math.DG/0209124

Yang-Mills connections over manifolds with Grassmann structure

Dmitri V. Alekseevsky^{1}, Vicente Cortés^{2,3}, Chandrashekar Devchand^{2}

1 Dept. of Mathematics, University of Hull, Cottingham Road, Hull, HU6 7RX, UK 2 Mathematisches Institut der Universität Bonn, Beringstraße 1, D-53115 Bonn, Germany

3 Institut Élie Cartan, Université Nancy 1, B.P. 239, F-54506 Vandoeuvre-lès-Nancy Cedex, France D.V.Alekseevsky@maths.hull.ac.uk , vicente@math.uni-bonn.de , devchand@math.uni-bonn.de

Abstract

Let M be a manifold with Grassmann structure, i.e. with an isomorphism of the cotangent bundle T^{∗}M≅E⊗H with the tensor product of two vector bundles E and H. We define the notion of a half-flat connection ∇^{W} in a vector bundle W →M as a connection whose curvature F ∈ S^{2}E ⊗ ∧^{2}H ⊗ End W ⊂ ∧^{2}T^{∗}M ⊗ End W. Under appropriate assumptions, for example, when the Grassmann structure is associated with a quaternionic Kähler structure on M, half-flatness implies the Yang-Mills equations. Inspired by the harmonic space approach, we develop a local construction of (holomorphic) half-flat connections ∇^{W} over a complex manifold with (holomorphic) Grassmann structure equipped with a suitable linear connection. Any such connection ∇^{W} can be obtained from a prepotential by solving a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kähler manifolds or more generally to hyper-Kähler manifolds with admissible torsion and to their higher-spin analogues. It yields solutions of the Yang-Mills equations.

This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft, MPI für Mathematik (Bonn) and the SFB 256 (University of Bonn).

Contents

1 Introduction 2

2 Generalised self-duality for manifolds of dimension greater than four 5 2.1 Yang-Mills data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Self-duality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Quaternionic Kähler case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Self-duality as half-flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Manifolds with half-flat holomorphic Grassmann structure 13 3.1 Grassmann structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Harmonic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Canonical distributions on harmonic space . . . . . . . . . . . . . . . . . . . . . . 15

4 Construction of half-flat connections over half-flat Grassmann manifolds 17 4.1 Half-flat connections over half-flat Grassmann manifolds . . . . . . . . . . . . . . 17 4.2 The construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 Construction of almost half-flat connections . . . . . . . . . . . . . . . . . 20 4.2.2 Transformation to the central frame . . . . . . . . . . . . . . . . . . . . . 23 4.2.3 Construction of half-flat connections . . . . . . . . . . . . . . . . . . . . . 24

4.3 Application to hyper-Kähler manifolds with admissible torsion . . . . . . . . . . 26

5 Generalisation to higher-spin Grassmann manifolds 27 5.1 Higher-spin Grassmann structures . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Partially flat connections over higher-spin Grassmann manifolds . . . . . . . . . . 28 5.3 Construction of partially flat connections over higher spin Grassmann manifolds 31

5.3.1 Construction of 0-partially flat connections . . . . . . . . . . . . . . . . . 31 5.3.2 Construction of 1-partially flat connections . . . . . . . . . . . . . . . . . 36

1 Introduction

The Yang-Mills self-duality equations have played an important role in field theory and in differential geometry. They are the main source of examples of solutions of the Yang-Mills equations on four-dimensional manifolds [AHS]. The self-duality equations ∗F^{∇}=F^{∇} mean that the curvature F^{∇} of a connection ∇ over a Riemannian 4-fold M is an eigenvector of the Hodge star operator, associated with the volume 4-form, which acts on two-forms. This apparently four-dimensional construction has an analogue in Riemannian manifolds M of arbitrary dimensions. Any 4-form Ω on M defines an endomorphism B_{Ω} of the space of 2-forms and one can define (Ω, λ)-self-duality as the condition, B_{Ω}F^{∇}=λF^{∇}, that the curvature is an eigenvector of B_{Ω} with eigenvalue λ = const ≠ 0. Under appropriate assumptions on Ω (for example, if it is co-closed) this implies the Yang-Mills equations, just as in four dimensions. For instance, this works for a constant Ω in flat space [CDFN, W] and for a parallel 4-form on a Riemannian manifold with special holonomy (some examples are discussed in [CS, N, BKS, DT, T]). If Ω is, for example, the canonical parallel 4-form associated to a quaternionic Kähler manifold M of dimension 4m, then the eigenspaces of B_{Ω} are the irreducible Sp(m) · Sp(1)–submodules of the space of two-forms. In terms of the associated locally defined Grassmann structure T^{∗ℂ}M = E ⊗ H, i.e. the identification of the complexified cotangent bundle T^{∗ℂ}M with a tensor product of two vector bundles E and H of rank 2m and 2 respectively, the eigenspace decomposition is given by

∧^{2}T^{∗ℂ}M = S^{2}E ⊗ ∧^{2}H ⊕ ∧_{0}^{2}E ⊗ S^{2}H ⊕ ω_{E} ⊗ S^{2}H ,

with corresponding B_{Ω}–eigenvalues λ_{1}=1, λ_{2}= − 1/3, λ_{3}= − (2m+1)/3 [W, CGK]. Here ω_{E} and ω_{H} are two-forms on E^{∗} and H^{∗} such that the complex metric on T^{ℂ}M is given by ω_{E} ⊗ ω_{H} and ∧_{0}^{2}E denotes the traceless part of ∧^{2}E with respect to ω_{E}. The eigenspaces of B_{Ω} can thus be described in terms of the Grassmann structure, which is a natural generalisation of the well-known spinor decomposition of a vector in four dimensions. A 2-form on any manifold with Grassmann structure is called half-flat if it belongs to the eigenspace S^{2}E ⊗ ∧^{2}H and a connection ∇ with half-flat curvature is called half-flat. If the Grassmann structure is associated with the quaternionic Kähler structure, then a half-flat connection is the same as an (Ω, λ_{1})-self-dual connection and hence satisfies the Yang-Mills equations. Inspired by the harmonic space approach [GIOS], we develop a construction of locally-defined holomorphic half-flat connections on a manifold M with holomorphic admissible half-flat Grassmann structure, namely, a holomorphic Grassmann structure T^{∗}M = E ⊗ H with holomorphic connections ∇^{E} and ∇^{H} in the bundles E and H respectively, such that ∇^{H} is flat and the torsion of the linear connection ∇ = ∇^{E} ⊗ Id + Id ⊗ ∇^{H} has no component in S^{3}H ⊗ E^{∗} ⊗ ∧^{2}E. The construction associates to a holomorphic prepotential a half-flat connection through the solution of a system of linear first order ODEs. The construction can be applied, for instance, to the complexification of hyper-Kähler manifolds or, more generally, to hyper-Kähler manifolds with admissible torsion. Our construction of gauge fields on such curved backgrounds extends that of [GIOS], where flat torsionfree backgrounds were considered.

We note that using analytic continuation any real analytic connection ∇ over a real analytic Grassmann manifold allows extension to a holomorphic connection ∇^{ℂ} over a holomorphic Grassmann manifold and ∇ can be reconstructed from ∇^{ℂ} in terms of some antiholomorphic involution.

The main idea of our construction is to pull-back a half-flat connection ∇ in a holomorphic vector bundle ν : W → M to the harmonic space S_{H}. The latter is the space of all symplectic frames h = (h_{+}, h_{−}) in the vector bundle H^{∗}. The group Sp(1, ℂ) acts freely on S_{H}, with the orbit space S_{H}/Sp(1, ℂ) = M. Hence, the projection π : S_{H} → M is an Sp(1, ℂ)-principal bundle. Choosing a (local) trivialisation, M ∋ x ↦ (h_{1}(x), h_{2}(x)) ∈ S_{H}, of π we can make the identification S_{H} = Sp(1, ℂ) × M. There exists a canonical decomposition,

T S_{H} = T^{v}S_{H} ⊕ 𝒟_{+} ⊕ 𝒟_{−} ,

of the (holomorphic) tangent bundle into the vertical sub-bundle T^{v}S_{H} and two (holomorphic) distributions 𝒟_{+} and 𝒟_{−} spanned respectively by vector fields X_{+}^{e} and X_{−}^{e} canonically associated with sections e of the bundle E^{∗}. If the Grassmann structure is admissible and half-flat, the distributions 𝒟_{+} and 𝒟_{−} are integrable. The vertical distribution T^{v}S_{H} is spanned by vector fields ∂_{0}, ∂_{++}, ∂_{−−}, which correspond to the standard generators of the Lie algebra 𝔰𝔭(1, ℂ). A half-flat connection ∇ in the bundle ν : W → M induces the pull-back connection π^{∗}∇ in the pull-back bundle π^{∗}ν : π^{∗}W → S_{H}. Since ∇ is half-flat, the curvature F of π^{∗}∇ satisfies certain equations (see Definitions 6 and 7). A connection in π^{∗}ν satisfying these equations is called a half-flat connection over S_{H} and is gauge equivalent to the pull-back of a half-flat connection over M. Any half-flat connection over S_{H} is flat along the leaves of the integrable distribution ⟨𝒟_{+}, ∂_{0}⟩ spanned by ∂_{0} and 𝒟_{+}. We can therefore choose a frame of the vector bundle π^{∗}ν which is parallel along its leaves. Such a frame is called an analytic frame. With respect to such a frame a half-flat connection has no potentials in the directions of the distribution ⟨𝒟_{+}, ∂_{0}⟩. Starting from a matrix-valued function (prepotential) A_{++} on S_{H}, which is constant along the leaves of the distribution 𝒟_{+} and satisfies the homogeneity condition ∂_{0}A_{++} = 2A_{++} , we construct a connection which satisfies almost all the conditions of half-flatness. We call such a connection an almost half-flat connection. It is half-flat if and only if its curvature satisfies the equation F (∂_{−−}, 𝒟_{−}) = 0. The construction of an almost half-flat connection reduces to the solution of first order linear ODEs. Assuming that the almost half-flat connection ∇ is defined globally along the fibres (over π^{−1}U, where U ⊂ M is a domain in M) we can modify ∇ to a half-flat connection over S_{H} which is the pull-back of a half-flat connection over M. In order to do this, we re-write ∇ with respect to a “central frame”, namely, a frame parallel along the fibres of π. The transformation from the analytic to the central frame reduces to the solution of the system of equations

∂_{++}Φ = −A_{++}Φ , ∂_{0}Φ = 0 .

With respect to the central frame the potential C(X_{+}^{e}) of the connection ∇ in the direction of the vector field X_{+}^{e} ∈ 𝒟_{+} has the form C(X_{+}^{e}) = u_{+}^{α}C_{α}^{e}, where C_{α}^{e} are matrix-valued functions on M = M × {Id} ⊂ M × Sp(1, ℂ) and u_{±}^{α} , α = 1, 2, are matrix coefficients of Sp(1, ℂ). The matrix-valued functions C_{1}^{e}, C_{2}^{e} define the desired half-flat connection on M given by

∇_{e⊗h_{1}}^{M} = e ⊗ h_{1} + C_{1}^{e} , ∇_{e⊗h_{2}}^{M} = e ⊗ h_{2} + C_{2}^{e} .

Moreover, any half-flat connection may be obtained in this way.

The above construction allows generalisation to manifolds with spin \frac{m}{2} Grassmann structure. This means that the cotangent bundle is identified as T^{∗}M = E⊗F = E⊗S^{m}H, where E and H are (holomorphic) vector bundles of rank p and 2 respectively. If a connection ∇^{E} on E and a flat connection ∇^{H} on H are given, then the Grassmann structure is called half-flat. The connection ∇^{H} defines a flat connection ∇^{F} on F = S^{m}H and the linear connection ∇ = ∇^{E} ⊗ Id + Id ⊗ ∇^{F}. The associated harmonic space π : S_{H} → M is defined as above, as the space of all symplectic frames h = (h_{+}, h_{−}) in H^{∗}. Its tangent space has decomposition

T S_{H} = T^{v}S_{H} ⊕ ⨁_{k=0}^{m} 𝒟_{k+} ⊕ ⨁_{k=1}^{m} 𝒟_{k−} .

Under certain conditions on the torsion of ∇ the distribution 𝒟_{(+)}^{k} := ⨁_{i=0}^{k} 𝒟_{(m−2i)+} , k ≤ m/2, is integrable. Such a half-flat Grassmann structure is called k-admissible. Generalising the notion of a half-flat connection, we may define a k-partially flat connection ∇ over a manifold with half-flat spin \frac{m}{2} Grassmann structure such that the pull-back connection π^{∗}∇ has no curvature in the directions of 𝒟_{(+)}^{k}. The harmonic space method can be applied to construct k-partially flat connections over k-admissible half-flat spin \frac{m}{2} Grassmann manifolds. In the final section we consider the case of m = 3 and sketch the construction of 0– and 1–partially flat connections. The latter are Yang-Mills connections.

We should like to thank Andrea Spiro for useful comments on the manuscript.

2 Generalised self-duality for manifolds of dimension

greater than four

2.1 Yang-Mills data

Let ν : W → M be a real vector bundle over M and ∇ a connection in ν, that is a bilinear map

\begin{matrix}∇ : 𝔛(M) × Γ(ν) & → & Γ(ν) \\ (X, σ) & ↦ & ∇_{X}σ \end{matrix}

which is C^{∞}(M)-linear in the vector field X ∈ 𝔛(M) and satisfies the Leibniz rule ∇_{X}(fσ) = (Xf)σ + f∇_{X}σ, for any function f ∈ C^{∞}(M) and any section, σ ∈ Γ(ν), of ν. The map ∇ can be extended to a complex bilinear map,

\begin{matrix}∇ : 𝔛^{ℂ}(M) × Γ(W^{ℂ}→M) & → & Γ(W^{ℂ}→M) \\ (X, σ) & ↦ & ∇_{X}σ , \end{matrix}  (2.1)

where 𝔛^{ℂ}(M) is the space of complex vector fields X+iY ; X, Y ∈𝔛(M) and W^{ℂ} → M is the complexification of the vector bundle ν. Note that ∇ satisfies the reality condition

∇\overline{X} \overline{σ} = \overline{∇_{X}σ} , X ∈ 𝔛^{ℂ}(M) , σ ∈ Γ(W^{ℂ}→M) ,  (2.2)

where the bar denotes complex conjugation. Conversely, any ℂ-bilinear map (2.1) which is 𝔛^{ℂ}(M)-linear and satisfies the Leibniz rule and the reality condition (2.2), defines a connection ∇ in the real vector bundle ν. If the reality condition (2.2) is dropped, then (2.1) defines a connection in the complex vector bundle W^{ℂ}→M.

