Special Geometric Structures in String Theory Bonn, 8th–11th September, 2001

PARTIALLY-FLAT GAUGE FIELDS ON MANIFOLDS OF DIMENSION GREATER THAN FOUR

DMITRI V. ALEKSEEVSKY, VICENTE CORTÉS AND CHANDRASHEKAR DEVCHAND

Abstract. We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Ω-self-duality, is based on the existence of an appropriate 4-form Ω on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The second is the notion of half-flatness, which is defined for manifolds with certain Grassmann structure T^{ℂ}M≅E ⊗ H. In some cases, for example for hyper-Kähler manifolds M, half-flatness implies Ω-self-duality. A construction of half-flat connections inspired by the harmonic space approach is described. Locally, any such connection can be obtained from a free prepotential by solving a system of linear first order ODEs.

1. Self-duality and half-flatness

The Yang-Mills self-duality equations have played a very important role in field theory and in differential geometry. This talk is about ongoing work on two generalisations of the notion of self-duality from four to higher dimensional manifolds M. The first generalisation is the notion of

Ω-self-duality.

This is based on the existence of an appropriate 4-form Ω on a pseudo-Riemannian manifold (M, g). For Ω ∈ Ω^{4}(M) we define a traceless symmetric endomorphism field B_{Ω} : ∧^{2}T^{∗}M → ∧^{2}T^{∗}M by

B_{Ω}ω := ∗(∗Ω ∧ ω) ,  (1)

where ω ∈ ∧^{2}T^{∗}M.

Talk given by C.D. on 11th September, 2001 at the Workshop on Special Geometric Structures in String Theory, Bonn. This work was supported by the ‘Schwerpunktprogramm Stringtheorie’ of the Deutsche Forschungsgemeinschaft.

2 D.V. ALEKSEEVSKY, V. CORTÉS AND C. DEVCHAND

Definition 1. A 4-form Ω ∈ Ω^{4}(M) on a pseudo-Riemannian manifold M is called appropriate if there exists a non-zero 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. We can now define a generalisation of the four dimensional notion of self-duality thus: Definition 2. 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-self-dual if its curvature F^{∇} satisfies the linear algebraic system

B_{Ω}F^{∇} = λF^{∇}  (2)
(d ∗ Ω) ∧ F^{∇} = 0 .  (3)

It is easy to see that an Ω-self-dual connection ∇ is a Yang-Mills connection, i.e. it satisfies the Yang-Mills equations:

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

in virtue of eq. (3) and the Bianchi identity d^{∇}F^{∇} = 0. This extension of selfduality is based on early work of [CDFN] on gauge fields in flat spaces. Interesting examples of manifolds with appropriate (and parallel) 4-forms are provided, for instance by manifolds with special holonomy groups. So for gauge fields on these manifolds, Ω-self-duality may be defined. Various examples have been discussed in the literature, e.g. in [CS, BKS, DT, T].

The second generalisation of self-duality is the notion of Half-flatness. This is defined for manifolds with certain Grassmann structure. In what follows M denotes a complex manifold. A Grassmann structure on M is an isomorphism of the (holomorphic) tangent bundle, T M ≅ E ⊗H, where E and H are two holomorphic vector bundles over M. This is just a generalised spinor decomposition, allowing representation of vector fields using two types of spinor indices, viz. X_{aα} := e_{a}⊗h_{α}, where (e_{a}) and (h_{α}) are frames of E and H respectively. If M is four dimensional, both E and H have rank 2 and a, α are the familiar 2-spinor indices. Manifolds with Grassmann structure provide interesting generalisations of four dimensional manifolds. A Grassmann connection ∇ is a linear connection which preserves the Grassmann structure:

∇(e ⊗ h) = ∇^{E}e ⊗ h + e ⊗ ∇^{H}h ,

PARTIALLY-FLAT GAUGE FIELDS 3

where ∇^{E}, ∇^{H} are connections in the bundles E, H and e, h are local sections of E, H respectively. A Grassmann structure with Grassmann connection ∇ is called half-flat if the connection ∇^{H} in H is flat. A manifold with such a half-flat Grassmann structure is called a half-flat Grassmann manifold. Here we assume that rank H = 2 and that a ∇^{H}-parallel non-degenerate fibre-wise 2-form ω_{H} ∈ Γ(∧^{2}H^{∗}) in the bundle H is fixed, so these manifolds are generalisations of hyper-Kähler manifolds. The torsion of the Grassmann connection lives in the space

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

where we omit the ⊗’s and we identify H^{∗} with H using ω_{H}. Notice that since the bundle H has rank 2, the line bundle ∧^{2}H is generated by the 2-form ω_{H}. (Choosing a frame (h_{1}, h_{2}) for H, we may write ω_{H}(h_{α}, h_{β}) := ϵ_{αβ}, a skew matrix with ϵ_{12} = 1.)

We now consider gauge fields on these manifolds. Let ∇ be a connection in a holomorphic vector bundle ν : W → M. (If ν has structure group G, we may choose a frame for it, in which the vector potential takes values in the gauge Lie algebra 𝔤). The curvature of the connection ∇ is a 2-form with values in the gauge algebra, F^{∇} ∈ ∧^{2}T^{∗}M ⊗ End W. Now, as above, in virtue of the Grassmann structure the space of bivectors decomposes as

∧^{2}T M = S^{2}E ⊗ ∧^{2}H ⊕ ∧^{2}E ⊗ S^{2}H = S^{2}E ⊗ ω_{H} ⊕ ∧^{2}E ⊗ S^{2}H .

If e, e′ ∈ Γ(E), we have the decomposition

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

where F^{(e,e′)} is symmetric in e, e′ and F_{(αβ)}^{[e,e′]} is skew in e, e′ and symmetric in αβ. In 4 dimensions both E and H have rank 2, so this corresponds to the familiar decomposition into self-dual and anti-self-dual parts of the curvature. Definition 3. Let M be a manifold with Grassmann structure, not necessarily half-flat. A connection ∇ in a vector bundle W → M is called half-flat if its curvature

F^{∇} ∈ ∧^{2}H ⊗ S^{2}E ⊗ End W = ω_{H} ⊗ S^{2}E ⊗ End W .

Quaternionic Kähler and hyper-Kähler manifolds are special cases of manifolds with (a locally defined) Grassmann structure, the Levi-Civita connection being the canonical Grassmann connection. In the hyper-Kähler case the Grassmann structure is half-flat. For these examples, half-flatness of a gauge connection implies

4 D.V. ALEKSEEVSKY, V. CORTÉS AND C. DEVCHAND

Ω-self-duality with respect to the canonical parallel 4-form Ω. In other words, a half-flat curvature is one of the eigenstates of the endomorphism field B_{Ω}, as in 4 dimensions. Therefore, in these cases, the two generalisations of the idea of self-duality mentioned above coincide.

There is an interesting special case of Grassmann structure, considered by [AG]:

Definition 4. 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 ∇^{T M} = ∇^{E} ⊗ Id + Id ⊗ ∇^{F}, where H is a rank 2 holomorphic vector bundle over M with symplectic form ω_{H} and holomorphic symplectic connection ∇^{H}, and ∇^{F} is the connection in F = S^{m}H induced by ∇^{H}. M is called half-flat spin \frac{m}{2} Grassmann manifold 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 multi-index A := α_{1}α_{2} … α_{m} , α_{i} = 1, 2. The symplectic form ω_{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 skewsymmetric 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 and we put X_{aA} := X_{A}^{e_{a}}.

Let (M, ∇^{TM}) be a half-flat spin \frac{m}{2} Grassmann manifold and ∇ a connection in a holomorphic vector bundle W → M. The space of bivectors ∧^{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. The corresponding decomposition of the curvature of ∇ is:

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′)}) ,

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^{∗})

are the curvature components in irreducible GL(E)·Sp(1, ℂ)-submodules.

