Elliptic Yang–Mills equation

Abstract

We discuss some recent progress on the regularity theory of the elliptic Yang–Mills equation. We start with some basic properties of the elliptic Yang–Mills equation, such as Coulomb gauges, monotonicity, and curvature estimates. Next we discuss singularity of stationary Yang–Mills connections and compactness theorems on Yang–Mills connections with bounded L² norm of curvature. We also discuss in some detail self-dual solutions of the Yang–Mills equation and describe a compactification of their moduli space.

The Yang–Mills equation has played a fundamental role in our study of physics and geometry and topology in last few decades. Its regularity theory is crucial to our understanding of its mathematical applications. The aim of this note is to give a brief tour of recent progress on regularity theory of the Yang–Mills equation in a Euclidean space or more generally, a Riemannian manifold.

In the following, unless specified, we assume for simplicity that M is an open subset in ℝⁿ with the Euclidean metric. Let G be a subgroup in SO(r) and g be its Lie algebra. But I should emphasize that all our discussions here are valid for any differential manifold with a Riemannian metric and any compact Lie group G.

1. Yang–Mills Connections

First we recall that a connection on M with values in g is of the form

1.1

where x₁, … , x_n are Euclidean coordinates. Its curvature can be computed as follows:

1.2

and

1.3

where ∂_i denotes the ith partial derivative and [A, B] = AB − BA is the Lie bracket of g.

The Yang–Mills functional is defined on the space of connections and given by

1.4

where |F_A|² = −∑_i,jtr(F_ijF_ij). The Yang–Mills equation is simply its Euler–Lagrange equation

1.5

If we denote by D_A the differential operator dB − [B, A] and D^*_A is its adjoint, then Eq. 1.5 can be written simply as D^*_AF_A = 0. On the other hand, as the curvature of a connection we have the second Bianchi identity D_AF_A = 0, that is,

1.6

We will call A a Yang–Mills connection if it satisfies Eqs. 1.5 and 1.6.

The gauge group 𝒢 consists of all smooth maps form M into G ⊂ SO(r). It acts on the space of connections by assigning A to σ(A) = σAσ⁻¹ − σdσ⁻¹ for each σ ∈ 𝒢. Clearly, the Yang–Mills functional is invariant under the action of 𝒢, and so is the Yang–Mills equation. In particular, it implies that the Yang–Mills equation is not elliptic. However, it is elliptic modulo gauge transformations. To see it, we assume that A is the so-called Columbus gauge, that is ∑_i ∂_iA_i = 0, then the Yang–Mills equation reads

Given any connection A, there is a gauge transformation σ ∈ 𝒢 such that σ(A) is in the Columbus gauge, that is

1.7

The local solvability of this equation on gauge transformations has been shown by K. Uhlenbeck (1). It follows from

Theorem 1.1.

(From ref. 1) Let A = A_idx_ibe any connection withA_i ∈ L^p(B₁(p),g) for some p ≥ n/2, whereB₁(p) is a unit ball inℝⁿ. Then there exists ɛ(n) > 0 and c(n) > 0 such that if∥F_A∥_n/2 ≤ ɛ(n), where ∥ ⋅ ∥_qdenotes theL^q-norm in B₁(p), then there is a gauge transformation σ satisfying Eq.1.7 and ∥σ(A)∥_p ≤ c(n)∥F_A∥_p.

In general, in the same way as we did above, one can introduce Yang–Mills connections for any vector bundle over any Riemannian manifold with structure group G. If G is U(1), then a Yang–Mills connection is simply a purely imaginary valued 1-form whose curvature is a harmonic 2-form. So the theory of Yang–Mills connections is reduced to the Hodge theory for 1-forms.

