Loop groups in Yang-Mills theory

We consider the Yang-Mills equations with a matrix gauge group $G$ on the de Sitter dS$_4$, anti-de Sitter AdS$_4$ and Minkowski $R^{3,1}$ spaces. On all these spaces one can introduce a doubly warped metric in the form $d s^2 =-d u^2 + f^2 d v^2 +h^2 d s^2_{H^2}$, where $f$ and $h$ are the functions of $u$ and $d s^2_{H^2}$ is the metric on the two-dimensional hyperbolic space $H^2$. We show that in the adiabatic limit, when the metric on $H^2$ is scaled down, the Yang-Mills equations become the sigma-model equations describing harmonic maps from a two-dimensional manifold (dS$_2$, AdS$_2$ or $R^{1,1}$, respectively) into the based loop group $\Omega G=C^\infty (S^1, G)/G$ of smooth maps from the boundary circle $S^1=\partial H^2$ of $H^2$ into the gauge group $G$. From this correspondence and the implicit function theorem it follows that the moduli space of Yang-Mills theory with a gauge group $G$ in four dimensions is bijective to the moduli space of two-dimensional sigma model with $\Omega G$ as the target space. The sigma-model field equations can be reduced to equations of geodesics on $\Omega G$, solutions of which yield magnetic-type configurations of Yang-Mills fields. The group $\Omega G$ naturally acts on their moduli space.


Introduction.
It is well known that the self-dual Yang-Mills equations in the Euclidean space R 4,0 have an infinite-dimensional algebra of "hidden symmetries" (see [1]- [6] for discovering, reviews and more references). For the Yang-Mills potentials with value in a Lie algebra g = Lie G, where G is a matrix gauge group, among these symmetries there is the Lie algebra of the loop group LG = C ∞ (S 1 , G). Here we shall show that the same group is a part of the moduli space of solutions to the Yang-Mills equations on the Lorentzian manifolds dS 4 , AdS 4 and R 3,1 of constant positive, negative and zero curvature.
We will use the adiabatic limit method which was applied to the first-order self-dual Yang-Mills equations on the product Σ 1 ×Σ 2 of two Riemann surfaces in [7]. It was shown that when the metric on the Riemann surface Σ 2 shrinks to a point, the Yang-Mills instantons converge to holomorphic maps from Σ 1 to the moduli space of flat connections on Σ 2 . In [8] this limit was discussed in the framework of topological Yang-Mills theories on Σ 1 ×Σ 2 . We will apply the adiabatic method to the second-order Yang-Mills equations on Lorentzian four-manifolds of constant curvature and describe how to construct approximate solutions of the Yang-Mills equations. It will be shown that these configurations become exact solutions in the adiabatic limit, when the Yang-Mills equations reduce to sigma-model equations describing harmonic maps from two-dimensional space-time (dS 2 , AdS 2 or R 1,1 ) into the based loop group ΩG. For static solutions of these equations, the moduli space is the tangent space T ΩG of the based loop group ΩG. Thus, ΩG is a "hidden symmetry group" not only of the first-order self-dual Yang-Mills equations but also of the second-order Yang-Mills equations on Lorentzian manifolds R 3,1 , AdS 4 and dS 4 .

Metrics.
It is known that on the de Sitter dS 4 and anti-de Sitter AdS 4 spaces one can introduce (local) coordinates such that the metrics on these spaces will be a double warped metrics of the form [10] dS 4 : ds 2 +1 = −du 2 + cosh 2 u dv 2 + sinh 2 u ds 2 where the first two terms are metrics on the spaces dS 2 and AdS 2 , respectively. Here, is the metric on the two-dimensional hyperbolic space H 2 . This space has two sheets H 2 = H 2 + ∪H 2 − with the common boundary S 1 at χ → +∞ for H 2 + and χ → −∞ for H 2 − . We introduce on the Minkowski space-time a metric similar to (1) and (2). In the Cartesian coordinates x µ , µ = 0, 1, 2, 3, the metric has the form Let us introduce coordinates u, χ and ϕ by and keep x 3 untouched. The coordinates (5) have a range They cover the interior of the light cone in R 2,1 and we denote this subset of R 2,1 by R 2,1 + . The region R 2,1 − = R 2,1 \ R 2,1 + can be covered by other choice of pseudospherical coordinates. In these coordinates the metric (4) acquires the form From (7) we recognize a cone over H 2 , i.e. R 2,1 + = C(H 2 ) and we restrict ourselves to the subset For the metrics (1) and (2) we also consider u > 0, since for u = 0 they degenerate.
