Flat bundles on affine manifolds

We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine manifolds is shown. Also shown, in the context of affine manifolds, is a bijective correspondence between polystable flat pairs and solutions of the vortex equation.


Introduction
An affine manifold is a smooth real manifold M equipped with a flat torsion-free connection D on its tangent bundle. It is well-known (see, e. g., [30]) that an n-dimensional real manifold M is an affine manifold if and only if M admits an atlas such that all the transition functions are affine maps of the form (1.1) If M is an affine manifold, then the total space of its tangent bundle T M can be given a complex structure in a canonical way. More precisely, given an atlas of M with transition functions as in (1.1), let {x i } be the corresponding coordinates defined on an open subset U ⊂ M; such coordinates {x i } are called local affine coordinates. Write y i for the fiber coordinates corresponding to the local trivialization of the tangent bundle T M over U given by { ∂ ∂ x i } n i=1 , meaning that every tangent vector y is written as Then on the open subset T U ⊂ T M, we have the holomorphic coordinate functions Every Riemannian metric g on M extends to a Hermitian metric g C on M C . The metric g is an affine Kähler metric if and only if g C is a Kähler metric. Cheng and Yau [10] proved the existence of affine Kähler-Einstein metrics on an affine manifold under suitable conditions. (See also [12] for related results.) An affine manifold M is called special if it admits a volume form (meaning a non-vanishing top-degree form) which is covariantly constant with respect to the flat connection D on T M. An affine manifold M is special if and only if it admits an atlas with transition functions as in (1.1) satisfying the additional condition that A ∈ SL(n, R). Special affine manifolds form an important class of affine manifolds. In particular, there is a famous conjecture of Markus (see [28]) stating that a compact affine manifold is special if and only if the flat connection D on T M is complete. Special affine manifolds also play a role in the Strominger-Yau-Zaslow conjecture; see [33]. Cheng and Yau showed that on a closed special affine manifold, an affine Kähler metric (if it exists) can be deformed to a flat metric by adding the Hessian of a smooth function; see [10]. Cheng and Yau's result shows that compact special affine Kähler manifolds are all torus quotients. On the other hand, there are many examples of compact special affine manifolds which do not admit any affine Kähler metric; see for example [16,17]. In order to work on the full class of special affine manifolds, we must consider affine Gauduchon metrics, which generalize the Kähler condition.
In the context of affine manifolds, the right analogue of a holomorphic vector bundle over a complex manifold is a flat vector bundle. Let E be a smooth complex vector bundle over an affine manifold M. Our aim here is to review some recent results in the theory of flat vector bundles and flat principal bundles over a compact connected special affine manifold equipped with an affine Gauduchon metric. More precisely, we will adapt various Donaldson-Uhlenbeck-Yau type correspondences, which relate stability conditions on a bundle to the existence of Hermitian-Einstein structures, from the complex case to the situation of affine manifolds. An important aspect of the proofs of these statements is that the above correspondence between flat vector bundles over an affine manifold M and holomorphic vector bundles over the complex manifold M C ensures that local calculations can be done exactly in the same way as in the complex case. When integrating by parts on an n-dimensional compact special affine manifold, one uses the covariant constant volume form to convert a 2n-form on M C to an n-form on M, which can be integrated.
In Sect. 2, we will introduce the relevant notions and basic results that are needed in the proofs of the Donaldson-Uhlenbeck-Yau type theorems. In particular, we will give some details on the affine Dolbeault complex and integration by parts on affine manifolds. The Donaldson-Uhlenbeck-Yau type correspondences for flat vector bundles and flat principal bundles over affine manifolds will be given in Sects. 3 and 4, respectively.
In Sect. 5, we will introduce Higgs fields on flat vector bundles and flat principal bundles over affine manifolds, and show that a flat Higgs bundle admits a Yang-Mills-Higgs metric if and only if it is polystable. If X is a compact connected Kähler manifold, then there is a natural bijective correspondence between the isomorphism classes of polystable Higgs vector bundles on X of vanishing Chern classes of positive degrees and the isomorphism classes of direct sums of irreducible flat connections on X [11,15,21] and [32,p. 19,Theorem 1]. This correspondence breaks down for a compact complex manifold with Gauduchon metric, in fact even with a balanced metric [2]. Therefore, we cannot expect the analogue of the correspondence to hold for a general affine manifold.
In Sect. 6, we will introduce τ -stability and the τ -vortex equation (where τ is a real number) for pairs consisting of a flat vector bundle over an affine manifold and a flat non-zero section of it, and give a bijective correspondence between polystable flat pairs and solutions of the vortex equation.
There are other generalizations of the Donaldson-Uhlenbeck-Yau correspondence [27]. It would be interesting to try to establish these in the frame-work of affine manifolds.

