Instantons in six dimensions and twistors

Recently, conformal field theories in six dimensions were discussed from the twistorial point of view. In particular, it was demonstrated that the twistor transform between chiral zero-rest-mass fields and cohomology classes on twistor space can be generalized from four to six dimensions. On the other hand, the possibility of generalizing the correspondence between instanton gauge fields and holomorphic bundles over twistor space is questionable. It was shown by Saemann and Wolf that holomorphic line bundles over the canonical twistor space Tw(X) (defined as a bundle of almost complex structures over the six-dimensional manifold X) correspond to pure-gauge Maxwell potentials, i.e. the twistor transform fails. On the example of X=CP^3 we show that there exists a twistor correspondence between Abelian or non-Abelian Yang-Mills instantons on CP^3 and holomorphic bundles over complex submanifolds of Tw(CP^3), but it is not so efficient as in the four-dimensional case because the twistor transform does not parametrize instantons by unconstrained holomorphic data as it does in four dimensions.


Introduction and summary
Let us consider an oriented real four-manifold X 4 with a Riemannian metric g and the principal bundle P (X 4 , SO(4)) of orthonormal frames over X 4 . The (metric) twistor space Tw(X 4 ) of X 4 can be defined as an associated bundle [1] Tw(X 4 ) = P × SO(4) SO(4)/U (2) (1.1) with the canonical projection Tw(X 4 ) → X 4 . This space parametrizes the almost complex structures on X 4 compatible with the metric g (almost Hermitian structures). It was shown in [1,2] that if the Weyl tensor of (X 4 , g) is anti-self-dual then the almost complex structure on the twistor space Tw(X 4 ) is integrable. Furthermore, it was proven that the rank r complex vector bundle E over X 4 with an anti-self-dual gauge potential A over such X 4 lifts to a holomorphic bundleÊ over complex twistor space Tw(X 4 ) [1,3].
The essence of the canonical twistor approach is to establish a correspondence between fourdimensional space X 4 (or its complex version) and complex twistor space Tw(X 4 ) of X 4 . Using this correspondence, one transfers data given on X 4 to data on Tw(X 4 ) and vice versa. In twistor theory one considers holomorphic objects h on Tw(X 4 ) (Čech cohomology classes, holomorphic vector bundles etc.) and transforms them to objects f on X 4 which are constrained by some differential equations [1]- [4]. Thus, the main idea of twistor theory is to encode solutions of some differential equations on X 4 in holomorphic data on the complex twistor space Tw(X 4 ) of X 4 .
The twistor approach was recently extended to maximally supersymmetric Yang-Mills theory on C 6 [5]. It was also generalized to Abelian [6,7] and non-Abelian [8] holomorphic principal 2-bundles over the twistor space Q ′ 6 ⊂ CP 7 \ CP 3 , corresponding to self-dual Lie-algebra-valued 3-forms on C 6 . These forms are the most important objects needed for constructing (2,0) superconformal field theories in six dimensions, which are believed to describe stacks of M5-branes in the low-energy limit of M-theory [9].
The papers [5]- [8] (see also references therein) show that the twistor methods can be useful in higher-dimensional Yang-Mills and superconformal field theories. However, there are some problems in generalizing the twistor approach to higher dimensions. Namely, let X 2n be a Riemannian manifold of dimension 2n. The metric twistor space of X 2n is defined as the bundle Tw(X 2n ) → X 2n of almost Hermitian structures on X 2n associated with the principal bundle of orthonormal frames of X 2n , i.e.
It is well known that Tw(X 2n ) can be endowed with an almost complex structure J , which is integrable if and only if the Weyl tensor of X 2n vanishes when n > 2 [10]. This is too strong a restriction. However, if the manifold X 2n has a G-structure (not necessarily integrable), then one can often find a subbundle Z of Tw(X 2n ) associated with the G-structure bundle P (X 2n , G) for G ⊂ SO(2n), such that an induced almost complex structure (also called J ) on Z is integrable. Many examples were considered in [10]- [14]. Another problem is that, in higher dimensions, solutions of differential equations do not always correspond to holomorphic objects on the reduced twistor space Z (even if Z is a complex manifold). In [15] this was shown for the example of Yang-Mills instantons on the six-sphere S 6 , which has the reduced complex twistor space Z = G 2 /U (2). For the definition of instanton equations in dimensions higher than four and for some instanton solutions see e.g. [16]- [23].
In this paper we discuss instantons in gauge theory on the complex projective space CP 3 by using twistor theory. Natural instanton-type equations in six dimensions are the Donaldson-Uhlenbeck-Yau (DUY) equations [17], which are SU(3) invariant but not invariant under the SO(6) Lorentz-type rotations of orthonormal frames. Hence, for their description one should consider reduced twistor spaces. The DUY equations are well defined on six-dimensional Kähler manifolds M (as well as on nearly Kähler spaces [24,25,26]), and their solutions are natural connections A on holomorphic vector bundles E → M [17]. On the example of M = CP 3 we will show that such bundles (E, A) are pulled back to holomorphic vector bundles (Ẽ,Ã) over the reduced twistor space Z ⊂ Tw(CP 3 ) trivial along the fibres of the fibration Z → CP 3 with Z = SU(4)/U(2)×U(1) or Z = Sp(2)/U(1)×U(1), depending on the chosen holonomy group. Note that this correspondence, valid for the reduced twistor spaces Z ֒→ Tw(CP 3 ), does not hold for the metric twistor space Tw(CP 3 ).
Coset representation of S 2 . Let us consider the Hopf bundle over the Riemann sphere S 2 ∼ = CP 1 and the one-monopole connection a on the bundle (2.13) having in the local complex coordinate ζ ∈ CP 1 the form Consider a local section of the bundle (2.13) given by the matrix and introduce the su(2)-valued one-form (flat connection) are the forms of type (1,0) and (0,1) on CP 1 and a is the one-monopole gauge potential (2.14).
Using the group element (3.23), we introduce a flat connection A ′ 0 on the trivial bundle Z ′ ×C 4 → Z ′ as The flat connectionÂ 0 is given in (2.20) and (3.9). Using (3.27), we obtain the connection