Let φ = (φ_{1}, …, φ_{r}) denote a local frame of ν such that for any section σ ∈ Γ(ν), σ = ∑ s^{i}φ_{i} =: φ · s, where s^{i} are the coordinates of σ with respect to the frame φ and s = (s^{1}, …, s^{r})^{t}. Then the connection ∇ in ν has local expression

∇_{X}σ = ∇_{X}(s^{i}φ_{i}) = φ · ∇_{X}s := (Xs^{i} + ∑_{j} A_{j}^{i}(X)s^{j})φ_{i} ,

where A_{j}^{i}(X) = (∇_{X}φ_{j}, φ^{i}) and φ^{∗} = (φ^{1}, …, φ^{r}) denotes the dual frame. The locally defined matrix-valued 1-form A = (A_{j}^{i}) : M → 𝔤𝔩(r, ℝ) is called the Yang-Mills potential with respect to the frame φ. If the vector bundle ν has structure group G, i.e. if it is a bundle associated with a principal G-bundle P → M and a representation ρ : G → GL(r, ℝ), such that W = P ×_{G} ℝ^{r}, then we may always choose a frame φ for which the potential takes values in the Lie algebra 𝔤 = Lie ρ(G) ⊂ 𝔤𝔩(r, ℝ). We will symbolically write ∇_{X} = X + A , A = A^{φ}. A change of frame (gauge transformation) φ′ = φU induces changes s′ = U^{−1}s and φ′(X + A′(X))s′ = φ′∇_{X}s′ = φ∇_{X}s = φ(X + A(X))s = φ′U^{−1}(X + A(X))Us′, yielding the transformation rule for the potential,

A′ = U^{−1}(XU) + U^{−1}A(X)U = U^{−1}∇_{X}U .  (2.3)

The curvature of the connection ∇, F = F^{∇} ∈ Ω_{M}^{2}(End W) = Γ(∧^{2}T^{∗}M ⊗ End W), is given by

F (X, Y) = [∇_{X}, ∇_{Y}] − ∇_{[X,Y]}
= XA(Y) − Y A(X) + [A(X), A(Y)] − A([X, Y]) .

The Jacobi identity for ∇_{X} is equivalent to the Bianchi identity, d^{∇}F^{∇} = 0. Here the covariant derivative d^{∇} : Ω^{p}(End W) → Ω^{p+1}(End W) is defined by

d^{∇}(ω ⊗ C) = dω ⊗ C + (−1)^{p}ω ∧ ∇C ,

where ω is a p-form and C is a section of End W. (The connection ∇ on W induces a connection on End W denoted by the same symbol.)

On any n-dimensional oriented pseudo-Riemannian (or complex Riemannian) manifold, (M, g) using the canonical volume form vol^{g} ∈ ∧^{n}T^{∗}M, we define the Hodge ∗ operator which interchanges forms of complementary degree, ∗ : ∧^{p}T^{∗}M → ∧^{n−p}T^{∗}M, by the relation: ⟨α, β⟩vol^{g} = α ∧ ∗β, where α, β ∈ ∧^{p}T^{∗}M and ⟨. , .⟩ is the natural scalar product on ∧^{p}T^{∗}M induced by the metric g. We define ∗ : ∧^{p}T^{∗}M⊗ End W → ∧^{n−p}T^{∗}M⊗ End W by ∗(ω ⊗ C) := (∗ω ⊗ C).

Definition 1 Let ν : W → M be a real vector bundle over a pseudo-Riemannian manifold (M, g). A YM connection ∇ in ν is one which satisfies the Yang-Mills equation

d^{∇} ∗ F^{∇} = 0 .

On a closed manifold this is the Euler-Lagrange equation for the YM functional

||F^{∇}||^{2} = ∫_{M} |F^{∇}|^{2} vol^{g},  (2.4)

where the norm on ∧^{2}T^{∗}M ⊗ End W is induced by the pseudo-Riemannian metric on M and the natural metric on End W.

2.2 Self-duality conditions

On a Riemannian four-manifold, the ∗ operator maps 2-forms to 2-forms and has eigenvalues ±1. The curvature tensor therefore has decomposition into the eigenspaces of the ∗ operator,

F^{∇} = F_{+1}^{∇} ⊕ F_{−1}^{∇} ∈ Ω_{M}^{+}(End W) ⊕ Ω_{M}^{−}(End W).

This splitting corresponds to the decomposition of the SO(4)-module ∧^{2}ℝ^{4} = ∧_{+}^{2} ⊕ ∧_{−}^{2} ≅ 𝔰𝔬(4) = 𝔰𝔭(1) ⊕ 𝔰𝔭(1) into its irreducible submodules. We call ∇ and F^{∇} self-dual or anti-self-dual if F_{−1}^{∇} := \frac{1}{2}(F^{∇} − ∗F^{∇}) = 0 or F_{+1}^{∇} := \frac{1}{2}(F^{∇} + ∗F^{∇}) = 0, respectively. For (anti-) self-dual connections, the YM equation, d^{∇} ∗F^{∇} = 0, is an immediate consequence of the Bianchi identity, d^{∇}F^{∇} = 0. On closed manifolds (anti-) self-dual connections in fact minimise the YM functional (2.4), since the inequality

||F^{∇}||^{2} = ||F_{+1}^{∇}||^{2} + ||F_{−1}^{∇}||^{2} ≥ |||F_{+1}^{∇}||^{2} − ||F_{−1}^{∇}||^{2}| = 8π^{2} |c_{2}(W)[M]|

is saturated. Here c_{2}(W)[M] = \frac{1}{8π^{2}} ∫_{M} tr F^{∇} ∧ F^{∇} is the evaluation of the second Chern class of the bundle W on the fundamental cycle.

The apparently four-dimensional notion of self-duality, has an analogue in higher dimensions. The construction originally given in [CDFN] for flat spaces, extends to arbitrary manifolds (M, g), of dimension greater than four, as follows.

For Ω ∈ Ω^{4}(M) we define a symmetric tracefree endomorphism field B_{Ω} : ∧^{2}T^{∗}M → ∧^{2}T^{∗}M by

B_{Ω}ω := ∗(∗Ω ∧ ω) ,  (2.5)

where ω ∈ ∧^{2}T^{∗}M. This endomorphism is zero if and only if the 4-form Ω is zero. Moreover, we have:

Lemma 1 Let

Ω = ∑ Ω_{ijkl}e^{i} ∧ e^{j} ∧ e^{k} ∧ e^{l}, ω = ∑ ω_{ij}e^{i} ∧ e^{j}

be the expressions for Ω and ω with respect to a frame e^{i} of T^{∗}M. Then B_{Ω} is given as the contraction

B_{Ω}ω = 12 ∑ g^{ii′}g^{jj′} Ω_{ijkl} ω_{i′j′} e^{k} ∧ e^{l} .

Proof: It is sufficient to check the above formula for decomposable forms Ω = e^{i}∧e^{j} ∧e^{k}∧e^{l} and ω = e^{m} ∧ e^{n}, where the e^{i} form an orthonormal basis of T^{∗}M.

Definition 2 A 4-form Ω ∈ Ω^{4}(M) on a pseudo-Riemannian manifold M is called appropriate if there exists a nonzero real constant eigenvalue λ of the endomorphism field B_{Ω}.

We note that on a Riemannian manifold the eigenvalues of B_{Ω} are real for any 4-form Ω. A generalisation of the four dimensional notion of self-duality may now be defined:

Definition 3 Let Ω be an appropriate 4-form on a pseudo-Riemannian manifold (M, g) and λ ≠ 0 ∈ ℝ. A connection ∇ in a vector bundle ν : W → M is (Ω, λ)-self-dual if its curvature F^{∇} satisfies the linear algebraic system

B_{Ω}F^{∇} = λF^{∇}  (2.6)
(d ∗ Ω) ∧ F^{∇} = 0 .  (2.7)

Theorem 1 Let (M, g) be a pseudo-Riemannian manifold with an appropriate 4-form Ω. Then any (Ω, λ)-self-dual connection ∇ is a YM connection.

Proof: Using (2.6) and (2.5) we obtain

d^{∇} ∗ F^{∇} = \frac{1}{λ}d^{∇} ∗ B_{Ω}F^{∇} = ±\frac{1}{λ}d^{∇}(∗Ω ∧ F^{∇})
= ±\frac{1}{λ} ((d ∗ Ω) ∧ F^{∇} + ∗Ω ∧ d^{∇}F^{∇})
= 0 ,

in virtue of (2.7) and the Bianchi identity d^{∇}F^{∇}=0. Examples of manifolds admitting appropriate 4-forms are easily obtained. Let V be a pseudo-Euclidean vector space and G ⊂ SO(V) be a linear group preserving a nonzero element Ω_{0} ∈ ∧^{4}V. Denote by Ω_{ijkl} the components of Ω_{0} with respect to an orthonormal basis of V. Given a manifold M with a G-structure, π : P → M, i.e. a principal Gsubbundle of the bundle of frames on M, we can define a 4-form Ω := ∑ Ω_{ijkl}e^{i}∧e^{j} ∧e^{k}∧e^{l}, where (e^{1}, …, e^{n}) is a coframe dual to a G-frame p = (e_{1}, …, e_{n}) ∈ P. Since G ⊂ SO(V), M has the structure of an oriented pseudo-Riemannian manifold and we can define the operator B_{Ω}. The matrix components of B_{Ω} = ∑ B_{ij}^{kl}e^{i} ∧ e^{j} ⊗ e_{k} ∧ e_{l} are constant for any G-frame and so are its eigenvalues. Hence Ω is appropriate if the endomorphism B_{Ω_{0}} ∈ ∧^{4}V has a nonzero real eigenvalue λ. This is automatic in the Riemannian case.

There exist many examples of subgroups G ⊂ SO(V) admitting nonzero G-invariant 4-forms, as shown by the following construction. Let G ⊂ SO(V) be a closed subgroup of the pseudo-orthogonal group SO(V) and 𝔤⊂𝔰𝔬(V)≅ ∧^{2} V^{∗} its Lie algebra. Assume that 𝔤 admits a G-invariant symmetric nondegenerate bilinear form B ∈ S^{2}(𝔤^{∗})^{G}, where W^{G} denotes the space of G-invariant elements of a G-module W. We can then identify 𝔤 with its dual 𝔤^{∗} via B and consider B as an element of (S^{2}(𝔤))^{G}⊂(S^{2} ∧^{2} V^{∗})^{G}. A Ginvariant 4-form is then defined by Ω_{0}^{G}:=alt B ∈ (∧^{4}V^{∗})^{G}, where alt :S^{2} ∧^{2} V^{∗}→ ∧^{4} V^{∗} denotes alternation. We denote the corresponding 4-form on a manifold with G-structure by Ω^{G}. The following variant of a theorem by Kostant [K] provides a wealth of examples of nonzero Ω_{0}^{G} ’s.

Theorem 2 Let G ⊂ SO(V) be a closed subgroup whose Lie algebra 𝔤 admits a nondegenerate G-invariant bilinear form B ∈ (S^{2}𝔤)^{G}. If the G-module V is not equivalent to the isotropy module of a pseudo-Riemannian symmetric space, then the four-form Ω_{0}^{G} := alt B ∈ (∧^{4}V)^{G} is nonzero.

Proof: Recall that the SO(V)-module S^{2} ∧^{2} V decomposes according to S^{2} ∧^{2} V = ℛ(𝔰𝔬(V)) + ∧^{4}V, where ℛ(𝔰𝔬(V)) denotes the space of curvature tensors of type 𝔰𝔬(V), i.e. the space of two-forms fulfilling the first Bianchi identity or the kernel of the map alt : S^{2} ∧^{2} V → ∧^{4}V. If Ω_{0}^{G} = alt B = 0, then B is a nonzero element of ℛ(𝔰𝔬(V)) ∩ S^{2}(𝔤)^{G} = ℛ(𝔤)^{G}. Since B is a G-invariant two-form on V with values in 𝔤 it can be used to define a Lie bracket [ · , ·] on the vector space 𝔩 = 𝔤 ⊕ V thus:

(i) 𝔤 is a subalgebra of 𝔩, (ii) V is a 𝔤-submodule with action defined by the inclusion 𝔤 ⊂ 𝔰𝔬(V), (iii) [u, v] := B(u, v) ∈ 𝔤 if u, v ∈ V. The Jacobi identity follows from the Bianchi identity and the G-invariance. Let L be the simply connected Lie group with Lie algebra 𝔩. Then L/G_{0} is a Riemannian symmetric space with V as its isotropy module, where G_{0} ⊂ L is the connected Lie subgroup with Lie G_{0} = 𝔤.

Clearly, (2.7) is automatic if the 4-form Ω is co-closed, d ∗ Ω = 0. This is the case, for example, if Ω is parallel. In the Riemannian case the Berger list of irreducible holonomy groups [B] and a theorem of Kostant [K] yield the following result.

Theorem 3 Let M be a complete simply connected irreducible Riemannian manifold of dimension n ≥ 4 with holonomy group Hol ⊂ SO(n), Hol ≠ SO(n). Then M admits a nontrivial parallel 4-form if one of the following holds: i) M is not a symmetric space or ii) M is a symmetric space and has a non-simple holonomy or, equivalently, isotropy group.

Proof: By Berger’s theorem on Riemannian irreducible holonomy groups [B], we have:

a) M is not a symmetric space and its holonomy group is one of: U(\frac{n}{2}), SU(\frac{n}{2}), Sp(\frac{n}{4})Sp(1), Sp(\frac{n}{4}), G_{2}, Spin(7), or

b) M is a symmetric space. All the groups in a) admit invariant 4-forms. These are given below. A theorem of Kostant [K] states that a simply-connected irreducible Riemannian symmetric space G/K has no nonzero parallel 4-form if and only if the isotropy group K is simple.

In the following examples we explicitly describe parallel (hence appropriate) 4-forms Ω on Riemannian n-manifolds with holonomy groups Hol ≠ SO(n) from Berger’s list.

1. Kähler manifolds, Hol ⊂ U(m) ⊂ SO(2m), n = 2m: Ω = ω ∧ ω, where ω is the Kähler form. One can check that this is proportional to Ω^{SU(m)} and that any parallel 4-form is proportional to ω∧ω if the holonomy group is SU(m) or U(m). If Hol ⊂ Sp(k) ⊂ SU(2k) ⊂ SO(4k), n = 4k > 4, i.e. if the manifold is hyper-Kähler, there exist three skewsymmetric parallel complex structures J_{α}, α = 1, 2, 3. Then there exist six independent parallel 4forms ω_{α} ∧ ω_{β}, α, β = 1, 2, 3, where ω_{α} is the Kähler form associated to J_{α}. For low dimensional examples, eigenvalues and eigenspaces of B_{Ω} are given in [CDFN].

2. Quaternionic Kähler manifolds, Hol ⊂ Sp(m)Sp(1) ⊂ SO(4m), n = 4m. In this case there exist 3 locally defined almost complex structures J_{α}, with corresponding Kähler forms ω_{α}, such that the 4-form Ω := ∑_{α} ω_{α} ∧ ω_{α} is globally defined and parallel. This will be discussed in more detail in section 2.3.

3. Hol ⊂ G_{2} ⊂ SO(7). Let V = 𝕆 = ℝ1 + Im𝕆 = ℝ ⊕ ℝ^{7} = ℝ^{8} be the algebra of octonions. Recall that G_{2} = Aut(𝕆) is the group of automorphisms of the octonions. We can decompose the product of two octonions a, b into its real and imaginary parts as follows:

ab = a · b = ⟨a, b⟩1 + \frac{1}{2}[a, b] ,

where ⟨a, b⟩ is the scalar product and [a, b] = ab − ba is the commutator. We define a 3-form φ and a 4-form ψ on Im 𝕆 = ℝ^{7} by the formulas

\begin{aligned}φ(x, y, z) & := & ⟨x · y, z⟩ = \frac{1}{2}⟨[x, y], z⟩ \\ ψ(x, y, z, w) & := & ⟨[x, y, z], w⟩ , \end{aligned}

where [x, y, z] = (xy)z − x(yz) is the associator. It is known that ψ = ∗φ. Notice that G_{2} is the group of isometries of 𝕆 = ℝ^{8} which fix the identity element 1 and preserve the 3-form φ (or equivalently the 4-form ψ) on Im 𝕆. The 4-form ψ defines a parallel 4-form on any Riemannian 7-fold with holonomy G_{2} ⊂ SO(7). It is known [OV] that ∧^{4}ℝ^{7} = ℝψ ⊕ V^{7}(π_{1}) ⊕ V^{27}(2π_{1}), where V^{d}(π) is the d-dimensional real irreducible representation of G_{2} with highest weight π and π_{i} denotes the i-th fundamental weight of G_{2}. From this it follows that the 4-form Ω_{0}^{G_{2}} coincides with ψ up to scaling. The corresponding endomorphism B_{ψ} of ∧^{2}ℝ^{7} = 𝔤_{2} ⊕ ℝ^{7} has two distinct eigenvalues which correspond to the two irreducible G_{2}-submodules 𝔤_{2} and ℝ^{7} ⊂ ∧^{2}ℝ^{7} [CDFN].

