valued differential forms.
Since the Hodge star operator
is defined on the base manifold
as it requires the metric
and the volume form
-Yang–Mills action functional is given by:[2][4] For a flat connection
is required to avert divergence for a non-compact manifold
, although this condition can also be left out as only the derivative
-Yang–Mills connection, if it is a critical point of the
-Yang–Mills action functional, hence if: for every smooth family
-Yang–Mills connection/field with:[1][2][4] Analogous to (weakly) stable Yang–Mills connections, one can define (weakly) stable
is called stable if: for every smooth family
It is called weakly stable if only
-Yang–Mills connection, which is not weakly stable, is called unstable.
[4] For a (weakly) stable or unstable
is furthermore called a (weakly) stable or unstable