Yang–Mills equations

Let ${\displaystyle X}$ be a compact, oriented, Riemannian manifold. The Yang–Mills equations can be phrased for a connection on a vector bundle or principal ${\displaystyle G}$-bundle over ${\displaystyle X}$, for some compact Lie group ${\displaystyle G}$. Here the latter convention is presented. Let ${\displaystyle P}$ denote a principal ${\displaystyle G}$-bundle over ${\displaystyle X}$. Then a connection on ${\displaystyle P}$ may be specified by a Lie algebra-valued differential form ${\displaystyle A}$ on the total space of the principal bundle. This connection has a curvature form ${\displaystyle F_{A}}$, which is a two-form on ${\displaystyle X}$ with values in the adjoint bundle ${\displaystyle \operatorname {ad} (P)}$ of ${\displaystyle P}$. Associated to the connection ${\displaystyle A}$ is an exterior covariant derivative ${\displaystyle d_{A}}$, defined on the adjoint bundle. Additionally, since ${\displaystyle G}$ is compact, its associated compact Lie algebra admits an invariant inner product under the adjoint representation.

Since ${\displaystyle X}$ is Riemannian, there is an inner product on the cotangent bundle, and combined with the invariant inner product on ${\displaystyle \operatorname {ad} (P)}$ there is an inner product on the bundle ${\displaystyle \operatorname {ad} (P)\otimes \Lambda ^{2}T^{*}X}$ of ${\displaystyle \operatorname {ad} (P)}$-valued two-forms on ${\displaystyle X}$. Since ${\displaystyle X}$ is oriented, there is an ${\displaystyle L^{2}}$-inner product on the sections of this bundle. Namely,

${\displaystyle \langle s,t\rangle _{L^{2}}=\int _{X}\langle s,t\rangle \,dvol_{g}}$

where inside the integral the bundle-wise inner product is being used, and ${\displaystyle dvol_{g}}$ is the Riemannian volume form of ${\displaystyle X}$. Using this ${\displaystyle L^{2}}$-inner product, the formal adjoint operator of ${\displaystyle d_{A}}$ is defined by

${\displaystyle \langle d_{A}s,t\rangle _{L^{2}}=\langle s,d_{A}^{*}t\rangle _{L^{2}}}$.

Explicitly this is given by ${\displaystyle d_{A}^{*}=\pm \star d_{A}\star }$ where ${\displaystyle \star }$ is the Hodge star operator acting on two-forms.

Assuming the above set up, the Yang–Mills equations are a system of (in general non-linear) partial differential equations given by

${\displaystyle d_{A}^{*}F_{A}=0.}$[3]

(1)

Since the Hodge star is an isomorphism, by the explicit formula for ${\displaystyle d_{A}^{*}}$ the Yang–Mills equations can equivalently be written

${\displaystyle d_{A}\star F_{A}=0.}$

(2)

A connection satisfying (1) or (2) is called a Yang–Mills connection.

Every connection automatically satisfies the Bianchi identity ${\displaystyle d_{A}F_{A}=0}$, so Yang–Mills connections can be seen as a non-linear analogue of harmonic differential forms, which satisfy

${\displaystyle d\omega =d^{*}\omega =0}$.

In this sense the search for Yang–Mills connections can be compared to Hodge theory, which seeks a harmonic representative in the de Rham cohomology class of a differential form. The analogy being that a Yang–Mills connection is like a harmonic representative in the set of all possible connections on a principal bundle.

The Yang–Mills equations are the Euler–Lagrange equations of the Yang–Mills functional, defined by

${\displaystyle \operatorname {YM} (A)=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}.}$

(3)

To derive the equations from the functional, recall that the space ${\displaystyle {\mathcal {A}}}$ of all connections on ${\displaystyle P}$ is an affine space modelled on the vector space ${\displaystyle \Omega ^{1}(P;{\mathfrak {g}})}$. Given a small deformation ${\displaystyle A+ta}$ of a connection ${\displaystyle A}$ in this affine space, the curvatures are related by

${\displaystyle F_{A+ta}=F_{A}+td_{A}a+t^{2}a\wedge a.}$

To determine the critical points of (3), compute

${\displaystyle {\begin{aligned}{\frac {d}{dt}}\left(\operatorname {YM} (A+ta)\right)_{t=0}&={\frac {d}{dt}}\left(\int _{X}\langle F_{A}+t\,d_{A}a+t^{2}a\wedge a,F_{A}+t\,d_{A}a+t^{2}a\wedge a\rangle \,d\mathrm {vol} _{g}\right)_{t=0}\\&={\frac {d}{dt}}\left(\int _{X}\|F_{A}\|^{2}+2t\langle F_{A},d_{A}a\rangle +2t^{2}\langle F_{A},a\wedge a\rangle +t^{4}\|a\wedge a\|^{2}\,d\mathrm {vol} _{g}\right)_{t=0}\\&=2\int _{X}\langle d_{A}^{*}F_{A},a\rangle \,d\mathrm {vol} _{g}.\end{aligned}}}$

The connection ${\displaystyle A}$ is a critical point of the Yang–Mills functional if and only if this vanishes for every ${\displaystyle a}$, and this occurs precisely when (1) is satisfied.

