Yang-Mills theory
Yang-Mills theory is defined on a four-dimensional smooth connected Riemannian manifold \( M \), physicists usually prefer a Minkowski or Euclidean space. Let \( G \) be a Lie group and \(T \) a \(G \)-connection on a principal G-bundle \(P \) over \( M \). The corresponding curvature \( F_T \in \Omega^2 (M) \) has values in \( ad(P) = P \times G\mathbb{g} \) the adjoint bundle and in a particular trivialization of \( P \) over \( M \) we have explicit formulas,
\( \quad T = T_0dx^0 + T_1dx^1 + T_2dx^2 + T_3dx^3 \)
\( \quad F_T = dT + T \wedge T \)
where \( Adx \wedge Bdy = [A,B]_\mathcal{g}(dx \wedge dy). \)
We have \( F_T = [D_{\mu}, D_{\nu}]dx^{\mu} \wedge dx^{\nu} \) for \( D_{\mu} = \partial_{\mu} + T_{\mu} \).
Notation convention …
Note that in physics literature \( D_{\mu} \) is often defined as \( \partial_{\mu} + i \tilde{T}_{\mu} \) and \( F_T \) changes accordingly.
Defining \( F_{\mu \nu} = [D_{\mu}, D_{\nu}] \), the free action Yang-Mills functional defining this physical theory is,
\( \quad \displaystyle S = -\frac{1}{4g^2} \int _M \langle F_{\mu \nu}, F_{\mu \nu} \rangle d^4x \)
where \( \langle , \rangle \) is a bi-invariant inner product of the Lie algebra \( g \) (usually taken to be the Cartan inner product) and g is the gauge coupling strength.
Calculus of variations leads to the Yang-Mills equations of motion corresponding to critical points of \( S \):
\( \quad D_T F_T = 0 \)
The Bianchi identity, for a connection on a bundle, is \( D_T \star F_T = 0\) (can be derived from Jacobi identity for example). Therefore solutions to the Yang-Mills equations of motion are:
\( \quad SD: F_T = \star F_T \)
\( \quad ASD: F_T = -\star F_T \)
These are the self-dual (SD) and anti-self-dual (ASD) equations.
Self-dual vs. Anti-self-dual …
The Hodge star operator \( \star \) naturally splits the exterior algebra of differential 2-forms into self-dual and anti-self-dual parts,
\( \quad \Omega^2 (M) = \Omega^+ (M) \oplus \Omega^-(M) \)
since the Hodge star operator applied twice equals the identity. Thus \( \Omega^+ (M) \) consists of the +1-eigenvalues of \( \star^2 \) and \( \Omega^- (M) \) the -1 -eigenvalues.
There is a trivial choice of orientation between the SD equations or the ASD equations. The self-dual viewpoint is more natural in the context of hyperkähler manifolds (see the hyperkähler quotient construction) since the quaternionic 2-form \( dx \wedge d\bar{x} \) is self-dual, while the anti-self-dual equations are simpler on holomorphic bundles. Donaldson and Kronheimer, in their book [DK], show this nicely in section 2.1.5. They use the fiber metric over the 2-dimensional complex surface along with the complex structure to define a (1,1) form \( \omega \) which splits the algebra of (1,1) forms:
\( \quad \Omega^{1,1}(M) = \Omega_0^{1,1}(M) + \Omega^{1,1}(M).\omega \)
where \(\rho \in \Omega_0^{1,1}(M) \Leftrightarrow \rho . \omega = 0 \).
\( \Omega_0^{1,1}(M) \) coincides with \( \Omega^- (M) \). As can be seen from [DK], the ASD equations take on a simpler (and more general) form. By considering a real four dimensional manifold locally parameterized by holomorphic or quaternionic coordinate functions, quick calculations can verify the above.
Dimensional reduction
In our local coordinates, the ASD equations are explicitly,
\( \quad F_{01} = – F_{23} \)
\( \quad F_{02} = -F_{31} \)
\( \quad F_{03} = -F_{12} \)
If we suppose that our connection matrices are independent of three dimensions i.e. \( T_\mu \rightarrow T_{\mu}(x_0) \) (this is the “dimensional reduction”) and change our notation slightly, \( x_0 :=s \), we get Nahm’s equations,
\( \quad [T_2, T_3] – \bigl [ \frac{d}{ds} + T_0,T_1 \bigr ] = 0 \)
\( \quad [T_3, T_1] – \bigl [ \frac{d}{ds} + T_0,T_2 \bigr ] = 0 \)
\( \quad [T_1, T_2] – \bigl [ \frac{d}{ds} + T_0,T_3 \bigr ] = 0 \)
Local form of Nahm equations …
By demanding that a section \( \phi \) of our principle bundle transforms covariantly under a gauge transformation \( g: M \rightarrow G \) i.e. \( D(g.\phi ) = g.D\phi \) leads to the transformation rule for connection T,
\( \quad T_\mu \rightarrow g^{-1} T_\mu g + g^{-1}dg \)
It is now a straightforward exercise to show that, using with an appropriate gauge transformation, Nahm’s equations may be expressed locally as,
\( \quad \displaystyle \frac{dT_i}{ds} = \epsilon_{ijk}[T_j, T_k] \)