For manifolds with spin \frac{m}{2} Grassmann structure it is natural to define half-flat connections as those satisfying

F^{(2i)}= 0 , for all i ∈ ℕ.

However, for these manifolds it is useful to consider a more refined notion: Definition 5. A connection ∇ in the vector bundle ν : W → M is called kpartially 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 in [W]. Some other k-partially flat connections on flat spaces were considered in [DN].

The penultimate case k = [\frac{m}{2}] is particularly interesting for odd m. Theorem 1. 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 a 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 ∇ on a vector bundle W over M is Ω-selfdual and hence it is a Yang-Mills connection, i.e. satisfies the Yang-Mills equations.

Sketch of 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 dual basis of the basis X_{aA}. 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). Contracting a tensor S = S_{ab}ω_{AB}e^{a}e^{b} ⊗ h^{A}h^{B} in S^{2}E^{∗} ⊗ ω_{H}^{m} with Ω, after some algebra, yields λS with λ = −4(m+1) ≠ 0. Hence any \frac{m−1}{2}-partially flat connection is Ω-selfdual and is a Yang-Mills connection.

2. Construction

We now briefly sketch a construction of half-flat connections on half-flat Grassmann manifolds using a variant of the harmonic space approach (c.f. [GIOS1]). The method affords application to a construction of partially flat connections on a half-flat spin \frac{m}{2} Grassmann manifold as well. Details and proofs will appear in [ACD].

We denote by S_{H} the Sp(1, ℂ)-principal holomorphic bundle over M consisting of symplectic bases of 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. A fixed parallel symplectic basis (h_{1}, h_{2}) of H, such that m ↦ s_{m} = (h_{1}(m), h_{2}(m)) ∈ S_{H} yields a trivialisations M × Sp(1, ℂ) ≅ S_{H} defined by

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

This is precisely the harmonic space of [GIKOS, GIOS2]. The matrix coefficients u_{±}^{α} of Sp(1, ℂ) are considered as holomorphic functions on S_{H} and together with local coordinates of M, we obtain a system of local coordinates on S_{H} constrained by the relation u_{+}^{1}u_{−}^{2} − u_{−}^{1}u_{+}^{2} = 1. We denote by ∂_{++}, ∂_{−−}, ∂_{0} the fundamental vector fields on S_{H} generated by the standard generators of Sp(1, ℂ), satisfying the relations

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

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 vector field Y on M. They satisfy the relations

[∂_{0}, X_{±}^{e}] = ±X_{±}^{e} , [∂_{±±}, X_{±}^{e}] = 0 , [∂_{±±}, X_{∓}^{e}] = X_{±}^{e} ,

\begin{aligned}[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′}) , \end{aligned}

where T is the torsion of the Grassmann connection and T̃(X, Y) := { \atop T (X, Y)} denotes the horizontal lift of the vector T (X, Y). We denote by 𝒟_{±} the distributions generated by vector fields of the form X_{±}^{e}, e ∈ Γ(E).

The holomorphic Grassmann structure is called 𝒟_{+}–admissible if the torsion of the Grassmann connection ∇ has no component in S^{3}H ⊗ E ⊗ ∧^{2}E^{∗}. This means, in particular, that the torsion can be written as a sum of tensors linear in ω_{H}. Now, it is not difficult to prove that if a Grassmann structure is 𝒟_{+}–admissible, the distribution 𝒟_{+} (associated to any parallel basis (h_{1}, h_{2})) on S_{H} is integrable.

The basic idea of the harmonic space construction is to lift the geometric data from M to S_{H} via the projection π : S_{H} → M. The pull back π^{∗}∇ of a half-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 half-flat gauge connection on S_{H}. One can also define the weaker notion of an almost half-flat connection in π^{∗}ν : π^{∗}W → S_{H} by considering only the equations on the curvature involving ∂_{0}, ∂_{++} and X_{+}^{e} in one of the arguments.