4. Hol ⊂ Spin(7) ⊂ SO(8). Using the 3- and 4-forms φ and ψ on ℝ^{7} introduced in the G_{2}-case, we construct the 4-form

Ω = dt ∧ φ + ψ ,

where t is the first coordinate on ℝ^{8} = ℝ1 + ℝ^{7}. In particular,

Ω(1, x, y, z) = φ(x, y, z) , Ω(x, y, z, w) = ψ(x, y, z, w) , x, y, z, w ∈ ℝ^{7} .

This 4-form Ω defines a parallel 4-form on any Riemannian 8-fold with holonomy Spin(7) ⊂ SO(8). It is known that ∧^{4}ℝ^{8} = ℝΩ⊕V^{7}(π_{1})⊕V^{27}(2π_{1})⊕∧^{4}ℝ^{7}. From this it follows that the 4-form Ω_{0}^{Spin(7)} coincides with Ω up to scaling. The corresponding endomorphism B_{Ω} of ∧^{2}ℝ^{8} = 𝔰𝔭𝔦𝔫_{7} ⊕ ℝ^{7} has two distinct eigenvalues which correspond to the two irreducible Spin(7)-submodules 𝔰𝔭𝔦𝔫_{7} and ℝ^{7} ⊂ ∧^{2}ℝ^{8} [CDFN].

2.3 Quaternionic Kähler case

Now we discuss in more detail the case of quaternionic Kähler manifolds (example 2 above). Riemannian manifolds (M, g) with holonomy group Hol ⊂ Sp(m)Sp(1) are called quaternionic Kähler manifolds. A quaternionic Kähler manifold with holonomy group Hol ⊂ Sp(m) is called hyper-Kähler. On any quaternionic Kähler manifold M, there exists a rank 3 vector subbundle Q ⊂ End T M, invariant under parallel transport, which is locally spanned by three almost complex structures (J_{α}) = (J_{1}, J_{2}, J_{3}=J_{1}J_{2}= − J_{2}J_{1}). The latter are in general only locally defined. The (globally defined) vector bundle Q is called the quaternionic structure of M. A local frame (J_{α}) as above is called a standard frame for Q. Similarly, a standard basis of Q at m ∈ M, is a triple I, J, K = IJ = −JI ∈ Q_{m} of complex structures on T_{m}M. A quaternionic Kähler manifold is hyper-Kähler if and only if there exists a globally defined parallel standard frame (J_{α}) = (J_{1}, J_{2}, J_{3}=J_{1}J_{2}= − J_{2}J_{1}).

Given a standard frame, we may locally define three nondegenerate 2-forms ω_{α} := g(J_{α}·, ·). The 4-form

Ω := ∑_{α} ω_{α} ∧ ω_{α}

is independent of the choice of standard frame and defines a global parallel 4-form.

To describe the eigenspace decomposition of Ω it is convenient to use the Grassmann structure (i.e. generalised spinor decomposition) of a quaternionic Kähler manifold. Recall that a Grassmann structure on a (real) manifold M is defined as an isomorphism T^{∗ℂ}M ≅ E ⊗H of the complexified cotangent bundle with the tensor product of two complex vector bundles E and H over M. Any quaternionic Kähler manifold admits a (locally defined) Grassmann structure T^{∗ℂ}M = E ⊗ H, where H has rank 2, such that the holonomy group Hol ⊂ Sp(E) ⊗ Sp(H). This follows from the fact that any complex irreducible representation of the group Sp(m)×Sp(1) is a tensor product of irreducible representations of its factors.

The complex extension g^{ℂ} of the Riemannian metric defines a complex bilinear metric on T^{ℂ}M, which locally factorises as g^{ℂ} = ω_{E} ⊗ ω_{H}, where ω_{E} and ω_{H} are sections of ∧^{2}E and ∧^{2}H, defining complex symplectic forms on the fibres of E^{∗} and H^{∗}, respectively. We call ω_{E} and ω_{H} the symplectic forms of the symplectic vector bundles E^{∗} and H^{∗}.

In terms of the Grassmann structure the eigenspaces V_{λ} of the endomorphism B_{Ω} on ∧^{2}T^{∗ℂ}M are given by [W, CGK]:

V_{λ_{1}} = S^{2}E ⊗ ω_{H} , V_{λ_{2}} = ∧_{0}^{2}E ⊗ S^{2}H , V_{λ_{3}} = ω_{E} ⊗ S^{2}H ,

where ∧_{0}^{2}E is the space of ω_{E}-traceless 2-forms and the eigenvalues are λ_{1} = 1, λ_{2} = −1/3 and λ_{3} = −(2m + 1)/3. In particular the λ_{1}-self-duality condition takes the form

F^{∇} ∈ S^{2}E ⊗ ω_{H} ⊗ End W .  (2.8)

Note that since Ω is parallel it is appropriate and co-closed and hence the (Ω, λ)-self-duality equations (Definition 3) reduce to (2.6), which implies the Yang-Mills equation. It is known (see Theorem 1 of [CS]) that λ_{1}- and λ_{3}-self-dual connections correspond to absolute minima of the Yang-Mills functional on compact quaternionic Kähler manifolds.

2.4 Self-duality as half-flatness

The λ_{1}-self-duality equation (2.8) in fact depends only the existence of the factorisation T^{∗ℂ}M ≅ E ⊗H and the symplectic structure in H^{∗}. A connection ∇ in a vector bundle W over a manifold M with a Grassmann structure is called half-flat if its curvature satisfies the condition

F^{∇} ∈ S^{2}E ⊗ ∧^{2}H ⊗ End W .  (2.9)

In general such half-flat connections are not YM connections (with respect to some metric), but it is possible to impose further conditions on F^{∇} in order to enforce the YM equation. In fact it is the half-flatness of the connection, rather than the YM property, which is crucial for our construction of solutions.

Proposition 1 A connection ∇ in a vector bundle W → M over a quaternionic Kähler manifold is half-flat if and only if it is λ_{1}-self-dual. Hence any such connection is a Yang-Mills connection.

Proof: The result follows from (2.8) and (2.9) since ∧^{2}H is the line bundle generated by ω_{H}. The Levi-Civita connection on a hyper-Kähler manifold is an example of a half-flat linear connection. Its complexification gives an example of what we call an admissible half-flat Grassmann structure in the next section.

3 Manifolds with half-flat holomorphic Grassmann

structure

Our goal is to give a construction of half-flat connections in a vector bundle ν : W → M over a manifold M. If all objects are real analytic, using analytic continuation we may obtain corresponding complex analytic objects. Specifically, assume that the manifold M and the bundle ν are real analytic. Then M is defined by an atlas of charts with analytic transition functions. Extending these functions to complex holomorphic functions, we may extend M to a complex manifold M^{ℂ} with antiholomorphic involution τ such that M = (M^{ℂ})^{τ}, the fixed point set of τ. Similarly, a real analytic vector bundle ν : W → M can be extended to a holomorphic vector bundle ν^{ℂ} : W^{ℂ} → M^{ℂ}. Moreover, an analytic connection ∇ in ν can be extended to a holomorphic connection ∇ in ν^{ℂ}. A holomorphic extension of a Yang-Mills connection is also a Yang-Mills connection. In the rest of this paper, we shall assume that all objects (manifolds, bundles and connections) are holomorphic. In section 4 we shall give a construction of half-flat connections in a holomorphic bundle W → M over a complex manifold M with holomorphic Grassmann structure. Now we describe the required geometrical notions.

3.1 Grassmann structure

Let M be a complex manifold with holomorphic Grassmann structure T^{∗}M = E ⊗ H, the isomorphism of the holomorphic cotangent bundle over M with the tensor product of holomorphic vector bundles E and H over M of rank p and q respectively. Then T M = E^{∗}⊗H^{∗}. A holomorphic linear connection ∇ on M is called a holomorphic Grassmann connection if it preserves the holomorphic Grassmann structure. This means that for any vector field X on M and local sections e ∈ Γ(E) and h ∈ Γ(H),

∇_{X}(e ⊗ h) = ∇_{X}^{E}e ⊗ h + e ⊗ ∇_{X}^{H}h

where ∇^{E}, ∇^{H} are connections in the bundles E, H respectively.

Definition 4 A holomorphic Grassmann structure, T^{∗}M = E ⊗ H, on a complex manifold M with a holomorphic Grassmann connection ∇ = ∇^{E} ⊗Id+Id⊗∇^{H} is called half-flat if the connection ∇^{H} in the holomorphic vector bundle H → M is flat. A manifold with such a half-flat holomorphic Grassmann structure is called a half-flat Grassmann manifold.

Assumption. In this section we assume that M is a manifold with a half-flat holomorphic Grassmann structure (T^{∗}M = E ⊗ H , ∇ = ∇^{E} ⊗ Id + Id ⊗ ∇^{H}), such that H has rank 2 and that a ∇^{H}-parallel nondegenerate fibre-wise 2-form ω_{H} ∈ Γ(∧^{2}H) in the bundle H^{∗} is fixed. If, in addition, a ∇^{E}-parallel nondegenerate two-form ω_{E} ∈ Γ(∧^{2}E) is fixed, then we can define a ∇-parallel complex Riemannian metric g = ω_{E} ⊗ ω_{H} on M. We do not assume, in general, that the linear connection ∇ is torsion-free.

The torsion of a linear connection belongs to T M ⊗ ∧^{2}T^{∗}M. Since T^{∗}M = E ⊗ H, we have the decomposition,

T M ⊗ ∧^{2}T^{∗}M = T M ⊗ (∧^{2}E ⊗ S^{2}H ⊕ S^{2}E ⊗ ∧^{2}H)
= E^{∗}H^{∗} (∧^{2}ES^{2}H ⊕ S^{2}Eω_{H})
≅ E^{∗} ∧^{2} E (S^{3}H ⊕ ω_{H}H) ⊕ E^{∗}S^{2}Eω_{H}H ,  (3.1)

where we omit the ⊗’s and we identify H^{∗} with H using ω_{H}.

Definition 5 A half-flat connection is called admissible if its torsion tensor has no component in E^{∗} ⊗ ∧^{2}E ⊗ S^{3}H. A half-flat Grassmann manifold (M, ∇) is called admissible if ∇ is admissible.

We remark that if the torsion of a half-flat connection is E-symmetric, i.e. if it belongs to T M ⊗ S^{2}E ⊗ ∧^{2}H = T M ⊗ S^{2}E ⊗ ω_{H}, then the connection is admissible. It follows from the above decomposition that the torsion tensor of any admissible connection can be written as

T (e⊗h, e′ ⊗h′) = T_{1}(e, e′)⊗ω_{H}(h, h′)h_{1} + T_{2}(e, e′)⊗ω_{H}(h, h_{2})h′ + T_{2}(e, e′)⊗ω_{H}(h′, h_{2})h

where e, e′ are sections of E^{∗}, h_{1}, h_{2} are fixed sections of H ≅ H^{∗}, T_{1} ∈ Γ(E^{∗} ⊗ S^{2}E) and T_{2} ∈ Γ(E^{∗} ⊗∧^{2}E). This shows that admissibility of the connection means that the torsion can be represented as the sum of two tensors linear in ω_{H}.

3.2 Harmonic space

Let M be a half-flat Grassmann manifold. We denote by S_{H} the Sp(1, ℂ)-principal holomorphic bundle over M consisting of symplectic bases of H_{m}^{∗} ≅ H_{m} ≅ ℂ^{2} , m ∈ M,

S_{H} = {s = (h_{+}, h_{−}) | ω_{H}(h_{+}, h_{−}) = 1} .

The bundle S_{H} → M is called harmonic space [GIOS]. A parallel (local) section

m ↦ s_{m} = (h_{1}(m), h_{2}(m)) ∈ S_{H}

defines a trivialisation

M × Sp(1, ℂ) ≅ S_{H} ,

given by

(m, 𝒰) ↦ s_{m}𝒰 = (h_{+} = ∑_{α=1}^{2} h_{α}u_{+}^{α} , h_{−} = ∑_{α=1}^{2} h_{α}u_{−}^{α}) , 𝒰 = (\begin{matrix}u_{+}^{1} & u_{−}^{1} \\ u_{+}^{2} & u_{−}^{2} \end{matrix}) ; det 𝒰 = 1 .

We denote by ∂_{++}, ∂_{−−}, ∂_{0} the fundamental vector fields on S_{H} generated by the standard generators of Sp(1, ℂ),

∂_{++} ∼ (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) , ∂_{−−} ∼ (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , ∂_{0} ∼ (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

They satisfy the relations

[∂_{++} , ∂_{−−}] = ∂_{0} , [∂_{0} , ∂_{++}] = 2∂_{++} , [∂_{0} , ∂_{−−}] = −2∂_{−−} .

Consider Mat(2, ℂ), the vector space of two by two matrices. The matrix coefficients u_{±}^{α} are coordinates on this vector space. One can easily check that the vector fields u_{+}^{α} \frac{∂}{∂u_{−}^{α}} , u_{−}^{α} \frac{∂}{∂u_{+}^{α}} and u_{+}^{α} \frac{∂}{∂u_{+}^{α}} − u_{−}^{α} \frac{∂}{∂u_{−}^{α}} annihilate the function det 𝒰 = ϵ_{βγ}u_{+}^{β}u_{−}^{γ}, where ϵ_{βγ} are the matrix coefficients of the standard symplectic form of ℂ^{2}. Therefore these vector fields are tangent to the submanifold Sp(1, ℂ) = {det 𝒰 = 1} ⊂ Mat(2, ℂ). One can easily prove:

Lemma 2 In terms of the identification, S_{H} ≅ M × Sp(1, ℂ), the fundamental vector fields on S_{H} generated by the standard generators of Sp(1, ℂ) may be written:

∂_{++} = u_{+}^{α} \frac{∂}{∂u_{−}^{α}} , ∂_{−−} = u_{−}^{α} \frac{∂}{∂u_{+}^{α}} , ∂_{0} = u_{+}^{α} \frac{∂}{∂u_{+}^{α}} − u_{−}^{α} \frac{∂}{∂u_{−}^{α}} .

We say that a function f on S_{H} has charge c if ∂_{0}f = cf. The charge measures the difference in the degrees of homogeneity in u_{+} and u_{−}.

Note that any frame (h_{+}, h_{−}) ∈ S_{H} defines an isomorphism ℂ^{2} →^{∼} H_{m}^{∗} given by (z^{1}, z^{2}) ↦ z^{1}h_{+} + z^{2}h_{−}. This induces an isomorphism

𝔰𝔭(1, ℂ) = 𝔰𝔭(ℂ^{2}) ≅ S^{2}ℂ^{2} →^{∼} S^{2}H_{m}^{∗} = span_{ℂ}{h_{+}^{2} , h_{−}^{2} , h_{+} ∨ h_{−}} ,

where we have identified 𝔰𝔭(ℂ^{2}) with S^{2}ℂ^{2} using the symplectic form of ℂ^{2}. The generators of 𝔰𝔭(1, ℂ) corresponding to h_{+}^{2} , −h_{−}^{2} , −h_{+} ∨ h_{−} under this identification are precisely ∂_{++} , ∂_{−−} , ∂_{0} respectively.

