Sigma-model limit of Yang-Mills instantons in higher dimensions

We consider the Hermitian Yang-Mills (instanton) equations for connections on vector bundles over a 2n-dimensional K\"ahler manifold X which is a product Y x Z of p- and q-dimensional Riemannian manifold Y and Z with p+q=2n. We show that in the adiabatic limit, when the metric in the Z direction is scaled down, the gauge instanton equations on Y x Z become sigma-model instanton equations for maps from Y to the moduli space M (target space) of gauge instantons on Z if q>= 4. For q<4 we get maps from Y to the moduli space M of flat connections on Z. Thus, the Yang-Mills instantons on Y x Z converge to sigma-model instantons on Y while Z shrinks to a point. Put differently, for small volume of Z, sigma-model instantons on Y with target space M approximate Yang-Mills instantons on Y x Z.


Introduction and summary
The Yang-Mills equations in two, three and four dimensions were intensively studied both in physics and mathematics. In mathematics, this study (e.g. projectively flat unitary connections and stable bundles in d = 2 [1], the Chern-Simons model and knot theory in d = 3, instantons and Donaldson invariants [2] in d = 4 dimensions) has yielded a lot of new results in differential and algebraic geometry. There are also various interrelations between gauge theories in two, three and four dimensions. In particular, Chern-Simons theory in d = 3 dimensions reduces to the theory of flat connections in d = 2 (see e.g. [3,4]). On the other hand, the gradient flow equations for Chern-Simons theory on a d = 3 manifold Y are the first-order anti-self-duality equations on Y × R, which play a crucial role in d = 4 gauge theory.
The program of extending familiar constructions in gauge theory, associated to problems in low-dimensional topology, to higher dimensions was proposed by Donaldson and Thomas in the seminal paper [5] (see also [6]) and developed in [7]- [14] among others. An important role in this investigation is played by first-order gauge-field equations which are a generalization of the anti-self-duality equations in d = 4 to higher-dimensional manifolds with special holonomy (or, more generally, with G-structure [15,16]). Such equations were first introduced in [17] and further considered in [18]- [22] (see also references therein).
Instanton equations on a d-dimensional Riemannian manifold X can be introduced as follows [17,5,10]. Suppose there exist a 4-form Q on X. Then there exists a (d−4)-form Σ := * Q, where * is the Hodge operator on X. Let A be a connection on a bundle E over X with curvature F = dA + A ∧ A. The generalized anti-self-duality (instanton) equation on the gauge field then is [10] * F + Σ ∧ F = 0 . (1.1) For d > 4 these equations can be defined on manifolds X with special holonomy, i.e. such that the holonomy group G of the Levi-Civita connection on the tangent bundle T X is a subgroup in SO(d). Solutions of (1.1) satisfy the Yang-Mills equation The instanton equation (1.1) is also well defined on manifolds X with non-integrable G-structures, i.e. when dΣ = 0. In this case (1.1) implies the Yang-Mills equation with (3-form) torsion T := * dΣ, as is discussed e.g. in [23]- [27].
Manifolds X with a (d−4)-form Σ which admits the instanton equation (1.1) are usually calibrated manifolds with calibrated submanifolds. Recall that a calibrated manifold is a Riemannian manifold (X, g) equipped with a closed p-form ϕ such that for any oriented p-dimensional subspace ζ of T x X, ϕ | ζ ≤ vol ζ for any x ∈ X, where vol ζ is the volume of ζ with respect to the metric g [28]. A p-dimensional submanifold Y of X is said to be a calibrated submanifold with respect to ϕ (ϕ-calibrated) if ϕ | Y = vol Y [28]. In particular, suitably normalized powers of the Kähler form on a Kähler manifold are calibrations, and the calibrated submanifolds are complex submanifolds. On a G 2 -manifold one has a 3-form which defines a calibration, and on a Spin(7)-manifold the defining 4-form (the Cayley form) is a calibration as well [5,6].