Affine Dolbeault complex
Let (M, D) be an affine manifold of dimension n, meaning that D is a flat torsion-free connection on the tangent bundle of M.
Define the bundle of ( p, q)-forms on M by Given local affine coordinates {x i } n i=1 on M, we will denote the induced frame on A p,q as where z i = x i + √ −1 y i are the complex coordinates on M C defined above; note that dz i = dz i = dx i on M. There is a natural restriction map from ( p, q)-forms on the complex manifold M C to ( p, q)-forms on M given in local affine coordinates on an open subset U ⊂ M by where φ i 1 ,...,i p , j 1 ,..., j q are smooth functions on T U ⊂ T M = M C , U is considered as the zero section of T U −→ U , and the sums are taken over all 1 ≤ i 1 < · · · < i p ≤ n and 1 ≤ j 1 < · · · < j q ≤ n. One can define natural operators given in local affine coordinates by if φ is a p-form, respectively by if ψ is a q-form. These operators are the restrictions of the corresponding operators on M C with respect to the restriction map given in (2.1).
Similarly, there is a wedge product defined by if φ i ⊗ ψ i are forms of type ( p i , q i ), i = 1, 2, and a conjugation map from ( p, q)-forms to (q, p)-forms given by if φ is a p-form and ψ is a q-form; as above, they are the restrictions of the corresponding operations on M C . A smooth Riemannian metric g on M gives rise to a natural (1, 1)-form expressed in local affine coordinates as it is the restriction of the corresponding (1, 1)-form on M C given by the extension of g to M C . The metric g is called an affine Gauduchon metric if ∂∂(ω n−1 g ) = 0 (recall that n = dim M). By [26,Theorem 5], on a compact connected special affine manifold, every conformal class of Riemannian metrics contains an affine Gauduchon metric, which is unique up to a positive scalar.

Integration by parts
The main difference between complex and affine manifolds is that on an n-dimensional complex manifold, an (n, n)-form is a top-degree form, which can be integrated, while on an n-dimensional affine manifold, an (n, n)-form is not a top-degree form. In this subsection, we will explain how to overcome this under the additional assumption that the affine manifold M is special.
Let (M, D, ν) be a special affine manifold, meaning that (M, D) is an affine manifold as above and ν is a D-covariant constant volume form on M. The volume form ν induces natural maps these maps are called division by ν. The choice of sign ensures that for every Riemannian metric g on M, the volume form ω n g ν induces the same orientation as the volume form ν. If M is compact, an (n, n)-form χ on M can be integrated by considering the integral M χ ν .
The fact that ν is covariant constant with respect to D ensures that the usual integration by parts formulas for ( p, q)-forms still work on the affine manifold M. More precisely, we have the following proposition from [26]. Also, if χ is an (n, n − 1)-form, we have

Hermitian-Einstein metrics on flat vector bundles
Let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g.
Let (E, ∇) be a flat complex vector bundle over M, meaning that E is a smooth complex vector bundle and ∇ is a flat connection on E. (In the following, unless otherwise stated, we will always be concerned with complex vector bundles.) As in the introduction, the pullback of Let h be a Hermitian metric on E; it defines a Hermitian metric on E C . Let d h be the Chern connection associated to this Hermitian metric on E C . Then d h corresponds to a pair is a flat real vector bundle over M and h is a real positive-definite metric on E, then the extended Hermitian connection of the complexified vector bundle E ⊗ C over M equipped with the flat connection induced by ∇ and the Hermitian metric induced by h has an interpretation in terms of the dual connection of ∇ with respect to h. Recall that the dual connection ∇ * on E is defined by for smooth sections s 1 and s 2 of E (see, e. g., [1]). Then we have the following statement from [26].