2. Monotonicity and Its Consequences

Given any vector field X on M with compact support, we can integrate it to get a one-parameter group of diffeomorphisms φ_t: M ↦ M. Put A_t = φ*_t(A). Then A₀ = A and A_t coincides with A near the boundary of M. If A is a smooth Yang–Mills connection, differentiating 𝒴(A_t) on t at t = 0, one can derive as Price did in ref. 2

2.1

where X = X^k∂_k. This is very important even though it is nothing but the first variation of 𝒴 along X. Let us derive some of its consequences. Let p ∈ M such that the ball B_ρ0(p) with radius ρ₀ and center p is contained inside M. Then taking X to be ξ(r)r∂_r, where r is the distance from p and ξ is a cut-off function in B_ρ0(p), we can get the monotonicity formula of Price.

Theorem 2.1.

(From ref. 2) Let A be any Yang–Mills connection on M. Then for any 0 ≤ σ ≤ τ ≤ ρ₀, we have

2.2

In particular,ρ⁴⁻ⁿ∫_Bρ(p)|F_A|²dV is nondecreasing with ρ.

An application of this monotonicity is the following curvature estimate, which was proved by K. Uhlenbeck [ref. 1; also see Nakajima (3)].

Theorem 2.2.

Let A be any Yang–Mills connection on U. Then there are ɛ = ɛ(n) > 0 andC = C(n) > 0, such that for anyB_ρ(p) ⊂ M, we have

2.3

whenever ρ⁴⁻ⁿ ∫_Bρ(p) |F_A|²dV ≤ ɛ.

We refer the readers to ref. 3 (also ref. 4) for its proof. This curvature estimate implies that a Yang–Mills connection is almost flat whenever its normalized action in a neighborhood ball is sufficiently small.

We can associate a measure μ_A to each connection A as follows: For any continuous function f with compact support, we define

2.4

We can simply write μ_A = |F_A|²dV. By the monotonicity, we have is a nondecreasing function ρ⁴⁻ⁿμ_A(B_ρ(p)).

Now we let {A_i} be a sequence of Yang–Mills connections such that for each compact subset K ⊂ M, μ_i(K) are uniformly bounded, where μ_i is the measure associated to A_i. Then a subsequence {μ_a} of {μ_i} converges weakly to a measure μ. Because of the monotonicity for μ_i, one can easily show that ρ⁴⁻ⁿμ(B_ρ(p)) is a nondecreasing function for each p ∈ M. Define the density function of μ by

2.5

Because of the monotonicity for μ, this density Θ_μ is well defined, nonnegative, and upper-semi-continuous. It follows that the support S of Θ_μ is a locally closed subset of M such that the Hausdorff measure ℋⁿ⁻⁴(S ∩ K) is finite for any compact subset K. Furthermore, it follows from Theorem 2.2 that Θ_μ(p) ≥ ɛ for any p ∈ S and the curvature of A_a is uniformly bounded on any compact subset in M∖S. Then, using Theorem 2.2, one can show the following theorem, which is due to Uhlenbeck.

Theorem 2.3.

(From ref. 5) Let A_a, μ_a, μ and S be as above. Then there are gauge transformations σ_a ∈ 𝒢such that by taking a subsequence if necessary, σ_a(A_a) converges smoothly to a Yang–Mills connection A defined onM∖S. Moreover, μ_A ≤ μ.

By an admissible Yang–Mills connection, we mean a smooth Yang–Mills connection A defined outside a locally closed subset S(A) in M, such that ℋⁿ⁻⁴(S(A) ∩ K) < ∞ and μ_A(K) < ∞ for any compact subset K ⊂ M. Clearly, the limiting connection in Theorem 2.3 is admissible. In fact, following Uhlenbeck (5), one can easily extend Theorem 2.3 to any sequence of admissible Yang–Mills connections. We will assume that S(A) is the singular set of an admissible Yang–Mills connection A. If S(A) = ⊘, then A is smooth.

3. Removable Singularity Theorem

Let A be an admissible Yang–Mills connection with singular set S(A). We say that A is stationary if Eq. 2.1 holds for any smooth vector field X with compact support. As we have shown in last section, any smooth Yang–Mills connection is stationary. However, not every admissible Yang–Mills connection is stationary.

Theorem 3.1.