It is not easy to construct solutions of (1.1) for d > 4 and to describe their moduli space. 1 It was shown by Donaldson, Thomas, Tian [5,10] and others that the adiabatic limit method provides a useful and powerful tool. The adiabatic limit refers to the geometric process of shrinking a metric in some directions while leaving it fixed in the others. It is assumed that on X there is a family Σ ε of (d−4)-forms with a real parameter ε such that Σ 0 = lim ε→0 Σ ε defines a calibrated submanifold Y of X. Then one can define a normal bundle N (Y ) of Y with a projection The metric on X induces on N (Y ) a Riemannian metric where Z ∼ = R 4 is a typical fibre. In fact, the fibres are calibrated by a 4-form Q ε dual to Σ ε . The metric (1.4) extends to a tubular neighborhood of Y in X, and (1.1) may be considered on this subset of X. Anyway, it was shown [5,10,6] that solutions of the instanton equation (1.1) defined by the form Σ ε on (X, g ε ) in the adiabatic limit ε → 0 converge to sigma-model instantons describing a map from the (d−4)-dimensional submanifold Y into the hyper-Kähler moduli space of framed Yang-Mills instantons on fibres R 4 of the normal bundle N (Y ).
The submanifold Y ֒→ X is calibrated by the (d−4)-form Σ defining the instanton equation (1.1). However, on X there may exist other p-forms ϕ and associated ϕ-calibrated submanifolds Y of dimension p = d−4. In such a case one can define a different normal bundle (1.3) with fibres R d−p and deform the metric as in (1.4). Also, one may take a direct product manifold X = Y ×Z with dim R Y = p and dim R Z = q = d−p with a p-form ϕ = vol Y , or consider non-flat manifolds Z and a (d−4)-form Σ defining (1.1). In string theory dim R X = 10, and calibrated submanifolds Y are identified with worldvolumes of p-branes where p varies from zero to ten.
In this short paper we explore the direct product case X = Y ×Z with dim R Y = p = d−4 for Kähler manifolds X and the adiabatic limit of the Hermitian Yang-Mills equations on bundles over X. We will show that for even p (and hence even q) the adiabatic limit of (1.1) yields sigmamodel instanton equations describing holomorphic maps from Y into the moduli space of Hermitian Yang-Mills instantons on Z. For odd p and q the consideration is more involved, and we describe only the case p=q=3 in which we obtain maps from Y into the moduli space of flat connections on Z. For the purpose of this paper, this special case sufficiently illustrates the main features of the odd-dimensional cases.

Moduli space of instantons in d ≥ 4
Bundles. Let X be an oriented smooth manifold of dimension d, G a semisimple compact Lie group, g its Lie algebra, P a principal G-bundle over X, A a connection 1-form on P and F = dA + A ∧ A its curvature. We consider also the bundle of groups IntP = P × G G (G acts on itself by internal automorphisms: h → ghg −1 , h, g ∈ G) associated with P , the bundle of Lie algebras AdP = P × G g and a complex vector bundle E = P × G V , where V is the space of some irreducible representation of G. All these associated bundles inherit their connection A from P .
Gauge transformations. We denote by A ′ the space of connections on P and by G ′ the infinitedimensional group of gauge transformations (automorphisms of P which induce the identity transformation of X), which can be identified with the space of global sections of the bundle IntP . Correspondingly, the infinitesimal action of G ′ is defined by global sections χ of the bundle AdP , Moduli space of connections. We restrict ourselves to the subspace A ⊂ A ′ of irreducible connections and to the subgroup G = G ′ /Z(G ′ ) of G ′ which acts freely on A. Then the moduli space of irreducible connections on P (and on E) is defined as the quotient A/G. We do not distinguish connections related by a gauge transformation. Classes of gauge equivalent connections are points Metric on A/G. Since A is an affine space, for each A ∈ A we have a canonical identification between the tangent space T A A and the space Λ 1 (X, AdP ) of 1-forms on X with values in the vector bundle AdP . We consider g as a matrix Lie algebra, with the metric defined by the trace. The metrics on X and on the Lie algebra g induce an inner product on Λ 1 (X, AdP ), This inner product is transferred to T A A by the canonical identification. It is invariant under the G-action on A, whence we get a metric (2.3) on the moduli space A/G.