Proposition 3.1 [26, Lemma 1] If (E, ∇) is a flat real vector bundle over an affine manifold M equipped with a real positive-definite metric h, then the extended Hermitian connection on E ⊗ C equipped with the flat connection induced by ∇ and the Hermitian metric induced by h is given by
where the dual connection ∇ * is defined in (3.2) and is flat.
Similarly, there are locally defined extended connection forms an extended curvature form an extended mean curvature and an extended first Chern form which are the restrictions of the corresponding objects on E C . Here tr g denotes contraction of differential forms using the Riemannian metric g, and tr denotes the trace homomorphism on the fibers of End E.
The extended first Chern form is given by The extended first Chern form and the extended mean curvature are related by

Definition 3.2 A Hermitian metric h on E is called a
Hermitian-Einstein metric (with respect to g) if its extended mean curvature K h is of the form The degree of (E, ∇) with respect to a Gauduchon metric g on M is defined to be To see that it is well-defined, observe that for any two Hermitian metrics h and h on E, we have which is ∂∂ of a function on M. Since by Proposition 2.1, we can integrate by parts as in the usual case, and g is a Gauduchon metric, it follows that the degree in (3.6) is well-defined. Note that even though E admits a flat connection ∇, there is no reason in general for the degree to be zero in the Gauduchon case. In particular, we can extend ∇ to a flat extended connection on E and then define an extended first Chern form c 1 (E, ∇). But is ∂-exact but not necessarily ∂∂-exact. Thus, by integration by parts, the Gauduchon condition is insufficient to force the degree to be zero. As usual, if rank(E) > 0, the slope of E with respect to g is defined to be rank(E) .
Note that in our situation of flat vector bundles over affine manifolds it suffices to consider subbundles rather than (singular) subsheaves in the definition of stability.
In [26], the following Donaldson-Uhlenbeck-Yau type correspondence was established. The main result of [10] provides an important antecedent and special case of Theorem 3.4. On a closed special affine Kähler manifold M, Cheng-Yau produce an affine Kähler metric g whose induced metric on M C is Kähler and Ricci-flat. In this case, the Kähler metric, considered as a bundle metric on the tangent bundle T (M C ), is Hermitian-Einstein with zero mean curvature. Therefore, g is affine Hermitian-Einstein with respect to its own Kähler form ω g . Cheng-Yau also show that such metrics are flat, and deduce by Bieberbach's Theorem that such an M is a finite quotient of a flat torus.  also produce affine Kähler-Einstein metrics with negative extended Ricci curvature on affine manifolds which are quotients of convex pointed cones. The induced metrics on the tangent bundles of these manifolds are not direct examples of the affine Hermitian-Einstein metrics we consider, however, as such affine manifolds do not admit parallel volume forms.) The proof of Theorem 3.4 is an adaptation to the affine situation of the methods of Uhlenbeck and Yau [34,35], for compact Kähler manifolds and their modification by Li and Yau [25], for the complex Gauduchon case. As mentioned above, all local calculations are identical to the complex case due to the correspondence between flat vector bundles on the affine manifold M and holomorphic vector bundles on the complex manifold M C which are constant along the fibers of M C = T M −→ M, and integration by parts is performed using Proposition 2.1. A major simplification compared to the complex case is the avoidance of the intricate proof of Uhlenbeck-Yau that a weakly holomorphic subbundle of a holomorphic vector bundle on a complex manifold is a reflexive analytic subsheaf. The corresponding regularity statement on affine manifolds is that a weakly flat subbundle of a flat Hermitian vector bundle (E, h 0 ), meaning a subbundle given by an h 0 -orthogonal L 2 1 projection π which satisfies (Id E −π) • ∂π = 0 in L 1 , is in fact a flat subbundle; this is much easier to prove (see [26,Proposition 27]).
We give an analogue of the above theorem for flat real vector bundles.

Definition 3.5
Let (E, ∇) be a flat real vector bundle over M.

Hermitian-Einstein connections on flat principal bundles
Let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g.
In this section, we will explain how to generalize the results of the previous section to flat principal bundles over M.