After denoting x 3 = v, we see that the metrics (1), (2) and (7) have the same form where f = cosh u and h = sinh u for dS 4 f = sin u and h = cos u for AdS 4 (9) Therefore, we will consider all three spaces together by using the metric (8), specifying f and h if necessary. Recall that we work in local coordinates which cover only part of any of the considered spaces. This is enough for our purposes. For further unification we introduce the coordinates (y µ ) = (y a , y i ) = (u, v, χ, ϕ), where µ = (a, i) with a, b, ... = 0, 1 and i, j, ... = 2, 3. Then metric (8) can be written as where ) is the metric on H 2 . Finally, as H 2 we will consider only the upper sheet of the two-dimensional hyperbolic space with χ ≥ 0 for all three metrics (1), (2), (7) and consider only y 0 = u > 0 in (8)- (10). All other regions of our spaces dS 4 , AdS 4 , and R 3,1 can be considered similarly.
3. Yang-Mills equations. So, we consider Yang-Mills theory on a Lorentzian 4-manifold M with local coordinates y µ and the metric given by (8)- (10). We start with the potential A = A µ dy µ with values in the Lie algebra g = Lie G having scalar product defined by the trace Tr. Here G is an arbitrary matrix gauge group. The field strength F = dA + A ∧ A is the g-valued two-form: The Yang-Mills equations on M with the metric given by (8)-(10) are where g = (g µν ) and indices are raised by g µν . We have the obvious splitting 4. Adiabatic limit. By using the adiabatic approach [7,8,9,11], which is based on the ideas of [12], we deform the metric (8) and introduce where ε is a real parameter. Then | det g ε | = ε 4 | det(g ab )| det(g ij ) and where indices of F µν are raised by the metric g µν from (10).
To avoid the divergent term ε −2 Tr(F ij F ij ) in the Lagrangian, we impose the flatness condition on the Yang-Mills curvature along H 2 in M . For the deformed metric (15) the Yang-Mills equations have the form and in the limit ε → 0 (after choosing F ij = 0) we have

Flat connections. Flat connection A H 2 = A i dy i on H 2 (upper sheet) has a simple form
where g = g(y a , y i ) is a smooth map from H 2 (for any given y a ) into the gauge group G. We consider smooth matrix-valued functions g with smooth boundary value on S 1 = ∂H 2 and impose additional condition g = Id at 1 ∈ S 1 (framing of flat connection on H 2 [9]). We denote by C ∞ 0 (H 2 , G) the space of all such g in (22). On H 2 , as on a manifold with the boundary, the group of gauge transformations is defined as [9] G H 2 = g : Hence the solution space of the equation F H 2 = 0 is the infinite-dimensional group N = C ∞ 0 (H 2 , G) and the moduli space is the based loop group [13] Recall that g and A H 2 depend on coordinates y a .
6. Moduli space. On the group manifold (24) we introduce local coordinates φ α with α = 1, 2, ... and assume that A µ 's depend on u and v only via the moduli parameters φ α = φ α (u, v). Then moduli of flat connections on H 2 define a map where by Σ we denote (a patch of) dS 2 , AdS 2 or R 1,1 depending on choice of M in (8)- (9). These maps are constrained by the equations (20) and (21). Since A H 2 is a flat connection for any y a ∈ Σ, the derivatives ∂ a A i have to satisfy the linearized around A H 2 flatness condition, i.e. ∂ a A i belong to the tangent space T A N of the space N = C ∞ 0 (H 2 , G) of flat connections on H 2 . Using the projection π on the moduli space, π : N → M, one can decompose ∂ a A i into the two parts where G is the gauge group (restricted to H 2 by fixing y a ∈ Σ, G| H 2 = G H 2 ), {ξ α = ξ αi dy i } is a local basis of tangent vectors on T A M (they form the Lie algebra Ωg) and ǫ a are g-valued gauge parameters (D i ǫ a are tangent vectors from T A G) which are determined by the gauge-fixing equations In fact, since φ α depend on y a ∈ Σ, we have N = N (y a ), G = G H 2 (y a ) and M = N /G = M(y a ).