3.3 Canonical distributions on harmonic space

Let S_{H} = {(h_{+} , h_{−})|h_{±} = u_{±}^{α}h_{α} , (u_{±}^{α}) ∈ Sp(1, ℂ)} be the harmonic space associated to a half-flat Grassmann manifold M. Here we have fixed a parallel symplectic frame (h_{1}, h_{2}) of H^{∗} which defines the trivialisation S_{H} = M × Sp(1, ℂ) of the holomorphic bundle S_{H}.

In particular, the matrix coefficients u_{±}^{α} of Sp(1, ℂ) will be considered as holomorphic functions on S_{H}. Together with local coordinates (x^{i}) of M, we obtain a system (x^{i}, u_{±}^{α}) of local (nonhomogeneous–homogeneous) coordinates on S_{H}.

For any section e ∈ Γ(E^{∗}) we define vector fields X_{±}^{e} ∈ 𝔛(S_{H}) by the formula

X_{±}^{e}|_{(h_{+},h_{−})} = { \atop e ⊗ h_{±}} ,

where Ỹ stands for the horizontal lift of a tangent vector Y on M with respect to the connection ∇^{H}. Since the frame h_{α} is parallel, this horizontal lift coincides with the horizontal lift with respect to the splitting S_{H} = M × Sp(1, ℂ). This shows that the vector fields X_{±}^{e} are tangent to M × {(h_{+}, h_{−})} and hence annihilate u_{±}^{α}. If h_{±} = u_{±}^{α}h_{α} then X_{±}^{e} = u_{±}^{α}X̃_{α}^{e}, where X_{α}^{e} := e ⊗ h_{α}.

There exists a canonical decomposition

T S_{H} = T^{v}S_{H} ⊕ 𝒟_{+} ⊕ 𝒟_{−}

of the (holomorphic) tangent bundle into the vertical sub-bundle T^{v}S_{H} and two (holomorphic) distributions 𝒟_{+} and 𝒟_{−} spanned respectively by vector fields X_{+}^{e} and X_{−}^{e} associated with sections e of the bundle E^{∗}. The vertical distribution T^{v}S_{H} is spanned by the vector fields ∂_{0} , ∂_{++} , ∂_{−−}, which correspond to the standard generators of the Lie algebra 𝔰𝔭(1, ℂ).

Lemma 3 The vector fields X_{±}^{e} ∈ 𝔛(S_{H}) satisfy the following commutation relations:

[∂_{0}, X_{±}^{e}] = ±X_{±}^{e} , [∂_{±±}, X_{±}^{e}] = 0 , [∂_{±±}, X_{∓}^{e}] = X_{±}^{e}
[X_{+}^{e}, X_{−}^{e′}] = X_{−}^{∇_{π∗X^{e}_{+}}e′} − X_{+}^{∇_{π∗X{e′ \atop −}} e} − T̃(π_{∗}X_{+}^{e}, π_{∗}X_{−}^{e′})
[X_{±}^{e}, X_{±}^{e′}] = X_{±}^{∇_{π∗X^{e}_{±}}e′} − X_{±}^{∇_{π∗X{e′ \atop ±}} e} − T̃(π_{∗}X_{±}^{e}, π_{∗}X_{±}^{e′}) ,  (3.2)

where T is the torsion of the Grassmann connection, T̃(X, Y) := { \atop T (X, Y)} denotes the horizontal lift of the vector T (X, Y) and we have used the abbreviation ∇_{X}e := ∇_{X}^{E}e.

Proof: The first three equations follow from X_{±}^{e} = u_{±}^{α}{ \atop e ⊗ h_{α}} and the expression for the fundamental vector fields given in Lemma 2. To prove the last equation we first compute the Lie bracket of two vector fields X = e ⊗ h and X′ = e′ ⊗ h on M, where h is parallel:

[X, X′] = ∇_{X}X′ − ∇_{X′}X − T (X, X′) = (∇_{X}e′ − ∇_{X′}e) ⊗ h − T (X, X′) .  (3.3)

Using this we calculate the commutator

[X_{±}^{e} , X_{±}^{e′}] = u_{±}^{α}u_{±}^{β} ({ \atop ∇_{X_{α}^{e}}X_{β}^{e′}} − { \atop ∇_{X_{β}^{e′}}X_{α}^{e}} − T̃(X_{α}^{e}, X_{β}^{e′}))
= (∇_{π_{∗}X_{±}^{e}}e′ ⊗ h_{±}) − (∇_{π_{∗}X_{±}^{e′}}e ⊗ h_{±}) − u_{±}^{α}u_{±}^{β}T̃(X_{α}^{e}, X_{β}^{e′})
= X_{±}^{∇_{π∗X^{e}_{±}}e′−∇_{π∗X{e′ \atop ±}} e} − T̃(π_{∗}X_{±}^{e}, π_{∗}X_{±}^{e′}) .

The expression for [X_{+}^{e}, X_{−}^{e′}] follows similarly. We shall use the abbreviation T (X_{±}^{e}, X_{±}^{e′}) := T̃(π_{∗}X_{±}^{e}, π_{∗}X_{±}^{e′}).

Proposition 2 The following conditions are equivalent:

i) For any parallel section h ∈ Γ(H^{∗}) the distribution E^{∗} ⊗ h on M is integrable. ii) The distribution 𝒟_{+} (associated to any parallel frame (h_{1}, h_{2})) on S_{H} is integrable. iii) The distribution 𝒟_{−} on S_{H} is integrable. iv) The holomorphic Grassmann structure is admissible, i.e. it has admissible connection.

Proof: The formula (3.3), where h is parallel shows that the distribution E^{∗}⊗h is integrable if and only if

T (E^{∗} ⊗ h, E^{∗} ⊗ h) ⊂ E^{∗} ⊗ h .  (3.4)

Using the decomposition (3.1), one can check that this condition is satisfied for all parallel sections h if and only if the connection is admissible. This proves the equivalence of i) and iv). Since π_{∗} (X_{+}^{e}|_{(h_{+},h_{−})}) = e ⊗ h_{+}, the last equation in (3.2) shows that the distribution 𝒟_{+} is integrable if and only if (3.4) holds for all h. Thus i) is equivalent to ii). The equivalence of i) and iii) is proved similarly.

4 Construction of half-flat connections over half-flat

Grassmann manifolds

4.1 Half-flat connections over half-flat Grassmann manifolds

In this section we describe the harmonic space method [GIOS] for constructing half-flat connections ∇ (Def. 6 below) in a holomorphic vector bundle ν : W → M over a complex manifold M with admissible half-flat holomorphic Grassmann structure. The basic ingredient of the construction is the lift of geometric data from M to S_{H} via π : S_{H} → M. Let ∇ be a holomorphic connection in a holomorphic vector bundle ν : W → M. Its curvature

F (e ⊗ h_{α} , e′ ⊗ h_{β}) = ω_{H}(h_{α}, h_{β})F^{(ee′)} + F_{αβ}^{[ee′]} ,  (4.1)

where (h_{1}, h_{2}) is the fixed parallel local frame of H^{∗} and e, e′ are local sections of E^{∗}. The curvature component F^{(ee′)} is symmetric in e, e′ and F_{αβ}^{[ee′]} is skew in e, e′ and symmetric in α, β. Lifting (4.1) to S_{H} we obtain the curvature of the pull-back connection π^{∗}∇ in π^{∗}ν : π^{∗}W → S_{H} with components, F (v , ·) = 0 , ∀v ∈ T^{v}S_{H}, together with

F (X_{±}^{e}, X_{±}^{e′}) = F_{±±}^{[ee′]} := u_{±}^{α}u_{±}^{β}F_{αβ}^{[ee′]}
F (X_{+}^{e}, X_{−}^{e′}) = F^{(ee′)} + F_{+−}^{[ee′]} := F^{(ee′)} + u_{+}^{α}u_{−}^{β}F_{αβ}^{[ee′]} .

Definition 6 A holomorphic connection ∇ in a holomorphic vector bundle ν : W → M over a complex manifold M with holomorphic Grassmann structure, T^{∗}M = E ⊗ H, is called half-flat if its curvature F satisfies the equation

F (e ⊗ h_{α} , e′ ⊗ h_{β}) = ω_{H}(h_{α}, h_{β})F^{(ee′)} ,  (4.2)

where (h_{1}, h_{2}) is a parallel local frame of H^{∗} and F^{(ee′)} is symmetric in the local sections e, e′ of E^{∗}.

Note that (4.2) is equivalent to (2.9). From this definition it follows that for any h ∈ H^{∗} we have F (e ⊗ h , e′ ⊗ h) = 0 .

Definition 7 A connection in a holomorphic vector bundle W → S_{H} over harmonic space S_{H} is called half-flat if its curvature F satisfies the equations

F (X_{+}^{e}, X_{+}^{e′}) = 0
F (X_{+}^{e}, X_{−}^{e′}) = F^{(ee′)}
F (X_{−}^{e}, X_{−}^{e′}) = 0
F (v , · ) = 0 , ∀v ∈ T^{v}S_{H} ,  (4.3)

where F^{(ee′)} is symmetric in the local sections e, e′ of E^{∗}.

Definition 8 Let ν : W → M be a holomorphic vector bundle and ∇ a connection in π^{∗}ν : π^{∗}W → S_{H}, where π : S_{H} → M. A local frame of π^{∗}ν defined on π^{−1}(U), where U is an open subset of M, is called a central frame with respect to ∇ if it is parallel along the fibres of the bundle π : S_{H} → M.

Remark: If χ = (χ_{1}, … χ_{r}) is a local frame of ν then π^{∗}χ will be a central frame with respect to the pull-back π^{∗}∇ of any connection ∇ in ν. The connection 1-form A of π^{∗}∇ with respect to the frame π^{∗}χ satisfies A(v) = 0 , A(X_{±}^{e}) = u_{±}^{α}A_{α}^{e}, where v is any vertical vector and A_{α}^{e} = A(X̃_{α}^{e}) is a matrix-valued function on M. Conversely, any connection satisfying these conditions is the pull-back of the connection over M with potential A(X_{α}^{e}) = A_{α}^{e}.

Proposition 3 Let π : S → M be any fibre bundle with simply connected fibres over a simply connected manifold M. There is a natural one-to-one correspondence between gauge equivalence classes of connections ∇^{M} in the trivial bundle ℂ^{r}×M and gauge equivalence classes of connections ∇^{S} in ℂ^{r}×S satisfying the curvature constraint F (v, ·) = 0 for all vertical vectors v.

Proof: It is clear that the pull-back ∇^{S} = π^{∗}∇^{M} to S of a connection ∇^{M} defined over M satisfies the curvature constraint. To prove the converse, we will apply the following elementary lemma to the connection 1-form A of a connection ∇ over N = S.

Lemma 4 Let π : N → M be a submersion with connected fibres and α a p-form on N. Then α is the pull-back π^{∗}β of a p-form β on M if and only if the inner products ι_{v}α = ι_{v}dα = 0 for all vertical tangent vectors v.

Since the connection ∇^{S} is flat along the (simply connected) fibres of π there exists a central frame ψ = (ψ_{1}, …, ψ_{r}) for ∇^{S}. Let A be the connection 1-form of ∇^{S} with respect to this central frame. We then have A(v) = 0 for any vertical vector v and the curvature condition F (v, ·) = 0 implies dA(v, ·) = 0. Now the above lemma shows that A is the pull-back of a 1-form B on M, which defines a connection ∇^{M} in the trivial bundle ℂ^{r}×M. Since any two central frames differ by a gauge transformation which is a matrix-valued function on M the connection ∇^{M} is well defined up to a gauge transformation. The pullback π^{∗}∇^{M} is gauge equivalent to ∇^{S} since it has the same expression with respect to the standard frame of ℂ^{r}×S (which is the pull-back of the standard frame of ℂ^{r}×M) as ∇^{S} with respect to the central frame ψ. It is clear that the pull-backs of gauge equivalent connections over M are gauge equivalent connections over S. Applying a gauge transformation to a connection ∇^{S} which has connection 1-form A with respect to a central frame ψ we get a new connection (∇^{S})′, which has the same connection form A with respect to the transformed frame ψ′. The frame ψ′ is therefore central with respect to (∇^{S})′ and the two connections ∇^{S} and (∇^{S})′ define the same gauge equivalence class of connections over M.

Proposition 4 Let ν : W = ℂ^{r} × M → M be a trivial vector bundle over a complex manifold M with admissible half-flat holomorphic Grassmann structure and π^{∗}ν : π^{∗}W = ℂ^{r} × S_{H} → S_{H} its pull-back to S_{H}. Then any half-flat connection over S_{H} is gauge equivalent to the pull-back of a half-flat connection over M.

Proof: It is clear that the pull-back of a half-flat connection is half-flat. To prove the converse, we apply Proposition 3, by which a half-flat connection ∇^{S} over S_{H} is gauge equivalent to a pull-back connection π^{∗}∇^{M}, which is necessarily half-flat. This implies that ∇^{M} is half-flat. In fact, if the connection ∇^{M} were not half-flat then it would have a nontrivial curvature component F_{αβ}^{[ee′]} which would imply that its pull-back π^{∗}∇^{M} has for instance a nonzero curvature component F_{++}^{[ee′]}. But this is impossible since π^{∗}∇^{M} is half-flat.

Corollary 1 The connection 1-form A of a half-flat connection over S_{H} with respect to a central frame ψ has the form

A(v) = 0 , A(X_{±}^{e}) = u_{±}^{α}A_{α}^{e} ,

where v is any vertical vector and A_{α}^{e} = A(X̃_{α}^{e}) is a matrix-valued function on M.

Remark: This shows that the half-flat connection is completely determined by the potential in the 𝒟_{+}–direction, A(X_{+}^{e}) = u_{+}^{α}A_{α}^{e}, with respect to a central frame. Proof: This follows from Proposition 4 and the remark following Definition 8.

4.2 The construction

In this section we construct half-flat connections in a bundle ν : W → M over a manifold M with a half-flat admissible Grassmann structure. First we define the weaker notion of an almost half-flat connection over S_{H} and show how to construct all such connections from appropriate prepotentials. Then we show that any almost half-flat connection over S_{H} may be used to construct a half-flat connection on M. Since our construction is local in M, we shall assume that the bundles π, ν and π^{∗}ν are trivial, i.e. π : M × Sp(1, ℂ) → M, ν : M × ℂ^{r} → M and π^{∗}ν : S_{H} × ℂ^{r} → S_{H}.

4.2.1 Construction of almost half-flat connections

The restriction of a half-flat connection to a leaf of the integrable distribution ⟨𝒟_{+}, ∂_{0}⟩ is clearly flat.

Definition 9 A frame φ_{1}, ..., φ_{r} in the holomorphic vector bundle π^{∗}ν : ℂ^{r} × S_{H} which is parallel along leaves of the integrable distribution ⟨𝒟_{+}, ∂_{0}⟩ is called an analytic frame.