Preliminaries
Let H be a Lie group. A principal H -bundle over M is a triple (E H , p, ψ) Let dp : T E H −→ p * T M be the differential of the projection p. A flat connection on E H is a smooth homomorphism for all locally defined smooth sections s and t of ad(E H ).

Stable and semistable principal bundles
Let G C be a complex reductive linear algebraic group. A real form on G C is an anti-holomorphic involution The real form σ G C is said to be of split type if there is a maximal torus T ⊂ G C such that σ G C (T ) = T and the fixed point locus of the involution σ G C | T of T is a product of copies of R * (the group of non-zero real numbers). Let G be a connected Lie group such that either it is a complex reductive linear algebraic group, or it is the fixed point locus of a split real form σ G C ∈ Aut(G C ), where G C and σ G C are as above.
If G is a complex reductive group, a connected closed algebraic subgroup P ⊂ G is called a parabolic subgroup if the quotient variety G/P is complete. So, in particular, G itself is a parabolic subgroup. Let P be a parabolic subgroup of G. A character χ of P is called strictly anti-dominant if the following two conditions hold: (1) the line bundle over G/P associated to the principal P-bundle G −→ G/P for χ is ample, and (2) the character χ is trivial on the connected component of the center of P containing the identity element.
Let R u (P) ⊂ P be the unipotent radical. The group P/R u (P) is called the Levi quotient of P. A Levi subgroup of P is a connected reductive subgroup L(P) ⊂ P such that the composition is an isomorphism. A Levi subgroup always exists (see [6, p. 158 If G is the fixed point locus of a real form (G C , σ G C ), by a parabolic subgroup of G we will mean a subgroup P ⊂ G such that there is a parabolic subgroup P ⊂ G C satisfying the conditions that σ G C (P ) = P and P ∩G = P. By a Levi subgroup of a parabolic subgroup P we will mean a subgroup L(P) ⊂ P such that there is a Levi subgroup L(P ) ⊂ P satisfying the conditions that σ G C (L(P )) = L(P ) and L(P ) ∩ G = L(P). (E G , ∇) is called semistable (respectively, stable) if for every triple (Q, E Q , λ), where Q ⊂ G is a proper parabolic subgroup, E Q ⊂ E G is a reduction of structure group of E G to Q compatible with ∇, and λ is a strictly anti-dominant character of Q, the inequality is the flat line bundle over M associated to the flat principal Q-bundle E Q for the character λ of Q, and the degree is defined in (3.6).
In order to decide whether (E G , ∇) is semistable (respectively, stable), it suffices to verify the above inequality (respectively, strict inequality) only for those Q which are proper maximal parabolic subgroups of G. More precisely, E G is semistable (respectively, stable) if and only if for every pair (Q, σ ), where Q ⊂ G is a proper maximal parabolic subgroup, and σ : M −→ E G /Q is a reduction of structure group of E G to Q compatible with ∇, the inequality  We note that a polystable flat principal G-bundle over M is semistable.

Hermitian-Einstein connections
Fix a maximal compact subgroup K ⊂ G of the reductive group G. Let E G be a flat principal G-bundle over M. A Hermitian structure on E G is a smooth reduction of structure group Recall that G is either the fixed point locus of a split real form on a complex reductive group G C , or G is complex reductive. In the second case, by G C we will denote G itself for notational convenience. By analogous arguments to those given in Sect. 3 for flat vector bundles, the flat principal G-bundle E G over M extends to a holomorphic principal G C -bundle E G C over the complex manifold M C . Also, given a Hermitian structure on E G , there is a naturally associated connection on the principal G C -bundle E G C over M C .
Any element z of the center of the Lie algebra g = Lie(G) defines a flat section of ad(E G ), because z is fixed by the adjoint action of G on g; this section of ad(E G ) given by z will be denoted by z.
Let E K ⊂ E G be a Hermitian structure on E G , and let d E K be the corresponding connection on E G C . The curvature of d E K will be denoted by R E K . So R E K is a smooth (1, 1)-form on M C with values in the adjoint vector bundle ad(E G C ). Contracting it using the metric g, we get a smooth section tr g R E K of ad(E G ).

