A note on the Hyper--CR equation, and gauged $N=2$ supergravity

We construct a new class of solutions to the dispersionless hyper--CR equation, and show how any solution to this equation gives rise to a supersymmetric Einstein--Maxwell cosmological space--time in $(3+1)$--dimensions.

In this note we shall construct all solutions to (1), where the linear and nonlinear terms in (1) vanish separately. This will be done in §2. In §3 we shall show how solutions to (1) lift to supersymmetric solutions to N = 2 pseudo-supergravity in 3 + 1 space-time dimensions.
Acknowledgements. MD is supported by the STFC consolidated grant ST/P000681/1. JG is supported by the STFC consolidated grant ST/L000490/1. WS is supported in part by the National Science Foundation under grant number PHY-1620505. MD and JG thank the American University of Beirut for hospitality when some of this work was undertaken.

Hyper-CR equation
Consider three one-forms on a three-dimensional manifold B where (x, y, t) is a local coordinate system on B, and u, w are two functions of (x, y, t). Let ω = u x dy + (uu x + 2u y )dt, and V = u x 2 be another one-form, and a function on B. The Gauduchon-Tod system of equations [8] holds where * is the Hodge operator 1 of a Lorentzian metric on B if the functions (u, w) satisfy a system of integrable equations of hydrodynamic type (the hyper-CR system) u t + w y + uw x − wu x = 0, u y + w x = 0.
In particular, it is possible to fix a conformal gauge such that V ≡ −2ℓ −1 is a constant so that the monopole equation reduces to In this gauge ω is divergence-free (so this is the Gauduchon gauge -note that the converse is not true. There is a residual gauge freedom if the Gauduchon gauge has been fixed which allows for non-constant V ).

2.2.
The ψ-equation. Let ψ be any p-form of conformal weight m, so that ψ → e mf ψ under (4). The weighted exterior derivative is a (p + 1)-form of weight m. Let us assume that ψ is a one-form. Using the conformal properties of the Hodge operator we verify that the equation is conformally invariant as the weight of V is −1. In [11] this equation has arisen in a gauge where V = −2ℓ −1 is a constant, and m = −1 where it becomes There is a particular solution to this equation given by ψ = cω, where c is a constant.

2.3.
Example. The Heisenberg group. Let us consider a particular solution of (3) given by u = 4ℓ −1 x, w = 0, where ℓ is a constant. The resulting Lorentizian Einstein-Weyl structure is defined on the nilpotent Lie group: and V = 2ℓ −1 is a constant. A MAPLE-aided computation shows that the most general solution to (5) which does not depend on y is of the form where c = c(x) and k = k(t) are arbitrary functions.
In general we can establish the following Then there exists a local coordinate system (p, y, t) on B such that (h, ω) takes one of the following three forms where β = β(y, t) satisfies β t + β yy = 0.
Proof. We rewrite the non-linear constraint in (8) as dH ∧ dH x ∧ dt = 0, and perform a Legendre transform G(p, t, y) = H − px, where H x = p and x = −G p , and the constraint can be solved as G(p, y, t) = A(p, t) + pB(y, t).
Imposing the wave equation (the linear constraint in (8)) yields Substituting (12) and differentiating the resulting expression with respect to y yields There are two cases to consider.
• If B yyy = 0 then we can solve (14) for A pp , and find where µ is a constant. Setting B = −µ ln β(y, t) − κ(t) removes κ(t) from G and reduces the equation (13) to β t + µβ yy = 0. Rescaling t and p in the resulting EW structure can be used to set µ = 1. The function ρ(t) does not appear in the EW structure, so can be set to zero. This yields (9).
• If B yyy = 0 then B = c 1 (t)y 2 + c 2 (t)y + c 3 (t) and (13) implies thaṫ The classification now branches. If c 1 = 0 then B has to be linear in y, and take the form B = cy + γ(t), where c is a constant. The equation (13) becomes where a, b are some arbitrary functions of one variable. Substituting this into G shows that γ(t) disappears from the EW structure, and b(t) can be set to zero. The constant c can also be set to zero by an affine transformation of the coordinate p in the EW structure. This yields (10), where F = a pp is an arbitrary function of p. The nilpotent example (6) belongs to this class, and corresponds to F = −ℓ/4, where ℓ is a constant.
Next consider the case where c 1 = −1/(4t) (the constant of integration in the denominator has been set to zero by shifting t). The function c 3 (t) can be absorbed into A(t, p), and (13) gives c 2 (t) = c/t, where c is a constant. The resulting equation for A is tpA pp − 2t 2 A pt + 2c 2 = 0, which can be solved in terms of an arbitrary function of tp 2 . The constant c can be set to zero by shifting y → y − 2c. The final expression for the EW structure takes the form (11). The solution (7) belongs to this class with A p = 2 −4/3 (tp 2 ) −1/3 .

Einstein-Maxwell cosmological space-times
It is known [10,6,12] that Riemannian solutions to the hyper-CR Einstein-Weyl equations lift to supersymmetric solutions to the minimal gauged supergravity in four dimensions (see also [14] where the EW geometry appears in supergravity in a rather different context). Here, following [11], we present an analytic continuation of these constructions where the underlying base Einstein-Weyl manifold is Lorentzian, and the resulting four-dimensional theory admits pseudo-supersymmetry. The bosonic content of the theory consist of a metric, and a one-form which satisfy the cosmological Einstein-Maxwell equations with non-standard coupling between the Maxwell and the Einstein terms. The metric and the one-form given by satisfy the Einstein-Maxwell equations R ab + 3ℓ −2 g ab + 2F ac F b c − 1 2 |F | 2 g ab = 0, d ⋆ g F = 0 iff equations (2) and (5) hold with V = −2ℓ −1 .