With respect to an analytic frame a connection in the vector bundle π^{∗}ν has components

∇_{∂_{0}}^{S} = ∂_{0}
∇_{X_{+}^{e}}^{S} = X_{+}^{e}
∇_{∂_{++}}^{S} = ∂_{++} + A_{++} := ∂_{++} + A(∂_{++})
∇_{∂_{−−}}^{S} = ∂_{−−} + A_{−−} := ∂_{−−} + A(∂_{−−})
∇_{X_{−}^{e}}^{S} = X_{−}^{e} + A(X_{−}^{e}) .

Definition 10 A connection ∇^{S} over S_{H} is called almost half-flat if its curvature satisfies the following equations:

F (X_{+}^{e} , X_{+}^{e′}) = F (X_{+}^{e} , v) = 0 , ∀v ∈ T^{v}S_{H}
F (∂_{++} , ·) = F (∂_{0} , ·) = 0 .  (4.4)

In fact, these equations are not independent; for instance the Bianchi identity with arguments (X_{+}, ∂_{++}, ∂_{−−}) together with F (∂_{++}, ∂_{−−}) = F (∂_{±±}, X_{+}^{e}) = 0 implies the equation F (∂_{++}, X_{−}^{e}) = 0.

Proposition 5 Any almost half-flat connection satisfies the following equation:

F (X_{+}^{e}, X_{−}^{e′}) = F (X_{+}^{e′}, X_{−}^{e}) .

Proof: Using the integrability of 𝒟_{+} and F (X_{+}^{e}, X_{+}^{e′}) = F (∂_{−−}, X_{+}^{e}) = 0, we obtain

0 = [∇_{∂_{−−}}^{S} , F (X_{+}^{e}, X_{+}^{e′})]
= [∇_{∂_{−−}}^{S} , [∇_{X_{+}^{e}}^{S} , ∇_{X_{+}^{e′}}^{S}]] − [∇_{∂_{−−}}^{S} , ∇_{[X_{+}^{e} , X_{+}^{e′}]}^{S}]
= [∇_{X_{−}^{e}}^{S} , ∇_{X_{+}^{e′}}^{S}] + [∇_{X_{+}^{e}}^{S} , ∇_{X_{−}^{e′}}^{S}] − ∇_{[X_{−}^{e} , X_{+}^{e′}]}^{S} − ∇_{[X_{+}^{e} , X_{−}^{e′}]}^{S}
= F (X_{−}^{e}, X_{+}^{e′}) − F (X_{−}^{e′}, X_{+}^{e}) .

It follows that an almost half-flat connection is a generalisation of a half-flat connection, satisfying only those equations in (4.3), which involve curvatures with ∂_{0}, ∂_{++} or X_{+}^{e} in one of the arguments.

Proposition 6 An almost half-flat connection is half-flat if and only if it satisfies F (∂_{−−}, X_{−}^{e}) = 0.

Proof: By Proposition 5 an almost half-flat connection is required to satisfy all the halfflatness equations (4.3) with the exception of

F (∂_{−−}, X_{−}^{e}) = 0 and F (X_{−}^{e}, X_{−}^{e′}) = 0.  (4.5)

The second equation here follows from the first in virtue of the Bianchi identity with arguments (X_{+}^{e}, X_{−}^{e′}, ∂_{−−}).

The following proposition shows that an almost half-flat connection is completely determined by the potentials A_{++} and A_{−−} with respect to an analytic frame.

Proposition 7 Let ∇^{S} be an almost half-flat connection in the vector bundle π^{∗}ν : ℂ^{r}×S_{H} → S_{H} with potentials A_{++}, A_{−−} and A(X_{−}^{e}) in an analytic frame. Then:

(i) The potential A_{++} is analytic and has charge +2, i.e.

X_{+}^{e}A_{++} = 0 , ∂_{0}A_{++} = 2A_{++} .  (4.6)

(ii) The potential A_{−−} satisfies

∂_{++}A_{−−} − ∂_{−−}A_{++} + [A_{++}, A_{−−}] = 0 , ∂_{0}A_{−−} = −2A_{−−} .  (4.7)

(iii) The potential A(X_{−}^{e}) is determined by A_{−−} and has charge −1:

A(X_{−}^{e}) = −X_{+}^{e}A_{−−} , ∂_{0}A(X_{−}^{e}) = −A(X_{−}^{e}) .  (4.8)

Conversely, any matrix-valued potentials A_{++}, A_{−−} and A(X_{−}^{e}) satisfying (4.6), (4.7) and (4.8) define an almost half-flat connection.

Proof: (i) The curvature constraints F (X_{+}^{e} , ∂_{++}) = 0 , F (∂_{0} , ∂_{++}) = 0, in an analytic frame, take the form (4.6). (ii) The further almost half-flatness conditions, F (∂_{++} , ∂_{−−}) = F (∂_{0} , ∂_{−−}) = 0 give equations (4.7) for the potential A_{−−}. (iii) Having obtained A_{−−}, we can find A(X_{−}^{e}) from the equations F (X_{+}^{e} , ∂_{−−}) = F (∂_{0} , X_{−}^{e}) = 0 , which take the form

X_{+}^{e}A_{−−} = A([X_{+}^{e}, ∂_{−−}]) = −A(X_{−}^{e}) , ∂_{0}A(X_{−}^{e}) = −A(X_{−}^{e}) .  (4.9)

The second equation follows from the first.

We can now write an algorithm for the construction of all almost half-flat connections:

Theorem 4 Let A_{++} be an analytic prepotential, i.e. a matrix-valued function on a domain U = π^{−1}(V) ⊂ S_{H}, where V ⊂ M is a simply connected domain, satisfying (4.6). Let Φ be an invertible matrix-valued function on U which satisfies the equations

∂_{++}Φ = −A_{++}Φ , ∂_{0}Φ = 0 .  (4.10)

It always exists. The pair (A_{++}, Φ) determines an almost half-flat connection ∇^{S} = ∇^{(A_{++},Φ)}. Its potentials with respect to an analytic frame are given by A_{++}, A_{−−} = −(∂_{−−}Φ)Φ^{−1} and A(X_{−}^{e}) = −X_{+}^{e}A_{−−}. Conversely, any almost half-flat connection is of this form.

Proof: We consider the connection defined by A_{++} and A(∂_{0}) = 0 along an orbit sB of the Borel subgroup of SL(2, ℂ),

B = \{(\begin{matrix}t_{0} & t_{1} \\ 0 & t_{0}^{−1} \end{matrix})| t_{0} ∈ ℂ^{∗}, t_{1} ∈ ℂ\} ≅ ℂ^{∗} × ℂ (diffeomorphic) .

It is flat since the second equation of (4.6) is equivalent to F (∂_{0} , ∂_{++})=0 (vanishing of the curvature along sB). Moreover it has trivial holonomy since the fundamental group of B ≅ ℂ^{∗}×ℂ coincides with the fundamental group of the ℂ^{∗}-factor and the potential is zero in direction of ∂_{0} which is tangent to ℂ^{∗}. An invertible solution to the system (4.10) exits and defines a parallel frame Φ with respect to the flat connection with trivial holonomy defined along each orbit of the Borel group. Since the space of Borel orbits in U is diffeomorphic to V × ℂP^{1} and is therefore simply connected, a solution Φ exists on the domain U. Now, given any such solution of (4.10), we define A_{−−} := −(∂_{−−}Φ)Φ^{−1}. This solves (4.7), since F (∂_{±±}, ∂_{0}) = F (∂_{++}, ∂_{−−}) = 0 is the integrability condition for the system ∂_{±±}Φ = −A_{±±}Φ, ∂_{0}Φ = 0. Finally, we define A(X_{−}^{e}) := −X_{+}^{e}A_{−−}, obtaining an almost half-flat connection by Proposition 7. Now the converse statement follows also from Proposition 7.

4.2.2 Transformation to the central frame

Since an almost half-flat connection ∇ = ∇^{S} is flat in vertical directions, it admits a central frame ψ. The following lemma shows that the solution Φ of equation (4.10) gives a gauge transformation from an analytic frame φ to a central frame ψ = φΦ for the almost half-flat connection ∇^{(A_{++},Φ)}.

Lemma 5 Let ∇ = ∇^{(A_{++},Φ)} be the almost half-flat connection associated to the analytic prepotential A_{++} with respect to the analytic frame φ and an invertible solution Φ of (4.10). Then the frame ψ := φΦ is a central frame for the connection ∇, i.e. the potentials C(∂_{±±}) and C(∂_{0}) with respect to that frame vanish.

Proof: The result follows from the pure gauge form of A(∂_{±±}) and A(∂_{0}) and the transformation law (2.3) for potentials. With respect to the central frame ψ, the almost half flat connection constructed above takes the form:

∇_{X_{+}^{e}}^{S} = X_{+}^{e} + C(X_{+}^{e}) = X_{+}^{e} + Φ^{−1} X_{+}^{e} Φ
∇_{X_{−}^{e}}^{S} = X_{−}^{e} + C(X_{−}^{e}) = X_{−}^{e} + Φ^{−1}X_{−}^{e}Φ + Φ^{−1}X_{+}^{e} (∂_{−−}ΦΦ^{−1}) Φ
∇_{∂_{++}}^{S} = ∂_{++} , ∇_{∂_{−−}}^{S} = ∂_{−−} , ∇_{∂_{0}}^{S} = ∂_{0} .

Moreover, the equations F (∂_{++} , X_{+}^{e}) = F (∂_{0} , X_{+}^{e}) = 0 imply that the potential C(X_{+}^{e}) satisfies the equations

∂_{++}C(X_{+}^{e}) = 0 , ∂_{0}C(X_{+}^{e}) = C(X_{+}^{e}) .  (4.11)

4.2.3 Construction of half-flat connections

We assume now that the analytic prepotential A_{++} is defined globally along the fibres of π : S_{H} → M. Then, restricting M to an appropriate domain, we may assume that A_{++} is defined globally on S_{H}. The previous construction then provides an almost half-flat connection over S_{H}. Using this connection, we may construct a half-flat connection on M. The crucial point is the following:

Proposition 8 The potential C(X_{+}^{e}) of an almost half-flat connection ∇ with respect to a central frame is linear in u_{+}^{α}, namely,

C(X_{+}^{e}) = u_{+}^{α}C(X̃_{α}^{e}) =: u_{+}^{α}C_{α}^{e} ,  (4.12)

where (x^{i}, u_{±}^{α}) are the local coordinates associated with the trivialisation S_{H} = M ×Sp(1, ℂ) and C_{α}^{e} = C_{α}^{e}(x^{i}) is a matrix-valued function on M.

Proof: Due to equations (4.11), the result follows from:

Lemma 6 i) If a holomorphic function f_{+} , defined on some domain

U ⊂ \{u_{+}^{2} ≠ 0\} ⊂ Sp(1, ℂ) = \{𝒰 = (\begin{matrix}u_{+}^{1} & u_{−}^{1} \\ u_{+}^{2} & u_{−}^{2} \end{matrix}) ; det 𝒰 = 1\}

satisfies

∂_{++}f_{+} = 0 , ∂_{0}f_{+} = f_{+} ,  (4.13)

then f_{+} = u_{+}^{α}f_{α}(\frac{u_{+}^{1}}{u_{+}^{2}} ). Here f_{α}(\frac{u_{+}^{1}}{u_{+}^{2}} ) are holomorphic functions on U invariant under the right action of the Lie algebra of upper-triangular matrices. ii) Moreover, if the function f_{+} is globally defined, then it is linear in u_{+}^{α}, i.e. f_{+} = u_{+}^{α}f_{α}, f_{α} =const.

Proof: i) One can immediately check that f_{+} = u_{+}^{α}f_{α}(\frac{u_{+}^{1}}{u_{+}^{2}} ) is a solution of (4.13). We note that the quotient of any two solutions of (4.13) is a solution of the corresponding homogeneous system,

∂_{++}f = 0 , ∂_{0}f = 0 .  (4.14)

It is sufficient to check that any solution of (4.14) is a function of u_{+}^{1}/u_{+}^{2} . To prove this we use the local factorisation of Sp(1, ℂ) into the product of a Borel subgroup ℬ and a nilpotent subgroup as follows:

(\begin{matrix}u_{+}^{1} & u_{−}^{1} \\ u_{+}^{2} & u_{−}^{2} \end{matrix}) = (\begin{matrix}1 & 0 \\ c^{−1} & 1 \end{matrix}) (\begin{matrix}a & b \\ 0 & a^{−1} \end{matrix}) , ℬ = \{(\begin{matrix}a & b \\ 0 & a^{−1} \end{matrix})\} .

Then c = u_{+}^{1}/u_{+}^{2} and ∂_{0}, ∂_{++} are generators of the right action of ℬ. This implies that the solutions of (4.14) are precisely the local functions on Sp(1, ℂ) invariant under the right action of ℬ. In terms of the local coordinate system (a, b, c) on Sp(1, ℂ) such functions are functions of c = u_{+}^{1}/u_{+}^{2} alone. ii) The restriction V |_{Sp(1)} to Sp(1) of any irreducible Sp(1, ℂ)-module V of holomorphic functions is a (finite dimensional) irreducible Sp(1)-module of smooth functions on Sp(1). The condition (4.13) shows that f_{+} is a highest weight vector with weight +1. Hence f_{+} generates a 2-dimensional submodule ⟨f_{+}⟩ = span{f_{+}, f_{−} := ∂_{−−}f_{+}} of holomorphic functions. It remains to show that any two dimensional module of holomorphic functions on Sp(1, ℂ) is spanned by linear functions. We know two such modules, generated by the highest weight vectors u_{+}^{1} and u_{+}^{2} respectively. On the other hand, by the Peter-Weyl Theorem the multiplicity of the two-dimensional irreducible representation of Sp(1) in L^{2}(Sp(1)) is 2.

Using Proposition 8, with respect to a central frame, we can write ∇_{X_{+}^{e}} = X_{+}^{e} + u_{+}^{α}C_{α}^{e} where the coefficients C_{α}^{e} = C_{α}^{e}(x^{i}) are matrix valued functions of coordinates x^{i} on M. Using them we define a new connection in the trivial bundle ℂ^{r} × S_{H} over S_{H} by

\{ \begin{matrix}ˆ∇_{X_{±}^{e}} & = & X_{±}^{e} + u_{±}^{α}C_{α}^{e} \\ ˆ∇_{∂_{±±}} & = & ∂_{±±} , ˆ∇_{∂_{0}} = ∂_{0} . \end{matrix}

Our main result now follows:

Theorem 5 Let M be a complex manifold with a half-flat admissible Grassmann structure. Let A_{++} be an analytic prepotential, i.e. a solution of (4.6), and Φ an invertible solution of (4.10). Then the connection ˆ∇ = ˆ∇^{(A_{++},Φ)} constructed from the data (A_{++}, Φ) is a half-flat connection in the trivial vector bundle ℂ^{r} × S_{H} → S_{H} and it is the pull-back of the following half-flat connection ∇^{M} in the bundle ℂ^{r} × M → M:

∇_{X_{α}^{e}}^{M} = X_{α}^{e} + C_{α}^{e} .  (4.15)

Conversely, any half-flat connection over S (or M) is gauge equivalent to one obtained from the above construction.