Let A be a stationary admissible Yang–Mills connection and smooth on M∖S, where S is a closed subset in M and has locally finite(n − 4)-dimensional Hausdorff measure. Then there is an ɛ > 0, which depends only onn, such that for any B_ρ(p) ⊂∈ S(A), if

3.1

then there is a gauge transformation σ near psuch that σ(A) extends to be a smooth connection near p.

When n ≤ 3, S(A) is empty. When n = 4, S(A) consists of finitely many points and Eq. 2.1 holds for any admissible Yang–Mills connections. Hence, this theorem reduces to the removable singularity theorem of K. Uhlenbeck for Yang–Mills connections on 4-manifolds (1). When n > 4, this theorem was first proved in ref. 4 under certain conditions on A and was proved in ref. 6 for general cases.

Corollary 3.1.

Let A be a stationary admissible Yang–Mills connection. Then there is a gauge transformation σ such that σ(A) is smooth outside a locally closed subsetS′ with vanishing (n − 4)-dimensional Hausdorff measure, that isℋⁿ⁻⁴(S′) = 0. Ifn = 4, then σ(A) is actually smooth.

We propose the following:

Conjecture 3.1.

Let A be a stationary admissible Yang–Mills connection, then there is a gauge transformation σ such that σ(A) extends to be a smooth connection outside a locally closed subset with locally finite Hausdorff measure of dimension n − 5.

4. Structure of Blow-up Loci

Let {A_i} be a sequence of smooth Yang–Mills connections such that its associated measures μ_i converge weakly to a measure μ. As before, we denote by Θ_μ the density and by S the support of μ. By Theorem 2.3 and taking a subsequence if necessary, we may assume that there are gauge transformations σ_i such that σ_i(A_i) converge to an admissible Yang–Mills connection A outside S.

Now we will examine the structure of S. Let μ_A be the measure associated to A. Define

4.1

This set is called the blow-up locus of {A_i}. If no confusion occurs, we will simply write S_b for this blow-up locus. It is easy to see that ℋⁿ⁻⁴() = 0. The following proposition was proved in ref. 4. It gives the first regularity on the blow-up locus.

Proposition 4.1.

Let {A_i} be the above sequence of Yang–Mills connections that converge to A. Then its blow-up locus S_bisℋⁿ⁻⁴-rectifiable; that is, forℋⁿ⁻⁴-a.e. p inS_b, there is a unique tangent spaceT_pS_bof S_bat p. Moreover, for any smooth function f with compact support, we have

4.2

Furthermore, there are constraints on the geometry of the blow-up loci.

Theorem 4.1.

(From ref. 4) For any vector field X with compact support in M, we have

4.3

where div_Sb X denotes the divergence of X along S_bandF_ijare the components ofF_A.

Corollary 4.1.

If A is stationary, thenS_bis stationary; that is, S_b has no boundary in M and its generalized mean curvature vanishes.

I doubt that A is stationary in general, but it is stationary when A is anti-self-dual (cf. next section). If A = 0, then S_b is stationary and the curvature of A_i concentrates near a minimal variety of codimension 4. It leads to the question: Let Sbe a minimal submanifold S of dimension n − 4 in general position; is S the limit of a sequence of Yang–Mills connections?

We will call the above (A, S_b, Θ_μ) a generalized Yang–Mills connection. Two generalized Yang–Mills (A, S_b, Θ) and (A′, S′_b, Θ′) if A and A′ are gauge equivalent on an open dense subset. The set of all generalized Yang–Mills connections modulo gauge transformations is precompact.

Theorem 4.1 can also be used to prove the existence of tangent cones for generalized Yang–Mills connections. Let A be a stationary admissible Yang–Mills connection with singular set S(A). For any λ > 0 and p ∈ S(A), we can define

where A = ∑_i A_idx_i. Then there are sequences {λ(i)} such that lim_i→∞λ(i) = 0 and A_λ(i) converge to a connection A^c outside S_c with ℋⁿ⁻⁴(S^c ∩ B_R(0)) < ∞ for any R > 0. Further, measures |F_Ai|²dV converge weakly to a measure μ_c with density Θ_c. From Theorem 4.1 follows

Corollary 4.2.