Definition 4.3
The Hermitian structure E K is called Hermitian-Einstein if there is an element z in the center of the Lie algebra g such that tr g R E K = z.
If E K is a Hermitian-Einstein structure, then the corresponding connection d E K is called a Hermitian-Einstein connection.

Theorem 4.4 [3, Theorem 1.2] A flat principal G-bundle E G over M admits a Hermitian-Einstein structure if and only if it is polystable. A polystable flat principal G-bundle admits a unique Hermitian-Einstein connection.
An important ingredient in the proof of Theorem 4.4 is the following analogue of the Harder-Narasimhan filtration for flat vector bundles.

Theorem 4.5 [3, Theorem 1.3] Let V be a flat vector bundle over M. Then there is a unique filtration of V by flat subbundles
such that for each i ∈ [1, ], the flat vector bundle F i /F i−1 is semistable, and The proof of Theorem 4.5 given in [3] makes use of a major simplification compared to the complex case. Namely, flatness implies that the space of all flat subbundles of some rank k of a given flat vector bundle V is a closed subset of the Grassmannian of k-dimensional subspaces of the fiber of V at a fixed point of M, and thus a simple compactness argument guarantees the existence of a flat subbundle with maximal slope. The corresponding argument in the complex Gauduchon case is more complicated; see [9].

A Bogomolov type inequality
We explain an application of Theorem 4.4 given in [3]. As before, let (M, D, ν) be a compact connected special affine manifold of dimension n equipped with an affine Gauduchon metric g. Recall that the Gauduchon condition says that ∂∂(ω n−1 g ) = 0, where ω g is the (1, 1)-form associated to g. The Gauduchon metric g is called astheno-Kähler if (see [24, p. 246]). Note that if n = 2, then g is automatically astheno-Kähler. If g is Kähler, then it is also astheno-Kähler.
For the rest of this subsection, we will assume that g is astheno-Kähler. If V is a flat vector bundle over M and h is a Hermitian metric on V , then the astheno-Kähler condition (4.2) implies that are independent of the choice of h.
The following Bogomolov type inequality for flat vector bundles was given in [3].

Proposition 4.6 [3, Lemma 6.1] Let V be a semistable flat vector bundle over M of rank r . Then
As before, let G be a connected Lie group which is either a complex reductive linear algebraic group, or the fixed point locus of a split real form on a complex reductive linear algebraic group. Then Proposition 4.6 has the following corollary.
Recently, this has been adapted for the compact Gauduchon case by Jacob [23]. As before, let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g. In this section, we will introduce Higgs fields on flat vector bundles over M and give a correspondence between polystable flat Higgs bundles and Yang-Mills-Higgs connections.
The flat connection D on T M induces a flat connection on T * M; this induced connection on T * M will be denoted by D * . Let (E, ∇) be a flat complex vector bundle over M. If ϕ is a flat Higgs field on (E, ∇), then (E, ∇, ϕ) (or (E, ϕ) if ∇ is understood from the context) is called a flat Higgs bundle.
The space of all connections on E is an affine space for the vector space of smooth sections of T * M ⊗End E; a family of connections {∇ t } t∈R is called affine if there is a smooth section α of T * M ⊗ End E such that ∇ t = ∇ 0 + t · α. The following equivalent definition of flat Higgs bundles was given in [4]. A Higgs field will always be understood as a section of A 1,0 (End E), meaning it is expressed in local affine coordinates as where ϕ i are locally defined flat sections of End E; note that dz i = dx i on M. Given a Hermitian metric h on E, the adjoint ϕ * of ϕ with respect to h will be regarded as an element of A 0,1 (End E). In local affine coordinates, this means that In particular, the Lie bracket [ϕ, ϕ * ] is an element of A 1,1 (End E). Locally, Let E be a flat vector bundle over M equipped with a flat Higgs field ϕ as well as a Hermitian metric h. As before, we write d h for the corresponding Chern connection on E C over M C and (∂ h , ∂) for the extended Hermitian connection of (E, h) (see (3.1)). The extended connection form θ ϕ of the Hermitian flat Higgs bundle (E, ϕ, h) is defined to be where θ is the extended connection form of (E, h) (see (3.3)) and ϕ * denotes the adjoint of ϕ with respect to h. This extended connection form corresponds to the connection form of d h + ϕ + ϕ * on E C −→ M C . Analogously, the extended curvature form R ϕ of (E, ϕ, h) is defined to be it corresponds to the curvature form of d h + ϕ + ϕ * on E C . As in the usual case, the extended mean curvature K ϕ of (E, ϕ, h) is obtained by contracting the (1, 1)-part of the extended curvature R ϕ using the Riemannian metric g, so Since tr[ϕ, ϕ * ] = 0, we have tr K ϕ = tr K , and so by (3.5), the extended mean curvature K ϕ of (E, ϕ, h) is related to the first Chern form c 1 (E, h) by (tr K ϕ ) ω n g = n · c 1 (E, h) ∧ ω n−1 g .

