The Twisted Photon Associated to Hyper-Hermitian Four-Manifolds

The Lax formulation of the hyper-Hermiticity condition in four dimensions is used to derive a potential that generalises Plebanski’s second heavenly equation for hyper-Kahler 4-manifolds. A class of examples of hyper-Hermitian metrics which depend on two arbitrary functions of two complex variables is given. The twistor theory of four-dimensional hyper-Hermitian manifolds is formulated as a combination of the Nonlinear Graviton Construction with the Ward transform for anti-self-dual Maxwell fields. 1 Complexified hyper-Hermitan manifolds A smooth manifold M equipped with three almost complex structures (I, J,K) satisfying the algebra of quaternions is called hypercomplex iff the almost complex structure Jλ = aI + bJ + cK is integrable for any (a, b, c) ∈ S. We shall use a stereographic coordinate λ = (a+ ib)/(c− 1) on S which will we view as a complex projective line CP. Let g be a Riemannian metric on M. If (M,Jλ) is hypercomplex and g(JλX,JλY ) = g(X,Y ) for all vectors X,Y on M then the triple (M, Jλ, g) is called a hyper-Hermitian structure. From now on we shall restrict ourselves to oriented four manifolds. In four dimensions a hyper-complex structure defines a conformal structure, which in explicit terms is represented by a conformal frame of vector fields (X, IX, JX,KX), for any X ∈ TM. It is well known that this conformal structure is anti self-dual (ASD) with the orientation determined by the complex structures. Let g be a representative of the conformal structure defined by Jλ, and let Σ ′B′ = (Σ ′ ,Σ ′ ,Σ ′ ) be a basis of the space of SD two forms Λ+(M) (see appendix for notation and conventions). The following holds Proposition 1 [1] The Riemannian four manifold (M, g) is hyper-Hermitian if there exists a one form A (called a Lee form) depending only on g such that dΣ ′B′ = −A ∧ Σ ′B′ . (1.1) Moreover if A is exact, then g is conformally hyper-Kähler. In Section 2 we shall express the hyper-Hermiticity condition on the metric in four dimensions in terms of Lax pairs of vector fields on M. The Lax formulation will be used to encode the hyperHermitian geometry in a generalisation of Plebański’s formalisms [14]. Some examples of hyperHermitian metrics are given in Section 3. In Section 4 we establish the twistor correspondence for the hyper-Hermitian four-manifolds. If M is real then the associated twistor space is identified with a sphere bundle of almost-complex structures and the resulting twistor theory is well-known [1, 13]. We will work with the complexified correspondence and assume that M is a complex four-manifold. ∗email: [email protected]