Recall that A i are fixed by (22) and A a are yet free. For the mixed components of the field strength we have It is natural to choose A a = ǫ a [12,14] and obtain On the other hand, since A i (φ α , y j ) depends on y a only via φ α , we have with the gauge parameters ǫ α defined by (27) via the equations are the Christoffel symbols and ∇ γ are the corresponding covariant derivatives on the moduli space ΩG of flat connections on H 2 . The equations are the Euler-Lagrange equations for the effective action of the Yang-Mills theory on M which appears from the term Tr(F aj F aj ) in the initial Yang-Mills Lagrangian in the adiabatic limit ε → 0. The equations (37) are the standard sigma-model equations defining harmonic maps from Σ (= dS 2 , AdS 2 or R 1,1 ) into the based loop group ΩG parameterized (locally) by coordinates φ α . Note that these equations are integrable and their solutions can be constructed similar to those for finite-dimensional target Lie groups (see e.g. [15]). If φ α do not depend on u then equations (37) reduce to the geodesic equations on the loop group ΩG and give static configurations of Yang-Mills fields. This case of magnetic configurations can be considered as supplementing [16], where only electric components of adiabatic Yang-Mills fields were nonvanishing. Note that any geodesic on ΩG is parametrized by the initial point φ 0 ∈ ΩG and by the velocity φ ′ 0 ∈ T 0 ΩG. Therefore, the moduli space of solutions (32) (with ∂ u φ = 0 and ∂ v φ =: φ ′ ) can be identified with the tangent bundle T ΩG of ΩG. The based loop group ΩG naturally acts on T ΩG which can be identified with the semi-direct product ΩG ⋉ g of ΩG and the Lie algebra g = LieG.
8. Concluding remarks. In conclusion we recall that in the Euclidean case Atiyah has shown [17] that the moduli space of instantons over R 4,0 is bijective to the moduli space of holomorphic maps from S 2 to ΩG. There is a conjecture (see e.g. [18]) that the moduli space of solutions of the second-order Yang-Mills equations on R 4,0 is bijective to the moduli space of harmonic maps from S 2 to ΩG. Our consideration in this paper can be repeated for the Euclidean space. In [19] it was observed that R 4 ∪ {∞} \ S 1 = S 4 \ S 1 is conformally diffeomorphic to the product manifold S 2 × H 2 . Considering M = S 2 × H 2 and literally repeating our calculations for this Euclidean manifold we will arrive to the equations (37) with Σ = S 2 . These equations will define harmonic maps from S 2 into the based loop group ΩG. Furthermore, from the implicit function theorem it follows that near every solution φ of (37) with Σ = S 2 (and the corresponding solution A ε=0 of the Yang-Mills equations) there exists a solution A ε>0 of the Yang-Mills equations on M for ε sufficiently small (cf. with the instanton case [7,9,19]). In other words, solutions of (37) with Σ = S 2 approximate solutions of the Yang-Mills equations on M = S 2 × H 2 (and on R 4,0 after some maps [9,19] from S 2 × H 2 to R 4,0 ) and one can conjecture that the moduli spaces for A ε=0 and A ε>0 are bijective. Our consideration shows that the same conjecture is valid for Lorentzian signature, i.e. it is reasonable to expect that the Yang-Mills model on Minkowski space R 3,1 with a gauge group G is equivalent to the sigma model on R 1,1 whose target space is the based loop group ΩG.