Proof: The remark after Definition 8 shows that the connection ˆ∇ is the pull-back of the connection ∇^{M}. It suffices now to show that ∇^{M} is half-flat. Note that the connections ∇ and ˆ∇ coincide in the direction of X_{+}^{e}. Hence, using C_{+}^{e} := u_{+}^{α}C_{α}^{e}, we have

0 = F^{∇}(X_{+}^{e}, X_{+}^{e′}) = F ˆ∇(X_{+}^{e}, X_{+}^{e′})
= X_{+}^{e}C_{+}^{e′} − X_{+}^{e′}C_{+}^{e} + [C_{+}^{e} , C_{+}^{e′}] − C([X_{+}^{e} , X_{+}^{e′}])
= u_{+}^{α}u_{+}^{β} (X_{α}^{e}C_{β}^{e′} − X_{β}^{e′}C_{α}^{e} + [C_{α}^{e} , C_{β}^{e′}] − C([X_{α}^{e} , X_{β}^{e′}]))
= u_{+}^{α}u_{+}^{β}F^{∇^{M}} (X_{α}^{e} , X_{β}^{e′}) ,

since X_{+}^{e}u_{+}^{β} = 0. This shows that the curvature F^{∇^{M}} (X_{α}^{e} , X_{β}^{e′}) is skewsymmetric in α, β, i.e. it belongs to ∧^{2}H ⊗ S^{2}E ⊗ End W. In other words, the connection ∇^{M} is half-flat.

Conversely, let ∇^{S} be a half-flat connection over S_{H}. By Proposition 4 we may assume that it is a pull-back of a half-flat connection ∇^{M} over M. Since the restriction of ∇^{S} to the leaves of ⟨𝒟_{+}, ∂_{0}⟩ is flat, there exists an analytic frame (i.e. a frame such that A(X_{+}^{e}) = A(∂_{0}) = 0, in which the potential A(∂_{++}) satisfies the equations (4.6)). Since ∇^{S} is flat along the (simply-connected) fibres, there exists an invertible solution Φ to the system

∂_{++}Φ + A_{++}Φ = ∂_{−−}Φ + A_{−−}Φ = ∂_{0}Φ = 0 .

This shows that ∇^{S} is gauge-equivalent to the almost half-flat connection ∇^{(A_{++},Φ)} = ˆ∇^{(A_{++},Φ)}.

4.3 Application to hyper-Kähler manifolds with admissible torsion

The above construction can be applied to the complexification of hyper-Kähler manifolds. Recall that any hyper-Kähler manifold admits a (locally defined) Grassmann structure T^{∗ℂ}M = E ⊗ H, such that the Levi-Civita connection on the cotangent bundle ∇ = ∇^{E} ⊗ Id + Id ⊗ ∇^{H} is half-flat, i.e. the connection ∇^{H} is flat. Since the hyper-Kähler metric is Ricci flat, hence analytic, we may, using analytic continuation, extend the manifold M to a complex manifold M^{ℂ} with holomorphic extension of the hyper-Kähler structure, in particular, we have a holomorphic Ricci flat metric on M^{ℂ} with holonomy in Sp(n, ℂ) and half-flat Grassmann structure. This Grassmann structure is admissible since the Levi-Civita connection on M^{ℂ} has no torsion. Hence we can apply the harmonic space method to construct half-flat connections on holomorphic vector bundles W → M^{ℂ}. The complex version of Proposition 1 shows that such connections are Yang-Mills connections. More generally, the method of construction of half-flat connections extends to real analytic (possibly indefinite) hyper-Kähler manifolds with admissible torsion, i.e. with torsion which has zero component in S^{3}H ⊗ E^{∗} ⊗ ∧^{2}E. A hyper-Kähler manifold with admissible torsion is defined as a pseudo-Riemannian manifold (M, g) with a linear metric connection ∇ with holonomy in Sp(k, l) which has admissible torsion. As in the (torsionfree) hyper-Kähler case there exists a parallel 4-form given by Ω = ∑_{α} ω_{α} ∧ ω_{α}, ω_{α} := g(J_{α}·, ·), and half-flat connections are characterised as connections with curvature in V_{λ_{1}} ⊗ End W, where V_{λ_{1}} is the λ_{1}-eigenspace of the endomorphism B_{Ω} associated to Ω. If the form Ω is co-closed then any half-flat connection will be (Ω, λ_{1})-self-dual and thus a Yang-Mills connection. We remark that co-closedness of Ω is equivalent to a linear Sp(k, l)-invariant condition on the torsion.

5 Generalisation to higher-spin Grassmann manifolds

5.1 Higher-spin Grassmann structures

The construction discussed in the previous section is in fact the m=1 specialisation of a more general construction of connections on spin \frac{m}{2} Grassmann manifolds, which we discuss in this section. These manifolds were considered in [AG].

Definition 11 A spin \frac{m}{2} Grassmann structure on a (complex) manifold M is a holomorphic Grassmann structure of the form T^{∗}M ≅ E ⊗ F = E ⊗ S^{m}H, with a holomorphic Grassmann connection ∇ = ∇^{E} ⊗ Id + Id ⊗ ∇^{F}, where H is a rank 2 holomorphic vector bundle over M with holomorphic symplectic connection ∇^{H} and symplectic form ω_{H} ∈ Γ(∧^{2}H), and ∇^{F} is the connection in F = S^{m}H induced by ∇^{H}. M is called half-flat if the connection ∇^{F} is flat.

The bundle S^{m}H is associated with the spin \frac{m}{2} representation of the group Sp(1, ℂ). Any frame (h_{1}, h_{2}) for H^{∗} defines a frame for S^{m}H^{∗}, (h_{A} := h_{α_{1}}h_{α_{2}} ⋯ h_{α_{m}}), where the multiindex A := α_{1}α_{2} … α_{m} , α_{i} = 1, 2. The ∇^{H}-parallel symplectic form ω_{H} on H^{∗} induces a bilinear form ω_{H}^{m} on F^{∗} = S^{m}H^{∗} given by

ω_{H}^{m}(h_{A}, h_{B}) := 𝔖_{A} 𝔖_{B} ω_{H}(h_{α_{1}}, h_{β_{1}})ω_{H}(h_{α_{2}}, h_{β_{2}}) ⋯ ω_{H}(h_{α_{m}}, h_{β_{m}}) ,

where 𝔖_{A} denotes the sum over all permutations of the α’s. This form is skew-symmetric if m is odd and symmetric if m is even. To any section e ∈ Γ(E^{∗}) and multi-index A we associate the vector field X_{A}^{e} := e ⊗ h_{A} on M.

The construction of half-flat connections described in section 4.2 may be adapted to obtain certain ‘partially flat’ connections in vector bundles W → M, provided that the torsion of ∇ obeys certain admissibility conditions.

Definition 12 Let (M, ∇) be a half-flat spin \frac{m}{2} Grassmann manifold. For any section e ∈ Γ(E^{∗}) we define vector fields

\begin{aligned}X_{(m−2i)+}^{e} & := & u_{−}^{α_{1}} ⋯ u_{−}^{α_{i}}u_{+}^{α_{i+1}} ⋯ u_{+}^{α_{m}}X_{A}^{e} if m − 2i ≥ 0 \\ X_{(2i−m)−}^{e} & := & u_{−}^{α_{1}} ⋯ u_{−}^{α_{i}}u_{+}^{α_{i+1}} ⋯ u_{+}^{α_{m}}X_{A}^{e} if m − 2i < 0 \end{aligned}

on the principal bundle S_{H} of symplectic frames in H. The distribution spanned by these vector fields is denoted by 𝒟_{k+} := ⟨X_{k+}^{E}⟩ for k≥0 , k ≡ m mod 2 and 𝒟_{k−} := ⟨X_{k−}^{E}⟩ for k>0 , k ≡ m mod 2. We define also

𝒟_{(±)}^{k} := ⨁_{i=0}^{k} 𝒟_{(m−2i)±} .

The Grassmann connection ∇ is called k–admissible if it preserves the distribution 𝒟_{(±)}^{k}, i.e.

T (𝒟_{(±)}^{k}, 𝒟_{(±)}^{k}) ⊂ 𝒟_{(±)}^{k} .  (5.1)

The Grassmann manifold (M, ∇) is called k–admissible if the connection ∇ is k–admissible.

For small m we shall write X_{0}^{e}, X_{+}^{e}, X_{−}^{e}, X_{++}^{e} etc. instead of X_{0+}^{e}, X_{1+}^{e}, X_{1−}^{e}, X_{2+}^{e} etc.

Proposition 9 Let (M, ∇) be a half-flat spin \frac{m}{2} Grassmann manifold. Then the distribution 𝒟_{(±)}^{k} is integrable if and only if the torsion of the Grassmann connection ∇ satisfies equation (5.1).

The proof is similar to that of Proposition 2.

5.2 Partially flat connections over higher-spin Grassmann manifolds

Let (M, ∇) be a half-flat spin \frac{m}{2} Grassmann manifold and ν : W → M a holomorphic vector bundle. Since our constructions are local we will assume that W is trivial. In the higher spin (m > 1) case, there exists, as a natural generalisation of the notion of a halfflat connection, the more refined notion of a k-partially flat connection in ν. The space of two-forms ∧^{2}T^{∗}M has the following decomposition into GL(E) ⊗ Sp(1, ℂ)-submodules:

∧^{2}T^{∗}M = ∧^{2}(E ⊗ S^{m}H) = ∧^{2}E ⊗ S^{2}S^{m}H ⊕ S^{2}E ⊗ ∧^{2}S^{m}H ,

where

S^{2}S^{m}H = S^{2m}H ⊕ ω_{H}^{2}S^{2m−4}H ⊕ ⋯ ⊕ ω_{H}^{2[ \frac{m}{2} ]}S^{2m−4[ \frac{m}{2} ]}H
∧^{2}S^{m}H = ω_{H}S^{2m−2}H ⊕ ω_{H}^{3}S^{2m−6}H ⊕ ⋯ ⊕ ω_{H}^{2[ \frac{m}{2} ]+1}S^{2m−4[ \frac{m}{2} ]−2}H .

Here we use the convention that S^{l}H = 0 if l < 0.

Let ∇ be a connection in the vector bundle W → M. Its curvature has the following decomposition, corresponding to the above decomposition of ∧^{2}T^{∗}M into irreducible GL(E)·Sp(1, ℂ)-submodules:

F (X_{A}^{e}, X_{B}^{e′}) = 𝔖_{A} 𝔖_{B} ∑_{k=0}^{[m/2]}(ω_{H}(h_{α_{1}}, h_{β_{1}}) ⋯ ω_{H}(h_{α_{2k}}, h_{β_{2k}}) F^{(2k)} _{α_{2k+1}...α_{m}β_{2k+1}...β_{m}}^{[ee′]}
+ω_{H}(h_{α_{1}}, h_{β_{1}}) ⋯ ω_{H}(h_{α_{2k+1}}, h_{β_{2k+1}}) F^{(2k+1)} _{α_{2k+2}...α_{m}β_{2k+2}...β_{m}}^{(ee′)}), (5.2)

where the tensors F^{(2k)}∈ Γ(∧^{2}E ⊗ S^{2m−4k}H) and F^{(2k+1)}∈ Γ(S^{2}E ⊗ S^{2m−4k−2}H).

We note that half-flat connections are those which satisfy the conditions

F^{(2i)}= 0 , for all i ∈ ℕ.  (5.3)

For m>1 these conditions are not suitable for application of the harmonic space method. However, the following more refined restrictions on the curvature are amenable to the method (c.f. [DN1]).

Definition 13 A connection ∇ in the vector bundle ν : W → M is called k-partially flat if F^{(i)}= 0 for all i ≤ 2k. Here 0 ≤ k ≤ [\frac{m+2}{2}].

Clearly, [\frac{m+2}{2}]-partially flat connections are simply flat connections. We note that for m=1 , 0-partially flat connections are precisely half-flat connections. For general odd m=2p+1 , 0-partially flat connections in a vector bundle ν over flat spaces with spin \frac{m}{2} Grassmann structure were considered by Ward [W]. He chose E to be a rank 2 flat bundle and showed that 0-partially flat connections, for m>1, do not correspond to Yang-Mills connections. Therefore, in our more general setting, we clearly cannot expect 0-partially flat connections to satisfy the Yang-Mills equations for m>1. On the other hand, the penultimate case, k = [\frac{m}{2}], is particularly interesting for odd m:

Theorem 6 Let M be a half-flat spin \frac{m}{2} Grassmann manifold M. If m is odd and the vector bundle E^{∗} → M admits a ∇^{E}-parallel symplectic form ω_{E}, then M has canonical Sp(E)·Sp(H)-invariant metric g = ω_{E} ⊗ ω_{H}^{m} and 4-form Ω ≠ 0. If Ω is co-closed with respect to the metric g, then any \frac{m−1}{2}-partially flat connection ∇ in a vector bundle W over M is (Ω, λ)-self-dual and hence it is a Yang-Mills connection.

Proof: To describe Ω we use the following notation: e_{a} is a basis of E^{∗}, h_{α} is a basis of H^{∗}, h_{A} is the corresponding basis of S^{m}H^{∗} and X_{aA} := e_{a} ⊗ h_{A} the corresponding basis of T M = E^{∗} ⊗ S^{m}H^{∗}. With respect to these bases, the skew symmetric forms ω_{E}, ω_{H} and ω_{H}^{m} are represented by the matrices ω_{ab}, ω_{αβ} and ω_{AB} respectively. We define Ω by

Ω := ∑ ω_{ab} ω_{cd} ω_{AC} ω_{BD} X^{aA} ∧ X^{bB} ∧ X^{cC} ∧ X^{dD} ,

where X^{aA} is the basis dual to X_{aA}. This form is obviously Sp(E)·Sp(H)-invariant since we used only ω_{E} and ω_{H} in the definition. One can easily check that Ω ≠ 0. The connection ∇ is \frac{m−1}{2}-partially flat if and only if its curvature F belongs to the space

S^{2}E⊗ω_{H}^{m}⊗ End W ⊂ S^{2}E⊗ ∧^{2} S^{m}H⊗ End W ⊂ ∧^{2} (E⊗S^{m}H) ⊗ End W .

Here we use the decomposition

∧^{2}S^{m}H = ω_{H}S^{2m−2}H ⊕ ω_{H}^{3}S^{2m−6}H ⊕ … ⊕ ℂω_{H}^{m} .

The Sp(E)·Sp(H)-submodule S^{2}E ⊗ω_{H}^{m} ⊂ ∧^{2}T^{∗}M is irreducible. Therefore it is contained in an eigenspace V_{λ} of the Sp(E)·Sp(H)-invariant operator B_{Ω} : ∧^{2}T^{∗}M → ∧^{2}T^{∗}M. It remains to check that λ ≠ 0. By Lemma 1 it suffices to compute the contraction K = K_{cCdD}X^{cC}X^{dD} of a tensor S = S_{ab}ω_{AB}e^{a}e^{b} ⊗ h^{A}h^{B} in S^{2}E ⊗ ω_{H}^{m} with Ω:

−K_{cCdD} = S^{ab}ω^{AB}(ω_{ab}ω_{cd}ω_{AC}ω_{BD} + ω_{ac}ω_{db}ω_{AD}ω_{CB} + ω_{ad}ω_{bc}ω_{AB}ω_{DC} − ω_{ba}ω_{cd}ω_{BC}ω_{AD}
−ω_{bc}ω_{da}ω_{BD}ω_{CA} − ω_{bd}ω_{ac}ω_{BA}ω_{DC} − ω_{ca}ω_{bd}ω_{CB}ω_{AD} − ω_{cb}ω_{da}ω_{CD}ω_{BA}
−ω_{cd}ω_{ab}ω_{CA}ω_{DB} + ω_{da}ω_{bc}ω_{DB}ω_{AC} + ω_{db}ω_{ca}ω_{DC}ω_{BA} + ω_{dc}ω_{ab}ω_{DA}ω_{CB})
= 4(m + 1)S_{cd}ω_{CD} .