Instantons. Suppose there exists a (d−4)-form Σ on X which allows us to introduce the instanton equation discussed in Section 1. We denote by N ⊂ A the space of irreducible connections subject to (2.4) on the bundle E → X. This space N of instanton solutions on X is a subspace of the affine space A, and we define the moduli space M of instantons as the quotient space According to the bundle structure (2.6), at any point A ∈ N , the tangent bundle T A N → N splits into the direct sum In other words, whereξ, ξ ∈ Λ 1 (X, AdP ) and χ ∈ Λ 0 (X, AdP ) = Γ(X, AdP ). The choice of ξ corresponds to a local fixing of a gauge.
Kähler forms on M. If X is Kähler with a complex structure J and a Kähler form ω(·, ·) = g(J·, ·), then the Kähler 2-form Ω = (Ω αβ ) on M is given by It is well known that the moduli space of framed instantons 2 on a hyper-Kähler 4-manifold X (with three integrable almost complex structures J i ) is hyper-Kähler, with three Kähler forms

Hermitian Yang-Mills equations
Instanton equations. On any Kähler manifold X of dimension d = 2n there exists an integrable almost complex structure J ∈ End(T X), J 2 = −Id, and a Kähler (1,1)-form ω(·, ·) = g(J·, ·) compatible with J. The natural 4-form and its dual Σ = * Q allow one to formulate the instanton equation (2.4) for a connection A on a complex vector bundle E over X associated to the principal bundle P (X, G). The fibres C N of E support an irreducible G-representation. For simplicity, we have in mind the fundamental representation of SU(N ). One can endow the bundle E with a Hermitian metric and choose A to be compatible with the Hermitian structure on E.
The instanton equations in the form (2.4) with Σ = 1 2 * (ω ∧ ω) may then be rewritten as the following pair of equations, and whereμ,ν, . . . = 1, . . . , 2n, and the notation ω exploits the underlying Riemannian metric of X for raising indices of ω. The equations (3.2)-(3.3) were introduced by Donaldson, Uhlenbeck and Yau [19] and are called the Hermitian Yang-Mills (HYM) equations. 3 The HYM equations have the following algebro-geometric interpretation. Equation (3.2) implies that the curvature F = dA + A ∧ A is of type (1,1) with respect to J, whence the connection A defines a holomorphic structure on E. Equation (3.3) means that E → X is a polystable vector bundle. The moduli space M X of HYM connections on E, the metric G = (G αβ ) and the Kähler form Ω = (Ω αβ ) on M X are introduced as described in Section 2 after specializing X to be Kähler.
Direct product of Kähler manifolds. The subject of this paper is the adiabatic limit of the HYM equations (3.2)-(3.3) on a direct product of Kähler manifolds Y and Z. The dimensions p and q of Y and Z are even, and p + q = 2n. Let {e a } with a = 1, . . . , p and {e µ } with µ = p+1, . . . , 2n be local frames for the cotangent bundles T * Y and T * Z, respectively. Then {eμ} = {e a , e µ } withμ = 1, . . . , 2n will be a local frame for the cotangent bundle T * X = T * Y ⊕ T * Z. We introduce on Y × Z the metric and an integrable almost complex structure whose components are defined by J Y e a = J a b e b and J Z e µ = J µ ν e ν . Likewise, the Kähler form ω(·, ·) = g(J·, ·) on Y × Z decomposes as with components ω Y = (ω ab ) and ω Z = (ω µν ).
