Yang-Mills equations

The Yang-Mills equation

Let $A$ be a connection on R^{d+1} which takes values in the Lie algebra g of a compact Lie group G. Formally, the connection A is said to obey the Yang-Mills equation if it is a critical point for the Lagrangian functional

$\int F^{ab}F_{ab}$

where $F:=dA+[A,A]$ is the curvature of the connection $A$. The Euler-Lagrange equations for this functional have the schematic form

$\Box A+\nabla (\nabla _{x,t}A)=[A,\nabla A]+[A,[A,A]]$

where $\nabla _{x,t}A=\partial _{a}A^{a}$ is the spacetime divergence of $A$. A more succinct (but less tractable) formulation of this equation is

$\partial _{a}F^{ab}=0$ .

It is often convenient to split $A$ into temporal and spatial components as $A=(A_{0},A_{i})$.

As written, the Yang-Mills equation is under-determined because of the gauge invariance

$A->U^{-1}dU+U^{-1}AU$

$<center>F->U^{-1}FU$

in the equation, where U is an arbitrary function taking values in $G$. In order to correctly formulate a Cauchy problem, one must impose a further constraint on the gauge. There are three standard ones:

Temporal gauge: $A^{0}=0$ Coulomb gauge: $\partial _{i}A_{i}=0$ Lorentz gauge: $\nabla _{x,t}A=0$

There are also several other useful gauges, such as the Cronstrom gauge Cs1980 centered around a point in spacetime.

The Lorentz gauge has the advantage of being invariant under conformal transformations, but it appears that the Yang-Mills equation is not well-behaved in this gauge for rough data. (For smooth data one can obtain local well-posedness in this gauge by energy estimates). The Coulomb gauge is the simplest to work with technically, and in this gauge the bilinear expression $[A,\nabla A]$ acquires a null structure KlMa1995 which allows for a satisfactory analysis of the equation. Unfortunately there are often difficulties in creating a global Coulomb gauge, and one often has to rely instead on local Coulomb gauges pieced together using finite speed of propagation; see KlMa1995. The Temporal gauge is fairly close to the Coulomb gauge, and one can develop a parallel theory for this gauge. The temporal gauge has the advantage of being easy to establish globally, but the null form structure is less obvious (one needs to partition the connection into divergence-free and curl-free components). See e.g. references:Ta-p3 Ta-p3.

In the Coulomb or Temporal gauges, one can create a model equation for the Yang-Mills system by ignoring cubic terms and any contribution from the "elliptic portion of the gauge ($A_{0}$ in the Coulomb gauge, or the curl-free portion of $A_{i}$ in the Temporal gauge). The resulting model equation is

$\Box A=\nabla ^{-1}Q(A,A)+Q(\nabla ^{-1}A,A)$

where $Q(A,A')$ is some null form such as

$Q(A,A'):=\partial _{i}A\partial _{j}A'-\partial _{j}A\partial _{i}A'$.

The results known for the model equation are slightly better than those known for the actual Yang-Mills or Maxwell-Klein-Gordon equations.

The Yang-Mills equations come with a positive definite conserved Hamiltonian

$\int |F_{0,i}|^{2}+|F_{i,j}|^{2}dx$

which mostly controls the $H^{1}$ norm of $A$ and the $L^{2}$ norm of $A_{t}$. However, there are some portions of the $H^{1}\times L^{2}$ norm which are not controlled by the Hamiltonian (in the Coulomb gauge, it is $\partial _{t}A_{0}$; in the Temporal gauge, it is the $H^{1}$ norm of the curl-free part of $A_{i}$). This causes some technical difficulties in the global well-posedness theory.

The Yang-Mills equations can also be coupled with a g-valued scalar field $f$, with the Lagrangian functional of the form

$\int F^{ab}F_{ab}+D_{a}f.D^{a}f+V(f)$

where $D_{a}=\partial _{a}+[A_{a},.]$ are covariant derivatives and $V$ is some potential function (e.g. $V(f)=|f|^{k+1})$. The corresponding Euler-Lagrange equations have the schematic form

$\Box A+\nabla (\nabla _{x,t}A)=[A,\nabla A]+[A,[A,A]]+[f,Df]<br/>D_{a}D^{a}f=V'(f)$

and are generally known as the Yang-Mills-Higgs system of equations. This system may be thought of as a Yang-Mills equation coupled with a semi-linear wave equation. [#mkg The Maxwell-Klein-Gordon system] is a special case of Yang-Mills-Higgs.

The theory of Yang-Mills connections is considerably more advanced in the elliptic case (when the Minkowski metric is replaced by a Riemannian one), especially in the critical case of four dimensions, but a discussion of this topic is beyond our expertise.

Attention has mostly focussed on the [#YM_on_R^3 three] and [#YM_on_R^4 four dimensional] cases; the one-dimensional case is trivial (e.g. in the temporal gauge it collapses to A_tt = 0). In higher dimensions n=5,7,9 singularities can develop from large smooth radial data CaSaTv1998 (see also [Biz-p]). Numerics suggest this phenomenon is generic, and also one appears to have blowup also at the critical dimension BizTb2001, [Biz-p].

The Yang-Mills equations can also be coupled with a spinor field. In the $U(1)$ case this becomes the Maxwell-Dirac equation.

The Yang-Mills equations in dimension n have many formal similarities with the wave maps equation at dimension n-2 (see e.g. CaSaTv1998 for a discussion)

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