Here ω^{ab} and ω^{AB} denote the inverses of ω_{ab} and ω_{AB} and S^{ab} = ω^{aa′}ω^{bb′}S_{a′b′}. We have used that S^{ab}ω^{AB} is skew symmetric under interchange of aA with bB and that ω^{AB}ω_{AB}= − (m+1). The above calculation shows that λ=−4(m+1) ≠ 0 and hence any \frac{m−1}{2}-partially flat connection is (Ω, λ)-self-dual and is a Yang-Mills connection by Theorem 1. The analogous result does not hold if m is even.

Proposition 10 If m is even and the vector bundle E^{∗}→M admits a ∇^{E}-parallel metric γ_{E}, then M has canonical SO(E)·Sp(H)-invariant metric g = γ_{E} ⊗ ω_{H}^{m} and 4-form Ω ≠ 0.

Proof: Analogously to the case of m odd, we can define Ω by

Ω := ∑ γ_{ab} γ_{cd} ω_{AC} ω_{BD} X^{aA} ∧ X^{bB} ∧ X^{cC} ∧ X^{dD} .

Here γ_{ab} = γ_{E}(e_{a}, e_{b}) and we recall that for even m the bilinear form ω_{H}^{m} is symmetric: ω_{AB} = ω_{BA}. For even m, a connection ∇ in a vector bundle W over M is \frac{m}{2}-partially flat if and only if its curvature F belongs to the space

(∧^{2}E⊗ω_{H}^{m} ⊕ S^{2}E⊗S^{2}H⊗ω_{H}^{m−1})⊗ End W
⊂ (∧^{2}E⊗S^{2}S^{m}H ⊕ S^{2}E⊗ ∧^{2} S^{m}H)⊗ End W = ∧^{2} (E⊗S^{m}H) ⊗ End W.

The SO(E)·Sp(H)-submodule ∧^{2}E ⊗ω_{H}^{m}⊕S^{2}E ⊗S^{2}H ⊗ω_{H}^{m−1} ⊂ ∧^{2}T^{∗}M is not irreducible, so unlike the odd m case we cannot conclude that it is contained in an eigenspace V_{λ} of the SO(E)·Sp(H)-invariant operator B_{Ω} : ∧^{2}T^{∗}M → ∧^{2}T^{∗}M. In fact, examples are known (see Appendix B of [DN2]) where B_{Ω} has different eigenvalues on each irreducible summand of ∧^{2}T^{∗}M. Therefore in the case of even m we cannot expect that \frac{m}{2}-partial flatness implies the Yang-Mills equations.

5.3 Construction of partially flat connections over higher spin Grassmann manifolds

Now we generalise the construction of half-flat connections over admissible half-flat Grassmann manifolds to the case of k–partially flat connections over k–admissible higher spin Grassmann manifolds M. The natural extension of the harmonic construction given in section 4.2 yields k-partially flat connections in the vector bundle ν over the k–admissible spin \frac{m}{2} Grassmann manifold M. Again, we lift the geometric data from M to S_{H} via the projection π : S_{H} → M. The pull back π^{∗}∇ of a k–partially flat connection ∇ in the trivial vector bundle ν : W = ℂ^{r} × M → M is a connection in the vector bundle π^{∗}ν : π^{∗}W → S_{H} which satisfies equations defining the notion of a k–partially flat gauge connection on S_{H}. One can also define the weaker notion of an almost k–partially flat connection in π^{∗}ν : π^{∗}W → S_{H}. The latter may be constructed from a prepotential and it affords the construction of a k-partially flat connection in the bundle W → M. To simplify our exposition we explain the construction in the m=3 case. Here the decomposition (5.2) of the curvature tensor takes the form

F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′}) = 𝔖_{A} 𝔖_{B} (F^{(0)} _{α_{1}α_{2}α_{3}β_{1}β_{2}β_{3}}^{[ee′]} + ω_{H}(h_{α_{1}}, h_{β_{1}}) F^{(1)} _{α_{2}α_{3}β_{2}β_{3}}^{(ee′)}

+ω_{H}(h_{α_{1}}, h_{β_{1}})ω_{H}(h_{α_{2}}, h_{β_{2}}) F^{(2)} _{α_{3}β_{3}}^{[ee′]}

+ω_{H}(h_{α_{1}}, h_{β_{1}})ω_{H}(h_{α_{2}}, h_{β_{2}})ω_{H}(h_{α_{3}}, h_{β_{3}}) F^{(3)} ^{(ee′)}) . (5.4)

In this case we have two nontrivial notions of partial flatness:

0–partial flatness : F^{(0)}= 0  (5.5)
1–partial flatness : F^{(0)}=F^{(1)}=F^{(2)}= 0 .  (5.6)

Clearly, 2-partial flatness is tantamount to flatness. By Theorem 6, a 1-partially flat connection is a Yang-Mills connection.

5.3.1 Construction of 0-partially flat connections

Let M be a 0–admissible spin \frac{3}{2} Grassmann manifold with a 0-partially flat connection ∇ (satisfying (5.5)) in a holomorphic vector bundle W → M. The pull-back of such a connection ∇ to a connection in the bundle π^{∗}W → S_{H}, where π : S_{H} → M, has curvature F with components given by:

F (X_{±±±}^{e}, X_{±±±}^{e′}) = 0
F (X_{±±±}^{e′}, X_{±}^{e}) = u_{±}^{α_{1}}u_{±}^{α_{2}}u_{±}^{α_{3}}u_{±}^{β_{1}}u_{±}^{β_{2}}u_{∓}^{β_{3}}F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′})
= ±12 F^{(1)} _{α_{1}α_{2}β_{1}β_{2}}^{(ee′)}u_{±}^{α_{1}}u_{±}^{α_{2}}u_{±}^{β_{1}}u_{±}^{β_{2}} =: ±12 F^{(1)} _{±±±±}^{(ee′)}
F (X_{±}^{e}, X_{±}^{e′}) = u_{±}^{α_{1}}u_{±}^{α_{2}}u_{∓}^{α_{3}}u_{±}^{β_{1}}u_{±}^{β_{2}}u_{∓}^{β_{3}}F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′})
= −8 F^{(2)} _{α_{1}β_{1}}^{[ee′]}u_{±}^{α_{1}}u_{±}^{β_{1}} =: −8 F^{(2)} _{±±}^{[ee′]}
F (X_{+++}^{e}, X_{−−−}^{e′}) = u_{+}^{α_{1}}u_{+}^{α_{2}}u_{+}^{α_{3}}u_{−}^{β_{1}}u_{−}^{β_{2}}u_{−}^{β_{3}}F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′})
= 36 (F^{(1)} _{α_{1}α_{2}β_{1}β_{2}}^{(ee′)}u_{+}^{α_{1}}u_{+}^{α_{2}}u_{−}^{β_{1}}u_{−}^{β_{2}}+ F^{(2)} _{α_{1}β_{1}}^{[ee′]}u_{+}^{α_{1}}u_{−}^{β_{1}}+ F^{(3)} ^{(ee′)})
=: 36 (F^{(1)} _{0}^{(ee′)}+ F^{(2)} _{0}^{[ee′]}+ F^{(3)} _{0}^{(ee′)})
F (X_{±±±}^{e}, X_{∓}^{e′}) = u_{±}^{α_{1}}u_{±}^{α_{2}}u_{±}^{α_{3}}u_{±}^{β_{1}}u_{∓}^{β_{2}}u_{∓}^{β_{3}}F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′})
= ±24 F^{(1)} _{±±}^{(ee′)} + 12 F^{(2)} _{±±}^{[ee′]}
F (X_{+}^{e}, X_{−}^{e′}) = u_{+}^{α_{1}}u_{+}^{α_{2}}u_{−}^{α_{3}}u_{+}^{β_{1}}u_{−}^{β_{2}}u_{−}^{β_{3}}F (X_{α_{1}α_{2}α_{3}}^{e}, X_{β_{1}β_{2}β_{3}}^{e′})
= 12 F^{(1)} _{0}^{(ee′)} − 4 F^{(2)} _{0}^{[ee′]} − 12 F^{(3)} _{0}^{(ee′)}
F (v , ·) = 0 ,  (5.7)

where v, v′ are vertical vector fields on S_{H}. The form of these components lead us to:

Definition 14 A connection in a holomorphic vector bundle W → S_{H} is 0-partially flat if its curvature satisfies the equations,

F (X_{±±±}^{e}, X_{±±±}^{e′}) = 0
F (X_{±±±}^{e}, X_{±}^{e′}) = F (X_{±±±}^{e′}, X_{±}^{e})
F (X_{±}^{e}, X_{±}^{e′}) = −F (X_{±}^{e′}, X_{±}^{e})
F (v , ·) = 0 , ∀v ∈ T^{v}S_{H} .

The restriction of a 0-partially flat connection to a leaf of the integrable distribution ⟨𝒟_{3+}, ∂_{0}⟩ is clearly flat. In this case, an analytic frame in the holomorphic vector bundle π^{∗}ν : ℂ^{r} ×S_{H} → S_{H} is a frame which is parallel along leaves of this integrable distribution.

With respect to such a frame, a connection in the vector bundle π^{∗}ν can be written as

\{ \begin{matrix}∇_{∂_{0}}^{S} & = ∂_{0} \\ ∇_{X_{+++}^{e}}^{S} & = X_{+++}^{e} \\ ∇_{X_{±}^{e}}^{S} & = X_{±}^{e} + A(X_{±}^{e}) \\ ∇_{∂_{±±}}^{S} & = ∂_{±±} + A(∂_{±±}) \\ ∇_{X_{−−−}^{e}}^{S} & = X_{−−−}^{e} + A(X_{−−−}^{e}) . \end{matrix}

Definition 15 A connection ∇^{S} over S_{H} is called almost 0-partially flat if its curvature satisfies the following equations:

F (X_{+++}^{e}, X_{+++}^{e′}) = F (X_{+++}^{e}, v) = F (X_{±}^{e}, v) = 0 , ∀v ∈ T^{v}S_{H} ,
F (∂_{++}, ·) = F (∂_{0}, ·) = 0 .  (5.8)

Following the construction of almost half-flat connections, we may construct almost 0-partially flat connections, which allow deformation to a 0-partially flat connection. As in the case of a half-flat connection (c.f. Proposition 7), an almost 0-partially flat connection is completely determined by the potentials A_{±±} =: A(∂_{±±}) with respect to an analytic frame.

Proposition 11 Let ∇ be an almost 0-partially flat connection in the vector bundle π^{∗}ν : ℂ^{r}×S_{H} → S_{H} with potentials A_{++}, A_{−−}, A(X_{±}^{e}) and A(X_{−−−}^{e}) in an analytic frame. Then:

(i) The potential A_{++} is analytic and has charge 2, i.e.

X_{+++}^{e}A_{++} = 0 , ∂_{0}A_{++} = 2A_{++} .  (5.9)

(ii) The potential A_{−−} satisfies

∂_{++}A_{−−} − ∂_{−−}A_{++} + [A_{++}, A_{−−}] = 0 , ∂_{0}A_{−−} = −2A_{−−} .  (5.10)

(iii) The potentials A(X_{±}^{e}) and A(X_{−−−}^{e}) are then recursively determined as follows:

A(X_{+}^{e}) = −\frac{1}{3}X_{+++}^{e}A_{−−}
A(X_{−}^{e}) = \frac{1}{2} (∂_{−−}A(X_{+}^{e}) − X_{+}^{e}A_{−−} + [A_{−−}, A(X_{+}^{e})])
A(X_{−−−}^{e}) = ∂_{−−}A(X_{−}^{e}) − X_{−}^{e}A_{−−} + [A_{−−}, A(X_{−}^{e})]  (5.11)

and they have charges +1, −1 and −3 respectively, i.e.

∂_{0}A(X_{±}^{e}) = ±A(X_{±}^{e}) , ∂_{0}A(X_{−−−}^{e}) = −3A(X_{−−−}^{e}) .  (5.12)

Conversely, any set of matrix-valued potentials A_{++}, A_{−−}, A(X_{±}^{e}) and A(X_{−−−}^{e}) satisfying (5.9)–(5.12) define an almost 0-partially flat connection.

Proof: (i) The curvature constraints F (X_{+}^{e} , ∂_{++}) = 0 , F (∂_{0} , ∂_{++}) = 0, in an analytic frame, take the form (5.9). (ii) The further almost 0-partial-flatness conditions, F (∂_{++} , ∂_{−−}) = F (∂_{0} , ∂_{−−}) = 0 give equations (5.10) for the potential A_{−−}. (iii) Having obtained A_{−−}, we can find A(X_{±}^{e}) and A(X_{−−−}^{e}) from the equations

F (∂_{−−}, X_{+++}^{e}) = 0 ⇔ −X_{+++}^{e}A_{−−} = A([∂_{−−}, X_{+++}^{e}]) = 3A(X_{+}^{e})
F (∂_{−−}, X_{+}^{e}) = 0 ⇔ ∂_{−−}A(X_{+}^{e})−X_{+}^{e}A_{−−}+[A_{−−}, A(X_{+}^{e})]=A([∂_{−−}, X_{+}^{e}]) = 2A(X_{−}^{e})
F (∂_{−−}, X_{−}^{e}) = 0 ⇔ ∂_{−−}A(X_{−}^{e})−X_{−}^{e}A_{−−}+[A_{−−}, A(X_{−}^{e})]=A([∂_{−−}, X_{−}^{e}]) = A(X_{−−−}^{e}) .

Equations (5.12) follow from (5.11). Now, using this proposition, a modification of Theorem 4 gives an algorithm for the construction of all almost 0-partially flat connections.

Theorem 7 Let A_{++} be an analytic prepotential, i.e. a matrix-valued function on a domain U = π^{−1}(V) ⊂ S_{H}, V ⊂ M a simply connected domain, satisfying (5.9), and Φ an invertible matrix-valued function on U which satisfies the equations

∂_{++}Φ = −A_{++}Φ , ∂_{0}Φ = 0 .  (5.13)

Such a function Φ always exists. Then the pair (A_{++}, Φ) determines an almost 0-partially flat connection ∇^{S} = ∇^{(A_{++},Φ)}. Its potentials with respect to an analytic frame are given by A_{++}, A_{−−} = −(∂_{−−}Φ)Φ^{−1} and (5.11). Conversely, any almost 0-partially flat connection is of this form.

The proof follows that for Theorem 4 and uses Proposition 11.

To deform an almost 0-partially flat connection into a 0-partially flat connection, we need to find a transformation from the above analytic frame to a central frame. Analogously to Lemma 5 we may prove:

Lemma 7 Let ∇ = ∇^{(A_{++},Φ)} be the almost 0-partially flat connection associated to the analytic prepotential A_{++} with respect to the analytic frame φ and an invertible solution Φ of (5.13). Then the frame ψ := φΦ is a central frame for the connection ∇, i.e. the potentials C(∂_{±±}) and C(∂_{0}) with respect to that frame vanish.

With respect to the central frame ψ, the above almost 0-partially flat connection then takes the form:

\{ \begin{matrix}∇_{X_{+++}^{e}}^{S} & = X_{+++}^{e} + C(X_{+++}^{e}) = X_{+++}^{e} + Φ X_{+++}^{e} Φ^{−1} \\ ∇_{X_{±}^{e}}^{S} & = X_{±}^{e} + C(X_{±}^{e}) \\ ∇_{X_{−−−}^{e}}^{S} & = X_{−−−}^{e} + C(X_{−−−}^{e}) \\ ∇_{∂_{++}}^{S} & = ∂_{++} , ∇_{∂_{−−}}^{S} = ∂_{−−} , ∇_{∂_{0}}^{S} = ∂_{0} . \end{matrix}

where in terms of the analytic frame potentials A(X), the central frame potentials C(X) are given by C(X) = Φ^{−1}A(X)Φ + Φ^{−1}(XΦ). Moreover, the equations F (∂_{++} , X_{+++}^{e}) = F (∂_{0} , X_{+++}^{e}) = 0 imply that the potential C(X_{+++}^{e}) satisfies the equations

∂_{++}C(X_{+++}^{e}) = 0 , ∂_{0}C(X_{+++}^{e}) = 3C(X_{+++}^{e}) .  (5.14)

The following proposition is analogous to Proposition 8 in the half-flat case.