Splitting of the HYM equations. We introduce on X = Y × Z local coordinates {y a } and {z µ } and choose e a = dy a , e µ = dz µ . Any connection on the bundle E → X is decomposed as where the components A a and A µ depend on (y, z) ∈ Y × Z. The curvature F of A has components F ab along Y , F µν along Z, and F aµ which we call "mixed".
Note that the holomorphicity conditions (3.2) may be expressed through the projector P = 1 2 (Id + iJ) ,P 2 =P (3.9) onto the (0,1)-part of the complexification of the cotangent bundle T * X = T * Y ⊕ T * Z as which in components reads δσ µ + iJσ µ δλ ν + iJλ ν Fσλ = 0 . From (3.6) it follows that these equations split into three parts: and Finally, with the help of (3.7) the stability equation (3.3) takes the form

Adiabatic limit of the HYM equations for even p and q
Moduli space M Z . In order to investigate the adiabatic limit of (3.12)-(3.15), we introduce on X = Y × Z the deformed metric and Kähler form while the complex structure J = J Y ⊕ J Z does not depend on ε according to (3.6). Since J Y and J Z are untouched, (3.12)-(3.14) keep their form in the adiabatic limit ε → 0. In particular, (3.12) implies that F 0,2 Y = 0, i.e. the bundle E → Y × Z is holomorphic along Y for any z ∈ Z. 4 On the other hand, (3.15) for ε → 0 becomes which together with (3.13) means that A Z is a HYM connection (framed instanton) on Z for any given y ∈ Y . We denote the moduli space of such connections by where N Z is the space of all instanton solutions on Z for a fixed y ∈ Y , and G Z consists of the elements of G with the same fixed value of y. We here suppress the y dependence in our notation. The moduli space M Z is a Kähler manifold on which we introduce the metric G and Kähler form Ω with components and similar to (2.9) and (2.10) but now with ξ α ∈ Λ 1 (Z, AdP ) and the Hodge operator * Z defined on Z. Note that for dim R Z = 2 the HYM equations (3.13) and (4.2) enforce F Z = 0, i.e. M Z becomes the moduli space of flat connections on bundles E(y) over a two-dimensional Riemannian manifold Z.
A map into M Z . The bundle E(y) is a HYM vector bundle over Z for any y ∈ Y . Letting the point y vary, the connection A Z = A µ (y, z)dz µ on E(y) defines a map where φ α with α = 1, . . . , dim R M Z are local coordinates on M Z . This map is constrained by our remaining set of equations, namely (3.14) for the mixed field-strength components Similarly to (2.7) and (2.8), ∂ a A µ decomposes into two parts, where {ξ α = ξ αµ dz µ } is a local basis of vector fields on M Z . Here, ǫ a are g-valued gauge parameters which are determined by the gauge-fixing equations Substituting (4.7) into (4.6), the mixed field-strength components simplify to Inserting this expression into our remaining equations (3.14), we obtain as a condition on the map φ.
Sigma-model instantons. In order to better interpret the above equations, we multiply both sides with dz µ ∧ * Z ξ β , take the trace over g, integrate over Z and recognize the integrals in (4.4).
The integral of the right-hand side of (4.10) vanishes due to (4.7)-(4.8) (orthogonality of ξ α ∈ T M Z and Dχ ∈ T G Z ), and we end up with Inverting the moduli-space metric G and introducing the almost complex structure J on M Z via its components J α β := Ω βγ G γα , (4.12) we rewrite (4.11) as and with the obvious definition for P.