Definition 5.3 A Yang-Mills-Higgs metric on (E, ϕ) is a Hermitian metric
h on E such that the extended mean curvature K ϕ of (E, ϕ, h) is of the form for some real constant γ . If h is a Yang-Mills-Higgs metric, then the corresponding connection d h is called a Yang-Mills-Higgs connection.

Definition 5.4
Let (E, ϕ) be a flat Higgs bundle over M.
(i) (E, ϕ) is called semistable (respectively, stable) (with respect to g) if for every proper non-zero flat subbundle E of E which is preserved by ϕ, meaning ϕ(E ) ⊂ T * M ⊗ E , we have where each (E i , ϕ i ) is a stable flat Higgs bundle with slope μ g (E i ) = μ g (E).

Remark 5.5
If {∇ t } t∈R is the family of flat connections on E satisfying the condition in Lemma 5.2 and corresponding to the flat Higgs bundle (E, ϕ), then Definition 5.4 (i) is equivalent to the condition that (5.1) holds for every proper non-zero smooth subbundle E of E which is preserved by ∇ t for all t.
The following Donaldson-Uhlenbeck-Yau type correspondence was proved in [4]. As in the usual case, there is an analogue of this theorem for flat real Higgs bundles.

Definition 5.7
Let (E, ϕ) be a flat real Higgs bundle over M.
(i) (E, ϕ) is called R-semistable (respectively, R-stable) (with respect to g) if for every proper non-zero flat real subbundle E of E which is preserved by ϕ, we have where each (E i , ϕ i ) is an R-stable flat real Higgs bundle with slope μ g (E i ) = μ g (E).

Flat Higgs principal bundles
As before, let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g. Let G be a reductive algebraic group, and let (E G , ∇) be a flat principal G-bundle over M (see Sect. 4.1). Denote by ad(E G ) the adjoint vector bundle of E G (see (4.1)), and recall that the flat connection ∇ on E G induces a flat connection ∇ ad on ad(E G ). Let Let (E G , ∇, ϕ) be a flat Higgs G-bundle over M. Fix a maximal compact subgroup K ⊂ G. Given a smooth reduction of structure group E K ⊂ E G to K , we have a natural connection d E K on the principal K -bundle E K constructed using ∇; the connection on E G induced by d E K will also be denoted by d E K . We may define as before the (1, 1)-part of the extended curvature which is a (1, 1)-form with values in ad(E G ); as before, θ is a (1, 0)-form with values in ad(E G ).
The reduction E K is called a Yang-Mills-Higgs reduction of (E G , ∇, ϕ) if there is an element z in the center of Lie(G) such that the section of ad(E G ) coincides with the one given by z. If E K is a Yang-Mills-Higgs reduction, then the connection d E K on E G is called a Yang-Mills-Higgs connection.
The proofs of Theorems 4.4 and 5.6 also give the following:

The vortex equation
A holomorphic pair on a compact Kähler manifold X is a pair (E, φ) consisting of a holomorphic vector bundle E over X and a holomorphic section φ of E which is not identically equal to zero. These objects were introduced by Bradlow [7,8] (see also [18,19]). He defined the notion of τ -stability, where τ is a real number, and established a Donaldson-Uhlenbeck-Yau type correspondence for holomorphic pairs. This correspondence relates τ -stability to the existence of a Hermitian metric solving the τ -vortex equation, which is similar to the Hermitian-Einstein equation but additionally involves the section φ. In [20], García-Prada showed that the vortex equation is a dimensional reduction of the Hermitian-Einstein equation for an SU(2)-equivariant holomorphic vector bundle over X × P 1 C , where SU(2) acts trivially on X and in the standard way on P 1 C . Let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g. In this section, we will introduce the vortex equation for a pair (E, φ) consisting of a flat vector bundle E over M and a flat non-zero section φ of E, and give a correspondence between polystability of such pairs and the existence of a solution to the vortex equation. (i) (E, φ) is called τ -stable (with respect to g) if the following two conditions are satisfied: (1) μ g (E ) < τ for every flat subbundle E of E with rank(E ) > 0.
(2) μ g (E/E ) > τ for every flat subbundle E of E with 0 < rank(E ) < rank(E) containing the image of the section φ. (ii) (E, φ) is called τ -polystable (with respect to g) if either it is τ -stable, or E decomposes as a direct sum of flat subbundles such that φ is a section of E , the flat pair (E , φ) is τ -stable, and the flat vector bundle E is polystable with slope μ g (E ) = τ n , where n = dim M.
where K h is the extended mean curvature of (E, h) (see (3.4)), φ is regarded as a homomorphism from the trivial Hermitian line bundle on M to E, and φ * denotes its adjoint with respect to h.
The following correspondence was established in [5].

Then E admits a smooth Hermitian metric satisfying the τ -vortex equation if and only if it is τ -polystable.
We give a brief review of the proof of Theorem 6.4. Let (M, D, ν) be a compact connected special affine manifold of dimension n. Denote by P 1 = P 1 C the complex projective line. Consider the product manifold which is a smooth real manifold of dimension n + 2. Let p : M × P 1 −→ M and q : M × P 1 −→ P 1 be the natural projections. The complexified tangent bundle T C X := T R X ⊗ C of X can be decomposed as T C X = p * T C M ⊕ q * T C P 1 = p * T C M ⊕ q * T 1,0 P 1 ⊕ q * T 0,1 P 1 .
Here T 1,0 P 1 and T 0,1 P 1 are respectively the holomorphic and anti-holomorphic tangent bundles of P 1 .
In [5], the theory of Hermitian-Einstein metrics on a flat vector bundle over the affine manifold M was adapted to a smooth complex vector bundle over the product manifold X = M × P 1 equipped with a flat partial connection in the direction of S 0,1 := p * T C M ⊕ q * T 0,1 P 1 ⊂ T C X (see [5,Subsection 2.2] for partial connections); such a bundle is called an S 0,1 -partially flat vector bundle.
A smooth complex vector bundle E over X admits a flat partial connection in the direction of S 0,1 if and only if it admits local trivializations with transition functions which are locally constant in the direction of M and holomorphic in the direction of P 1 . Denote by E C the pullback of E to M C × P 1 by the natural projection The transition functions for E C are obtained by extending the transition functions of E in a constant way along the fibers of T M. Consequently, E C is a holomorphic vector bundle if and only if E is an S 0,1 -partially flat vector bundle. Therefore, the map E −→ E C gives a bijective correspondence between S 0,1 -partially flat vector bundles over X and holomorphic vector bundles over M C × P 1 that are constant along the fibers of T M.
Let g M be an affine Gauduchon metric on M, and let g P 1 be the Fubini-Study metric on P 1 with Kähler form ω P 1 , normalized so that P 1 ω P 1 = 1.
Define a Riemannian metric g on X by g := p * g M ⊕ q * g P 1 . (6. 2) The following analogue of Theorem 3.4 was given in [5].
Theorem 6.5 [5,Theorem 8] Let (M, D, ν) be a compact connected special affine manifold equipped with an affine Gauduchon metric g M , and consider X := M × P 1 together with the Riemannian metric g defined in (6.2). Let (E, ∇) be an S 0,1 -partially flat vector bundle over X . Then E admits a Hermitian-Einstein metric with respect to g if and only if it is polystable.
Using this, Theorem 6.4 was proved in [5] by showing that the vortex equation for a flat pair on M is a dimensional reduction of the Hermitian-Einstein equation for an SU(2)-equivariant S 0,1 -partially flat vector bundle over M × P 1 .