Let A_λ(i), A_c, S_c, Θ_cbe as above. Then we have that∂_rΘ_c = 0, a ⋅ S_c = S_candF_Ac(∂_r, ⋅) = 0.

5. Anti-Self-Dual Instantons

Anti-self-dual instantons provide special solutions of the Yang–Mills equation. They are widely used in physics, geometry, and topology.

Let Ω be a closed differential form on M of degree n − 4. Let us introduce Ω-anti-self-dual instantons, or simply asd instantons if no possible confusion may occur. For simplicity, we assume that G = SU(r). Recall that the Hodge operator ∗ on differential forms is defined by

where σ is any permutation of {1, … , n}. We say that a SU(r)-connection A is an Ω-anti-self-dual instanton if its curvature F_A = ∑ F_ijdx_i ∧ dx_j satisfies

5.1

or equivalently

Using the closedness of Ω and the second Bianchi identity, one can easily show that any asd instantons are Yang–Mills connections. An asd instanton is an absolute minimizer of the Yang–Mills functional 𝒴 if Ω has conorm no more than one.

The following theorem shows advantages of using asd instantons. We call A an admissible asd instanton if it is an admissible Yang–Mills connection and anti-self-dual wherever it is well defined.

Theorem 5.1.

(From ref. 4) Assume that Ω is a parallel form of degree n − 4. Let A be an admissible Ω-anti-self-dual instanton on M. Then Ais stationary.

In particular, combining this theorem with the Removable Singularity Theorem, we see that if A is an admissible asd instanton, then there is a gauge transformation τ such that the singular set S of τ(A) is of ℋⁿ⁻⁴(S) = 0. In fact, we propose

Conjecture 5.1.

If A is an admissible asd instanton, then there is a gauge transformation τ such thatℋⁿ⁻⁶(S(τ(A)) ∩ K) < ∞for any compact K ⊂ M.

Now we assume that {A_i} is a sequence of Ω-anti-self-dual instantons that converge to an admissible Ω-anti-self-dual instanton A (cf. Theorem 2.3), where Ω is a form on M of degree n − 4. Let S_b ⊂ M be the blow-up locus of {A_i} with the density Θ_μ. Note that μ_i is the measure associated to A_i and lim_i→∞μ_i = μ.

For any admissible connection A′, we can associate a current C₂(A′) as follows: For any smooth form ϕ with compact support in M, we define

5.2

Clearly, if A′ is smooth, it is nothing else but the current represented by the Chern–Weil form defining the second Chern class, so it is closed. In general, it was proved in ref. 4 that C₂(A′) is closed in M.

Since S_b is rectifiable (Proposition 4.1), we can also define a current C₂(S_b, Θ_μ) by

5.3

Theorem 5.2.

(From ref. 4) Let A_i, A, et al. be as above. Then(1/8π²)Θ_μis integer-valued and S_bis calibrated by Ω; that is, for ℋⁿ⁻⁴-a.e. p ∈ S_bwhereT_pS_bexists, the restriction of Ω to T_pS_bcoincides with the induced volume form. Moreover, we have

5.4

In particular, for any compact K, we have

A simplified situation of Theorem 5.2 can be described as follows: Let π : ℝⁿ ↦ ℝ⁴ be an orthogonal projection and B be an asd instanton on ℝ⁴. Then the pull-back A = π*B is Ω-asd if and only if L = π⁻¹(0) is an Ω-calibrated subspace. This can be checked directly. As before, we ask if an Ω-calibrated submanifold is the limit of a sequence of Ω-asd instantons.

It is well known (cf. ref. 7) that if |Ω| ≤ 1, then any integral current calibrated by Ω is minimizing in its homology class. The corollary follows

Corollary 5.1.