Proposition 12 The potential C(X_{+++}^{e}) of an almost 0-partially flat connection ∇ with respect to a central frame is cubic in u_{+}^{α},

C(X_{+++}^{e}) = u_{+}^{α}u_{+}^{β}u_{+}^{γ}C({ \atop X_{αβγ}^{e}}) = u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e} ,  (5.15)

where the coefficients C_{αβγ}^{e} = C_{αβγ}^{e}(x^{i}), symmetric in α, β, γ, are matrix valued functions of coordinates x^{i} on M and (x^{i}, u_{±}^{α}) are the local coordinates associated with the trivialisation S_{H} = M × Sp(1, ℂ).

With respect to a central frame, we can therefore write ∇_{X_{+++}^{e}} = X_{+++}^{e} + u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e}. Using C_{αβγ}^{e}, we now define a new connection in π^{∗}ν over S_{H} by

\{ \begin{matrix}ˆ∇_{X_{+++}^{e}} & = X_{+++}^{e} + u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{+}^{e}} & = X_{+}^{e} + u_{+}^{α}u_{+}^{β}u_{−}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{−}^{e}} & = X_{−}^{e} + u_{−}^{α}u_{−}^{β}u_{+}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{−−−}^{e}} & = X_{−−−}^{e} + u_{−}^{α}u_{−}^{β}u_{−}^{γ}C_{αβγ}^{e} \\ ˆ∇_{∂_{++}} & = ∂_{++} , ˆ∇_{∂_{−−}} = ∂_{−−} , ˆ∇_{∂_{0}} = ∂_{0} . \end{matrix}

The following theorem is the analogue of Theorem 5 in the half-flat case.

Theorem 8 The constructed connection ˆ∇ is a 0-partially flat connection in π^{∗}ν over S_{H} and it is the pull-back of the following 0-partially flat connection ∇^{M} in ν over M:

∇_{X_{αβγ}^{e}}^{M} = X_{αβγ}^{e} + C_{αβγ}^{e} .

Proof: As in Lemma 1 we may show that the connection ˆ∇ is the pull-back of the connection ∇^{M}. It then suffices to show that ∇^{M} is 0-partially flat. The connections ∇ and ˆ∇ coincide in the direction of X_{+++}^{e}. Hence, using C_{+++}^{e} := u_{+}^{α_{1}}u_{+}^{α_{2}}u_{+}^{α_{3}}C_{α_{1}α_{2}α_{3}}^{e}, we have

0 = F^{∇}(X_{+++}^{e}, X_{+++}^{e′})
= F ˆ∇(X_{+++}^{e}, X_{+++}^{e′})
= X_{+++}^{e}C_{+++}^{e′} − X_{+++}^{e′}C_{+++}^{e} + [C_{+++}^{e} , C_{+++}^{e′}] − C([X_{+++}^{e} , X_{+++}^{e′}])
= u_{+}^{α_{1}}u_{+}^{α_{2}}u_{+}^{α_{3}}u_{+}^{β_{1}}u_{+}^{β_{2}}u_{+}^{β_{3}} (X_{α_{1}α_{2}α_{3}}^{e}C_{β_{1}β_{2}β_{3}}^{e′} − X_{β_{1}β_{2}β_{3}}^{e′}C_{α_{1}α_{2}α_{3}}^{e} + [C_{α_{1}α_{2}α_{3}}^{e} , C_{β_{1}β_{2}β_{3}}^{e′}]
−C([X_{α_{1}α_{2}α_{3}}^{e} , X_{β_{1}β_{2}β_{3}}^{e′}]))
= u_{+}^{α_{1}}u_{+}^{α_{2}}u_{+}^{α_{3}}u_{+}^{β_{1}}u_{+}^{β_{2}}u_{+}^{β_{3}} F^{∇^{M}} (X_{α_{1}α_{2}α_{3}}^{e} , X_{β_{1}β_{2}β_{3}}^{e′}) ,

since X_{+++}^{e}u_{+}^{β} = 0. This shows that the component F^{(0)}= 0 in the decomposition (5.4), i.e. the connection ∇^{M} is 0-partially flat.

5.3.2 Construction of 1-partially flat connections

Let M be a 1–admissible spin \frac{3}{2} Grassmann manifold with a 1-partially flat connection ∇ (satisfying (5.6)) in a holomorphic vector bundle W → M. The pull-back of such a connection to a connection in the bundle π^{∗}W → S_{H}, where π : S_{H} → M, has curvature F with components given by:

F (X_{±±±}^{e}, X_{±±±}^{e′}) = 0
F (X_{±±±}^{e′}, X_{±}^{e}) = 0
F (X_{±}^{e}, X_{±}^{e′}) = 0
F (X_{+++}^{e}, X_{−−−}^{e′}) = 36 F^{(3)} ^{(ee′)}
F (X_{±±±}^{e}, X_{∓}^{e′}) = 0
F (X_{+}^{e}, X_{−}^{e′}) = −12 F^{(3)} ^{(ee′)}
F (v , ·) = 0 ,  (5.16)

where v is any vertical vector field on S_{H}. A connection in a holomorphic vector bundle W → S_{H} is 1-partially flat if its curvature satisfies the above equations. The restriction of a 1-partially flat connection to a leaf of the integrable distribution ⟨𝒟_{3+}, 𝒟_{+}, ∂_{0}⟩ is clearly flat. In this case, an analytic frame in the holomorphic vector bundle π^{∗}ν : ℂ^{r} × S_{H} → S_{H} is a frame which is parallel along leaves of this distribution. With respect to such a frame, a connection in the vector bundle π^{∗}ν can be written as

\{ \begin{matrix}∇_{∂_{0}}^{S} & = ∂_{0} \\ ∇_{X_{+++}^{e}}^{S} & = X_{+++}^{e} \\ ∇_{X_{+}^{e}}^{S} & = X_{+}^{e} \\ ∇_{∂_{++}}^{S} & = ∂_{++} + A_{++} \\ ∇_{∂_{−−}}^{S} & = ∂_{−−} + A_{−−} \\ ∇_{X_{−−−}^{e}}^{S} & = X_{−−−}^{e} + A(X_{−−−}^{e}) \\ ∇_{X_{−}^{e}}^{S} & = X_{−}^{e} + A(X_{−}^{e}) , \end{matrix}

with potentials A(X_{+++}^{e}) = A(X_{+}^{e}) = A(∂_{0}) = 0. We look for solutions of the system (5.16) in this analytic gauge.

Definition 16 A connection ∇^{S} over S_{H} is called almost 1-partially flat if its curvature satisfies the equations:

F (X_{+++}^{e}, X_{+++}^{e′}) = F (X_{+++}^{e}, X_{+}^{e′}) = F (X_{+}^{e}, X_{+}^{e′}) = 0
F (X_{+++}^{e}, v) = F (X_{±}^{e}, v) = F (∂_{++}, ·) = F (∂_{0}, ·) = 0 , ∀v ∈ T^{v}S_{H} . (5.17)

In virtue of these equations, the potentials A_{±±} =: A(∂_{±±}) determine all other potentials:

Proposition 13 Let ∇ be an almost 1-partially flat connection in the vector bundle π^{∗}ν : ℂ^{r} × S_{H} → S_{H} with potentials A_{++}, A_{−−}, A(X_{−}^{e}) and A(X_{−−−}^{e}) in an analytic frame. Then:

(i) The potential A_{++} is analytic and has charge 2, i.e.

X_{+++}^{e}A_{++} = 0 , X_{+}^{e}A_{++} = 0 , ∂_{0}A_{++} = 2A_{++} .  (5.18)

(ii) The potential A_{−−} satisfies

∂_{++}A_{−−} − ∂_{−−}A_{++} + [A_{++}, A_{−−}] = 0 , ∂_{0}A_{−−} = −2A_{−−} .  (5.19)

(iii) The potentials A(X_{−}^{e}) and A(X_{−−−}^{e}) are then recursively determined as follows:

A(X_{−}^{e}) = −\frac{1}{2}X_{+}^{e}A_{−−}
A(X_{−−−}^{e}) = ∂_{−−}A(X_{−}^{e}) − X_{−}^{e}A_{−−} + [A_{−−}, A(X_{−}^{e})]  (5.20)

and they have charges −1 and −3 respectively; i.e.

∂_{0}A(X_{−}^{e}) = ±A(X_{−}^{e}) , ∂_{0}A(X_{−−−}^{e}) = −3A(X_{−−−}^{e}) .  (5.21)

Conversely, any set of matrix-valued potentials A_{++}, A_{−−}, A(X_{−}^{e}) and A(X_{−−−}^{e}) satisfying (5.18)–(5.21) define an almost 1-partially flat connection.

Proof: (i) Equations (5.18) are equivalent to F (∂_{++}, X_{+++}^{e}) = F (∂_{++}, X_{+}^{e}) = F (∂_{0}, ∂_{++})=0. (ii) The further almost 1-partial-flatness conditions, F (∂_{++} , ∂_{−−}) = F (∂_{0} , ∂_{−−}) = 0 give equations (5.19). (iii) Having obtained A_{−−}, we can find A(X_{−}^{e}) and A(X_{−−−}^{e}) from the equations

F (∂_{−−}, X_{+}^{e}) = 0 ⇔ −X_{+}^{e}A_{−−} = A([∂_{−−}, X_{+}^{e}]) = 2A(X_{−}^{e})
F (∂_{−−}, X_{−}^{e}) = 0 ⇔ ∂_{−−}A(X_{−}^{e}) − X_{−}^{e}A_{−−} + [A_{−−}, A(X_{−}^{e})] = A([∂_{−−}, X_{−}^{e}]) = A(X_{−−−}^{e}) .

The equations (5.21) follow from (5.20). Now, starting from a prepotential A_{++} , which solves (5.18), we may construct an almost 1-partially flat connection. The potential A_{−−} = −(∂_{−−}Φ)Φ^{−1} is determined, as before, from a solution Φ of (5.13). Then, with the remain potentials in an analytic frame being given by (5.20) and satisfying (5.21), all the other equations in (5.17) follow. This shows that a almost 1-partially flat connection is determined by an arbitrary analytic prepotential A_{++} and an invertible solution Φ of (5.13). As before, Φ is a transition function from an analytic frame to a central frame, in which the above almost 1-partially flat connection takes the form:

\{ \begin{matrix}∇_{X_{+++}^{e}}^{S} & = X_{+++}^{e} + C(X_{+++}^{e}) = X_{+++}^{e} + Φ X_{+++}^{e} Φ^{−1} \\ ∇_{X_{+}^{e}}^{S} & = X_{+}^{e} + C(X_{+}^{e}) = X_{+}^{e} + Φ X_{+}^{e} Φ^{−1} \\ ∇_{X_{−}^{e}}^{S} & = X_{−}^{e} + C(X_{−}^{e}) \\ ∇_{X_{−−−}^{e}}^{S} & = X_{−−−}^{e} + C(X_{−−−}^{e}) \\ ∇_{∂_{++}}^{S} & = ∂_{++} , ∇_{∂_{−−}}^{S} = ∂_{−−} , ∇_{∂_{0}}^{S} = ∂_{0} . \end{matrix}

Moreover, the equations F (∂_{++} , X_{+++}^{e}) = F (∂_{0} , X_{+++}^{e}) = 0 imply that the potential C(X_{+++}^{e}) satisfies the equations

∂_{++}C(X_{+++}^{e}) = 0 , ∂_{0}C(X_{+++}^{e}) = 3C(X_{+++}^{e}) .

Proposition 14 The potentials C(X_{+++}^{e}) and C(X_{+}^{e}) of an almost 1-partially flat connection ∇ with respect to a central frame have the form,

C(X_{+++}^{e}) = u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e} , C(X_{+}^{e}) = u_{+}^{α}u_{+}^{β}u_{−}^{γ}C_{αβγ}^{e} ,

where C_{αβγ}^{e} is a function on M, symmetric in α, β, γ.

With respect to a central frame, we can therefore write

∇_{X_{+++}^{e}} = X_{+++}^{e} + u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e} , ∇_{X_{+}^{e}} = X_{+}^{e} + u_{+}^{α}u_{+}^{β}u_{−}^{γ}C_{αβγ}^{e} .

We define a modified connection in the bundle π^{∗}ν over S_{H} by

\{ \begin{matrix}ˆ∇_{X_{+++}^{e}} & = X_{+++}^{e} + u_{+}^{α}u_{+}^{β}u_{+}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{+}^{e}} & = X_{+}^{e} + u_{+}^{α}u_{+}^{β}u_{−}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{−}^{e}} & = X_{−}^{e} + u_{−}^{α}u_{−}^{β}u_{+}^{γ}C_{αβγ}^{e} \\ ˆ∇_{X_{−−−}^{e}} & = X_{−−−}^{e} + u_{−}^{α}u_{−}^{β}u_{−}^{γ}C_{αβγ}^{e} \\ ˆ∇_{∂_{++}} & = ∂_{++} , ˆ∇_{∂_{−−}} = ∂_{−−} , ˆ∇_{∂_{0}} = ∂_{0} . \end{matrix}  (5.22)

As in the 0-partially flat case, we have:

Theorem 9 The constructed connection ˆ∇ is a 1-partially flat connection in π^{∗}ν over S_{H} and it is the pull-back of the following 1-partially flat connection ∇^{M} in ν over M:

∇_{X_{αβγ}^{e}}^{M} = X_{αβγ}^{e} + C_{αβγ}^{e} .  (5.23)

Proof: As before one shows that the connection ˆ∇ is the pull-back of the connection ∇^{M}. It remains to show that ∇^{M} is 1-partially flat. Since any almost 1-partially flat connection is almost 0-partially flat we have F^{(0)}= 0 by Theorem 8. Next we show that F^{(1)}= 0. The connections ∇ and ˆ∇ coincide in the direction of X_{+++}^{e} and X_{+}^{e}. Hence, using equation (5.7), which holds for 0-partially flat connections, we have

0 = F^{∇}(X_{+++}^{e}, X_{+}^{e′})
= F ˆ∇(X_{+++}^{e}, X_{+}^{e′})
= 12 F^{(1)} _{α_{1}α_{2}β_{1}β_{2}}^{(ee′)}u_{+}^{α_{1}}u_{+}^{α_{2}}u_{+}^{β_{1}}u_{+}^{β_{2}} .

This shows that the component F^{(1)} in the decomposition (5.4) vanishes. Similarly,

0 = F^{∇}(X_{+}^{e}, X_{+}^{e′})
= F ˆ∇(X_{+}^{e}, X_{+}^{e′})
= −8 F^{(2)} _{α_{1}β_{1}}^{[ee′]}u_{+}^{α_{1}}u_{+}^{β_{1}} ,

implies F^{(2)} ^{[ee′]} = 0, and hence that ˆ∇ is 1-partially flat. By Theorem 6, the 1-partially flat connection ∇_{X_{αβγ}^{e}}^{M} in (5.23) is a Yang-Mills connection.

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