Now, the main result, for which the integrability of 𝒟_{+} is crucial, can be stated: Theorem 2. Let M be a manifold with a half-flat 𝒟_{+}–admissible Grassmann structure. Consider a matrix-valued function A_{++} on S_{H} (the “analytic prepotential”), constant along the leaves of 𝒟_{+} with the homogeneity property ∂_{0}A_{++} = 2A_{++} , and an invertible matrix-valued solution of the system of first-order linear ODE’s, ∂_{++}Φ = −A_{++}Φ , ∂_{0}Φ = 0 . The pair (A_{++}, Φ) defines an almost half-flat connection ∇^{(A_{++},Φ)} in the bundle π^{∗}ν : π^{∗}W → S_{H}, which depends only on A_{++}, Φ and their first and second partial derivatives. The potential A(X_{+}^{e}) of this connection with respect to the frame eΦ = e_{i}Φ_{j}^{i}, where e = (e_{i}) is the standard frame of W = ℂ^{r} × M, determines a half-flat connection in the bundle ν : W → M. Conversely, any half-flat connection over M can be obtained using this construction.

The proof and further details may be found in [ACD].

References

[ACD] D. V. Alekseevsky, V. Cortes and C. Devchand, Yang-Mills connections over manifolds with Grassmann structure, in preparation

[AG] D. V. Alekseevsky and M.M. Graev, Grassmann and hyperkähler structures on some spaces of sections of holomorphic bundles in Manifolds and geometry, Ed. de Bartolomeis et al, Cambridge Univ. Press, U.K., 1996

[BKS] L. Baulieu, H. Kanno and I. M. Singer, Special quantum field theories in eight and other dimensions, Commun. Math. Phys. 194 (1998) 149–175 [hep-th/9704167]

[CS] M. M. Capria and S. M. Salamon, Yang-Mills fields on quaternionic spaces, Nonlinearity 1 (1988), 517–530

[CDFN] E. Corrigan, C. Devchand, D.B. Fairlie and J. Nuyts, First order equations for gauge fields in spaces of dimension greater than four, Nucl. Phys. B214 (1983) 452–464

[DN] C. Devchand and J. Nuyts, Supersymmetric Lorentz-covariant hyperspaces and selfduality equations in dimensions greater than (4|4), Nucl. Phys. B503 (1997) 627–656 [hep-th/9704036]

[DT] S.K. Donaldson and R.P. Thomas, Gauge theory in higher dimensions, in The geometric universe: science, geometry, and the work of Roger Penrose, ed. S.A. Huggett, et al., Oxford Univ. Press, 31-47 (1998)

[GIKOS] A. Galperin, E. Ivanov, S. Kalitzin, V. Ogievetsky and E. Sokatchev, Unconstrained N=2 matter, Yang-Mills and supergravity theories in harmonic superspace, Class. Quant. Grav. 1 (1984) 469–498

[GIOS1] A. Galperin, E. Ivanov, V. Ogievetsky and E. Sokatchev, Gauge field geometry from complex and harmonic analyticities. I. Kähler and selfdual Yang-Mills cases, Ann. Phys. 185 (1988) 1–21

[GIOS2] A. Galperin, E. Ivanov, V. Ogievetsky and E. Sokatchev, Harmonic superspace, Cambridge Univ. Press, U.K., 2001

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[T] G. Tian, Gauge theory and calibrated geometry. I., Ann. Math. (2) 151 (2000) 193–268

Dept. of Mathematics, University of Hull, Hull, HU6 7RX, UK

E-mail address: D.V.Alekseevsky@maths.hull.ac.uk

Mathematisches Institut, Universität Bonn, Beringstr. 1, D-53115 Bonn

E-mail address: vicente@math.uni-bonn.de
E-mail address: devchand@math.uni-bonn.de