Assume that |Ω| ≤ 1. LetS_bbe the blow-up locus of a sequence of asd instantons A_iconverging to A andΘ_μbe its associated density. Then C₂(S_b, Θ_μ)is an area-minimizing integral current.

The support S_b of C₂(S_b, Θ_μ) may not be smooth. However, one can show that a dense open subset of S_b is smooth. Further, we do expect

Conjecture 5.2.

Let Ω be any closed differential form with|Ω| ≤ 1. Then Ω-calibrated integral current is supported on the closure N of a smooth manifold N₀such that N∖N₀is of codimension at least two.

We end this section with an example. Assume that n = 2m. Fix an identification ℝⁿ = ℂ^m. Let ω be given in complex coordinates z₁, … , z_m by

Put Ω = ω^m−2/(m − 2)!. Then an Ω-asd instanton A is simply a Hermitian–Yang–Mills connection; that is, F = 0 and F⋅ω = 0, where F is the (k, l)-part of F_A. Moreover, a subspace L ⊂ ℝⁿ of codimension 4 is Ω-calibrated if and only if L is a complex subspace in ℂ^m. Let S be the blow-up locus of a sequence of Hermitian–Yang–Mills connections and Θ be its associated density. Then C₂(S, Θ) is a closed integral current whose tangent spaces are complex subspaces. It follows from a result of J. King (8) that there are positive integers m_a and irreducible complex subvarieties V_a such that for any smooth ϕ with compact support in M,

It can be also proved that if A is an admissible asd instanton with respect to ω^m−2/(m − 2)!, then there is a gauge transformation τ such that τ(A) extends to be a smooth connection outside a complex subvariety of codimension greater than 2. For more details, see Tian and Yang (9).

6. Compactifying Moduli Spaces

In this section, I give an application. I will give a natural compactification of the moduli space of asd instantons.

Now we let M be a compact n-manifold with a Riemannian metric g and Ω be a closed form of degree n − 4. Let E be a unitary vector bundle over M. Recall that 𝔐_Ω,E consists of all gauge equivalence classes of Ω-asd instantons of E over M. In general, 𝔐_Ω,E may not be compact. So we will compactify it.

A generalized Ω-asd instanton is made of an admissible Ω-asd instanton A of E, which extends to a smooth connection over M∖S(A) for a closed subset S(A) with ℋⁿ⁻⁴(S(A)) = 0, and a closed integral current C = C₂(S, Θ) calibrated by Ω, such that cohomologically,

6.1

where C₂(E) denotes the second Chern class of E. Two generalized Ω-asd instantons (A, C), (A′, C′) are equivalent if and only if C = C′ and there is a gauge transformation σ on M∖S(A) ∪ S(A′), such that σ(A) = A′ on M∖S(A) ∪ S(A′). We denote by [A, C] the gauge equivalence class of (A, C). We identify [A, 0] with [A] in ℳ_Ω,E if A extends to a smooth connection of E over M modulo a gauge transformation. We define 𝔐_Ω,E to be set of all gauge equivalence classes of generalized Ω-asd instantons of E over M.

The topology of _Ω,E can be defined as follows: a sequence [A_i, C_i] converges to [A, C] in _Ω,E if and only if there are representatives (A_i, C_i) such that their associated currents C₂(A_i, C_i) converge weakly to C₂(A, C) as currents, where

It is not hard to show that, by taking a subsequence if necessary, τ_i(A_i) converges to A outside S(A) and the support of C for some gauge transformations τ_i.

Theorem 6.1.

(From ref. 4) For any M, G, Ω, and E as above,_Ω,E is compact with respect to this topology.

When M is an m-dimensional compact Kähler manifold with Kähler form ω, Ω-asd instantons are Hermitian–Yang–Mills connections, where Ω = ω^m−2/(m − 2)!. A generalized Ω-asd instanton consists of a holomorphic cycle together with a Hermitian–Yang–Mills connection of a reflexive sheaf. In particular, the compactification 𝔐_Ω,E can be explicitly described (cf. ref. 9).

Abbreviations

• asd instantons, Ω-anti-self-dual instantons