These equations mean that φ 1 + iφ 2 , φ 3 + iφ 4 , . . . are holomorphic functions of complex coordinates on Y , i.e. φ is a holomorphic map. It is clear that our equations (4.15) are BPS-type (instanton) first-order equations for the sigma model on Y with target space M Z , whose field equations define harmonic maps from Y into M Z . For dim R Y = dim R Z = 2 these equations have appeared in [29] as the adiabatic limit of the HYM equations on the product of two Riemann surfaces. Our (4.15) generalize [29] to the case dim R Y > 2 and dim R Z ≥ 2. From the implicit function theorem it follows that near every solution φ of (4.15) there exists a solution A ε of the HYM equations (3.2)-(3.3) for ε sufficiently small. In other words, solutions of (4.15) approximate solutions of the HYM equations on X.

Adiabatic limit of gauge instantons for p = q = 3
If the Kähler manifold X is a direct product of two odd-dimensional manifolds Y and Z, i.e. if p = dim R Y and q = dim R Z are both odd, then we may need to impose conditions on the geometry of Y and Z for X = Y × Z to be Kähler. However, we are not aware of these demands outside of special cases, such as products of tori. Therefore, we restrict ourselves to tori Y and Z with p = q = 3 since already this case illustrates essential differences from the case of even p and q. More general situations demand more effort and will be considered elsewhere.
The combined torus T 3 × T 3 r supports an integrable almost complex structure J satisfying Jθ j = iθ j for j = 1, 2, 3, which determines its components, with a, b = 1, 2, 3 and µ, ν = 4, 5, 6, as well as In the adiabatic limit ε → 0 the first two lines of (5.6) reduce to while the mixed-component part of (5.6) together with (5.7) produces Recall that A = A Y + A Z = A a (y, z)dy a + A µ (y, z)dz µ (5.10) is a connection on a vector bundle E over X = T 3 × T 3 r . From (5.8) we learn that A Z is a flat connection on Z = T 3 r for any y ∈ Y = T 3 . We denote by N Z the space of solutions to (5.8) and by M Z the moduli space of all such connections. From (5.9) we see that in the adiabatic limit there are no restrictions on A Y , since the components A a and F ab no longer appear.
Sigma-model equations. For the mixed components F aµ of the field strength we have where, as in Section 4, we used for ∂ a A µ the decomposition formula (4.7) and introduced the map Let us, for a short while, relax the gauge fixing (4.8) and allow φ(y) to take values in the full solution space N T 3 r . Correspondingly ξ α = ξ αµ dz µ will be momentarily a basis of all vector fields on N T 3 r , and ǫ a are undetermined. Substituting (5.11) into (5.9), we obtain the equations (5.14) Multiplying both sides with ξ βµ for µ = 4, 5, 6 and integrating tr (ξ αµ ξ βν ) over T 3 r , the above four equations yield the 3 dim R N T 3 r relations The right-hand side of (5.15) is given by The (1,1) tensors π a = (ε b ac ), a = 1, 2, 3, on T 3 and the (1,1) tensors Π a = (δ ab Π b α β ) on N T 3 r satisfy the identities π 3 a + π a = 0 and Π 3 a + Π a = 0 , i.e. they define three so-called f -structures [30] correspondingly on T 3 and on N T 3 r . To clarify their meaning we observe that (5.19) defines orthogonal projectors which defines on T 3 two distributions L a and N a of rank two and one, respectively, and decomposes the 3-torus in three different ways. Analogously, the projector P a yields a splitting T (N T 3 r ) = L a ⊕ N a (5.23) In the adiabatic limit of ε → 0 with the deformed metric g ε = g Y + ε 2 g Z the G 2 -instanton equations become ∂ a φ α + ε b ac (∂ b φ β ) J c α β = 0 . (5.29) This looks similar to (5.15) with j α a = 0 and features three complex structures J c = (J c α β ) (instead of f -structures Π c ) on the hyper-Kähler moduli space M Z of framed Yang-Mills instantons on the hyper-Kähler 4-manifold Z. These equations were discussed e.g. in [6,13] in the form of Fueter equations. In the above case (5.28) they define maps φ : T 3 → M Z which are sigma-model instantons minimizing the standard sigma-model energy functional.