30)
32) From the flatness F ′ 0 = 0 of the connection (3.29) we obtain the Maurer-Cartan equations Its integrability follows from the vanishing (0,2)-type components of the torsion on the right hand side of (3.35).

Twistor description of instanton bundles over CP 3
Instanton bundles over CP 3 . Consider a complex vector bundle E over CP 3 with a connection one-form A having the curvature F. Recall that (E, A) is called an instanton bundle if A satisfies the Hermitian Yang-Mills (HYM) equations, 1 which on CP 3 can be written in the form where the notationω exploits the underlying Riemannian metric g = δâbeâeb on CP 3 ,â,b, . . . = 1, . . . , 6. Here,ω given in (2.28) is a (1,1)-form, andΩ :=θ 1 ∧θ 2 ∧θ 3 is a locally defined (3,0)-form on CP 3 . Recall that, from the point of view of algebraic geometry, (4.1) means that the bundle E → CP 3 is holomorphic and (4.2) means that E is a polystable vector bundle [17]. In fact, in the right hand side of (4.2) one can add the term βω ∧ω ∧ω with β proportional to the first Chern number c 1 (E), but we assume c 1 (E) = 0 since for a bundle with field strength F of non-zero degree one can obtain a degree-zero bundle by consideringF = F − 1 r (tr F) · 1 r , where r = rank E.
Pull-back to Z. Consider the twistor fibration (3.6). Let ( E, A) = (π * E, π * A) be the pulled-back instanton bundle over Z with the curvature F = d A + A ∧ A. We have with a, b, ... = 1, ..., 5. Using the relation betweenθ a andθ a described in Section 3, we obtain where C =V † with , (4.5) andC is the complex conjugate matrix. Thus, more explicitly, we get The vanishing of F23 for all values of (λ 1 , λ 2 ) ∈ CP 2 is equivalent to the holomorphicity equation (4.1). In homogeneous coordinates y i on CP 2 (λ 1 = y 2 /y 1 , λ 2 = y 3 /y 1 , y 1 = 0), this condition can be written as where the indicesī,, . . . are raised with the metric δ i . From (4.6)-(4.9) we see that the bundle E is holomorphic for holomorphic E as well as polystable due to (4.2), (4.10) and it is holomorphically trivial after restricting to the fibres CP 2 x ֒→ Z of the projection π for each x ∈ CP 3 .
We have seen that solutions of the Hermitian Yang-Mills equations on the complex projective space CP 3 , represented either as the symmetric space SU(4)/U(3) or as the nonsymmetric homogeneous space Sp(2)/Sp(1)×U(1), have a twistor description similar to the four-dimensional case but valid for the reduced twistor spaces Z and Z ′ with fibres CP 2 and CP 1 , respectively, instead of the metric twistor space Tw(CP 3 ) = P (CP 3 , SO(6))× SO(6) SO(6)/U(3) with fibres SO(6)/U(3) ∼ = CP 3 .