The Yang–Mills equations are gauge invariant. Mathematically, a gauge transformation is an automorphism ${\displaystyle g}$ of the principal bundle ${\displaystyle P}$, and since the inner product on ${\displaystyle \operatorname {ad} (P)}$ is invariant, the Yang–Mills functional satisfies

${\displaystyle \operatorname {YM} (g\cdot A)=\int _{X}\|gF_{A}g^{-1}\|^{2}\,d\mathrm {vol} _{g}=\int _{X}\|F_{A}\|^{2}\,d\mathrm {vol} _{g}=\operatorname {YM} (A)}$

and so if ${\displaystyle A}$ satisfies (1), so does ${\displaystyle g\cdot A}$.

There is a moduli space of Yang–Mills connections modulo gauge transformations. Denote by ${\displaystyle {\mathcal {G}}}$ the gauge group of automorphisms of ${\displaystyle P}$. The set ${\displaystyle {\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}}$ classifies all connections modulo gauge transformations, and the moduli space ${\displaystyle {\mathcal {M}}}$ of Yang–Mills connections is a subset. In general neither ${\displaystyle {\mathcal {B}}}$ or ${\displaystyle {\mathcal {M}}}$ is Hausdorff or a smooth manifold. However, by restricting to irreducible connections, that is, connections ${\displaystyle A}$ whose holonomy group is given by all of ${\displaystyle G}$, one does obtain Hausdorff spaces. The space of irreducible connections is denoted ${\displaystyle {\mathcal {A}}^{*}}$, and so the moduli spaces are denoted ${\displaystyle {\mathcal {B}}^{*}}$ and ${\displaystyle {\mathcal {M}}^{*}}$.

Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances. Michael Atiyah and Raoul Bott studied the Yang–Mills equations for bundles over compact Riemann surfaces.[4] There the moduli space obtains an alternative description as a moduli space of holomorphic vector bundles. This is the Narasimhan–Seshadri theorem, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson.[5] In this setting the moduli space has the structure of a compact Kähler manifold. Moduli of Yang–Mills connections have been most studied when the dimension of the base manifold ${\displaystyle X}$ is four.[3][6] Here the Yang–Mills equations admit a simplification from a second-order PDE to a first-order PDE, the anti-self-duality equations.

When the dimension of the base manifold ${\displaystyle X}$ is four, a coincidence occurs: the Hodge star operator maps two-forms to two-forms,

${\displaystyle \star$ :\Omega ^{2}(X)\to \Omega ^{2}(X)} .

The Hodge star operator squares to the identity in this case, and so has eigenvalues ${\displaystyle 1}$ and ${\displaystyle -1}$. In particular, there is a decomposition

${\displaystyle \Omega ^{2}(X)=\Omega _{+}(X)\oplus \Omega _{-}(X)}$

into the positive and negative eigenspaces of ${\displaystyle \star }$, the self-dual and anti-self-dual two-forms. If a connection ${\displaystyle A}$ on a principal ${\displaystyle G}$-bundle over a four-manifold ${\displaystyle X}$ satisfies either ${\displaystyle F_{A}={\star F_{A}}}$ or ${\displaystyle F_{A}=-{\star F_{A}}}$, then by (2), the connection is a Yang–Mills connection. These connections are called either self-dual connections or anti-self-dual connections, and the equations the self-duality (SD) equations and the anti-self-duality (ASD) equations.[3] The spaces of self-dual and anti-self-dual connections are denoted by ${\displaystyle {\mathcal {A}}^{+}}$ and ${\displaystyle {\mathcal {A}}^{-}}$, and similarly for ${\displaystyle {\mathcal {B}}^{\pm }}$ and ${\displaystyle {\mathcal {M}}^{\pm }}$.

The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where ${\displaystyle G=\operatorname {SU} (2)}$ and ${\displaystyle X}$ is simply-connected.[7][8][9] In this setting, the principal ${\displaystyle \operatorname {SU} (2)}$-bundle is classified by its second Chern class, ${\displaystyle c_{2}(P)\in H^{4}(X,\mathbb {Z} )\cong \mathbb {Z} }$.[Note 1] For various choices of principal bundle, one obtains moduli spaces with interesting properties. These spaces are Hausdorff, even when allowing reducible connections, and are generically smooth. It was shown by Donaldson that the smooth part is orientable. By the Atiyah–Singer index theorem, one may compute that the dimension of ${\displaystyle {\mathcal {M}}_{k}^{-}}$, the moduli space of ASD connections when ${\displaystyle c_{2}(P)=k}$, to be

${\displaystyle \dim {\mathcal {M}}_{k}^{-}=8k-3(1-b_{1}(X)+b_{+}(X))}$

where ${\displaystyle b_{1}(X)}$ is the first Betti number of ${\displaystyle X}$, and ${\displaystyle b_{+}(X)}$ is the dimension of the positive-definite subspace of ${\displaystyle H_{2}(X,\mathbb {R} )}$ with respect to the intersection form on ${\displaystyle X}$.[3] For example, when ${\displaystyle X=S^{4}}$ and ${\displaystyle k=1}$, the intersection form is trivial and the moduli space has dimension ${\displaystyle \dim {\mathcal {M}}_{1}^{-}(S^{4})=8-3=5}$. This agrees with existence of the BPST instanton, which is the unique ASD instanton on ${\displaystyle S^{4}}$ up to a 5 parameter family defining its centre in ${\displaystyle \mathbb {R} ^{4}}$ and its scale. Such instantons on ${\displaystyle \mathbb {R} ^{4}}$ may be extended across the point at infinity using Uhlenbeck's removable singularity theorem.

• Connection (vector bundle)
• Connection (principal bundle)
• Donaldson theory
• Hermitian Yang–Mills equations
• Deformed Hermitian Yang–Mills equations
• Yang–Mills–Higgs equations