Elliptic Yang–Mills equation
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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 L^{2} norm of curvature. We also discuss in some detail selfdual 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 ℝ^{n} 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 x_{1}, … , 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,j}tr(F_{ij}F_{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^{*}_{A}F_{A} = 0. On the other hand, as the curvature of a connection we have the second Bianchi identity D_{A}F_{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 socalled Columbus gauge, that is ∑_{i} ∂_{i}A_{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_{i}dx_{i}be any connection withA_{i} ∈ L^{p}(B_{1}(p),g) for some p ≥ n/2, whereB_{1}(p) is a unit ball inℝ^{n}. Then there exists ɛ(n) > 0 and c(n) > 0 such that if∥F_{A}∥_{n/2} ≤ ɛ(n), where ∥ ⋅ ∥_{q}denotes theL^{q}norm in B_{1}(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 1form whose curvature is a harmonic 2form. So the theory of Yang–Mills connections is reduced to the Hodge theory for 1forms.
2. Monotonicity and Its Consequences
Given any vector field X on M with compact support, we can integrate it to get a oneparameter group of diffeomorphisms φ_{t}: M ↦ M. Put A_{t} = φ*_{t}(A). Then A_{0} = 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 ρ_{0} 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 cutoff 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 ≤ ρ_{0}, we have2.2 In particular,ρ^{4−n}∫_{Bρ}_{(p)}F_{A}^{2}dV 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 have2.3whenever ρ^{4−n} ∫_{Bρ}_{(p)} F_{A}^{2}dV ≤ 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}^{2}dV. By the monotonicity, we have is a nondecreasing function ρ^{4−n}μ_{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 ρ^{4−n}μ(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 uppersemicontinuous. It follows that the support S of Θ_{μ} is a locally closed subset of M such that the Hausdorff measure ℋ^{n−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 ℋ^{n−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), if3.1then 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 4manifolds (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 isℋ^{n−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 Blowup 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 blowup locus of {A_{i}}. If no confusion occurs, we will simply write S_{b} for this blowup locus. It is easy to see that ℋ^{n−4}() = 0. The following proposition was proved in ref. 4. It gives the first regularity on the blowup locus.
Proposition 4.1.
Let {A_{i}} be the above sequence of Yang–Mills connections that converge to A. Then its blowup locus S_{b}isℋ^{n−4}rectifiable; that is, forℋ^{n−4}a.e. p inS_{b}, there is a unique tangent spaceT_{p}S_{b}of S_{b}at p. Moreover, for any smooth function f with compact support, we have4.2 Furthermore, there are constraints on the geometry of the blowup loci.
Theorem 4.1.
(From ref. 4) For any vector field X with compact support in M, we have4.3 where div_{Sb} X denotes the divergence of X along S_{b}andF_{ij}are the components ofF_{A}.
Corollary 4.1.
If A is stationary, thenS_{b}is 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 antiselfdual (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_{i}dx_{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 ℋ^{n−4}(S^{c} ∩ B_{R}(0)) < ∞ for any R > 0. Further, measures F_{Ai}^{2}dV 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_{c}be as above. Then we have that∂_{r}Θ_{c} = 0, a ⋅ S_{c} = S_{c}andF_{Ac}(∂_{r}, ⋅) = 0.
5. AntiSelfDual Instantons
Antiselfdual 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 Ωantiselfdual 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 Ωantiselfdual instanton if its curvature F_{A} = ∑ F_{ij}dx_{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 antiselfdual 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 Ωantiselfdual 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 ℋ^{n−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 thatℋ^{n−6}(S(τ(A)) ∩ K) < ∞for any compact K ⊂ M.
Now we assume that {A_{i}} is a sequence of Ωantiselfdual instantons that converge to an admissible Ωantiselfdual instanton A (cf. Theorem 2.3), where Ω is a form on M of degree n − 4. Let S_{b} ⊂ M be the blowup 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 C_{2}(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 C_{2}(A′) is closed in M.
Since S_{b} is rectifiable (Proposition 4.1), we can also define a current C_{2}(S_{b}, Θ_{μ}) by 5.3
Theorem 5.2.
(From ref. 4) Let A_{i}, A, et al. be as above. Then(1/8π^{2})Θ_{μ}is integervalued and S_{b}is calibrated by Ω; that is, for ℋ^{n−4}a.e. p ∈ S_{b}whereT_{p}S_{b}exists, the restriction of Ω to T_{p}S_{b}coincides with the induced volume form. Moreover, we have5.4 In particular, for any compact K, we have A simplified situation of Theorem 5.2 can be described as follows: Let π : ℝ^{n} ↦ ℝ^{4} be an orthogonal projection and B be an asd instanton on ℝ^{4}. Then the pullback A = π*B is Ωasd if and only if L = π^{−1}(0) is an Ωcalibrated subspace. This can be checked directly. As before, we ask if an Ωcalibrated submanifold is the limit of a sequence of Ωasd 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_{b}be the blowup locus of a sequence of asd instantons A_{i}converging to A andΘ_{μ}be its associated density. Then C_{2}(S_{b}, Θ_{μ})is an areaminimizing integral current.
The support S_{b} of C_{2}(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 withΩ ≤ 1. Then Ωcalibrated integral current is supported on the closure N of a smooth manifold N_{0}such that N∖N_{0}is of codimension at least two.
We end this section with an example. Assume that n = 2m. Fix an identification ℝ^{n} = ℂ^{m}. Let ω be given in complex coordinates z_{1}, … , z_{m} by Put Ω = ω^{m−2}/(m − 2)!. Then an Ωasd 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 ⊂ ℝ^{n} of codimension 4 is Ωcalibrated if and only if L is a complex subspace in ℂ^{m}. Let S be the blowup locus of a sequence of Hermitian–Yang–Mills connections and Θ be its associated density. Then C_{2}(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 nmanifold 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 Ωasd instantons of E over M. In general, 𝔐_{Ω,E} may not be compact. So we will compactify it.
A generalized Ωasd instanton is made of an admissible Ωasd instanton A of E, which extends to a smooth connection over M∖S(A) for a closed subset S(A) with ℋ^{n−4}(S(A)) = 0, and a closed integral current C = C_{2}(S, Θ) calibrated by Ω, such that cohomologically, 6.1 where C_{2}(E) denotes the second Chern class of E. Two generalized Ωasd 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 Ωasd 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 C_{2}(A_{i}, C_{i}) converge weakly to C_{2}(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 mdimensional compact Kähler manifold with Kähler form ω, Ωasd instantons are Hermitian–Yang–Mills connections, where Ω = ω^{m−2}/(m − 2)!. A generalized Ωasd 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).
Acknowledgments
This work was supported partially by National Science Foundation grants and the Simons Fund.
Footnotes

↵† Email: tian{at}math.mit.edu.

This paper results from the National Academy of Sciences colloquium, “Nonlinear Partial Differential Equations and Applications,” held January 4–19, 1999, at the Arnold and Mabel Beckman Center of the National Academies of Science and Engineering in Irvine, CA.
Abbreviations

asd instantons, Ωantiselfdual instantons
 Copyright © 2002, The National Academy of Sciences
References
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