Special Geometry and Space-time Signature

We construct N=2 four and five-dimensional supergravity theories coupled to vector multiplets in various space-time signatures (t,s), where t and s refer, respectively, to the number of time and spatial dimensions. The five-dimensional supergravity theories, t+s=5, are constructed by investigating the integrability conditions arising from Killing spinor equations. The five-dimensional supergravity theories can also be obtained by reducing Hull's eleven-dimensional supergravities on a Calabi-Yau threefold. The dimensional reductions of the five-dimensional supergravities on space and time-like circles produce N=2 four-dimensional supergravity theories with signatures (t-1,s) and (t,s-1) exhibiting projective special (para)-K\"ahler geometry.


Introduction
The study of the geometry of the scalar manifolds of Euclidean N = 2 vector and hypermultiplets with or without coupling to supergravity has recently been considered in [1][2][3][4]. Our present work will only focus on theories with vector multiplets coupled to supergravity. In the standard supergravity theories with Lorentzian signature, it is well known that the scalar manifold is described by a projective special Kähler manifold in four dimensions and by a projective special real manifold in five dimensions. In [3], four-dimensional Euclidean supergravity theories were obtained by dimensionally reducing the five-dimensional Lorentzian theories of [5] over a time-like circle. It was established that the scalar geometries of the four-dimensional Euclidean vector multiplets can be obtained by replacing complex structures by para-complex structures. Four-dimensional Euclidean supergravity theories were also obtained as a dimensional reduction of the Euclidean ten-dimensional supergravity on a Calabi-Yau three-fold [6]. The Killing spinor equations of the Euclidean four-dimensional supergravities were also obtained in [7] and their gravitational solutions admitting Killing spinors were analysed in [8].
The theories of N = 2, five-dimensional Euclidean vector multiplets coupled to supergravity has recently been constructed in [9]. The Lagrangian of the Euclidean theory is the same as in the Lorentzian theory except that the gauge fields terms appear with the opposite sign. Multi-centered solutions of the gauged versions of these theories were recently studied in [10]. The dimensional reduction of the five-dimensional Euclidean theory on a circle produces the N = 2 four-dimensional Euclidean supergravity of [3] but with the non-conventional signs of the gauge terms.
In this work, our aim is to obtain N = 2 four and five-dimensional supergravity theories coupled to vector multiplets in various space-time signatures (t, s) , where t and s refer, respectively, to the number of time and spatial dimensions. Space-times with various signatures are of mathematical and physical interest. For instance, spaces with (2, 2) signature have applications to string theory, M-theory, cosmology and twistor theory [11]. Moreover, the (2, 2) theory and its solutions without matter fields, have been considered in [12]. More recently, a classification of solutions with Killing spinors for the (2, 2) Einstein-Maxwell the-ory with a cosmological constant was given in [13]. We organise our work as follows. In Sec.
2, the five-dimensional supergravity theories are constructed through the analysis of the integrability conditions arising from generalised Killing spinor equations. The theories with various signatures are then obtained by reducing Hull's eleven-dimensional supergravity [14] on Calabi-Yau threefold. In Sec. 3, we obtain new N = 2 four-dimensional supergravity theories via the dimensional reductions of the five-dimensional supergravities on space and time-like circles. In particular, we obtain N = 2 supergravity with signature (2, 2) with scalar manifold described by a projective special para-Kähler manifold. We also obtain the Killing spinor equations for the reduced four-dimensional N = 2 supergravity theories. We end with a summary of our results.

(t, s) Five-Dimensional Supergravity
We start our analysis with the original theory of N = 2, D = 5 supergravity theory coupled to Abelian vector multiplets constructed in [5]. The theory contains the gravity multiplet and n vector multiplets and its bosonic Lagrangian is given bŷ where C ijk are real constants symmetric in i, j, k. The dynamics of (2.1) is encoded in the cubic potential where the very special coordinates h i are functions of the n real scalar fields belonging to the vector multiplets. The scalar manifold is described by the very special geometry The gauge coupling metric takes the form where the dual fields h i are given by The Killing spinor equations arising from the vanishing of the fermionic fields and their supersymmetry transformations can be written in the form 1 In what follows we obtain five-dimensional theories with various space-time signatures.
One method to do so is through the analysis of the integrability of the Killing spinor equations. We start by allowing for a slight modification of the Killing spinor equations and After some calculation one can derive the following integrability condition The vanishing of the second and third lines in the above equation constitute the equations of motion for the scalar fields in a theory where the gauge kinetic terms coefficient is α 2 4 . Assuming that this is the case, then (2.8) reduces to If we modify the action (2.1) to take the form (2.10) 1 Our conventions are as follows: The Clifford algebra is {Γ a , Γ b } = 2η ab . The covariant derivative on spinors isDm = ∂m + 1 4 ωm ,ab Γ ab where ωm ,ab is the spin connection. Finally, antisymmetrization is with weight one, so Γ a1a2···an then the equations of motion for the gauge fields derived from (2.10) are given bŷ In theories with (1,4), (3,2) and (5, 0) and signature (1, 10) on a Calabi-Yau three-fold, CY 3 [16]. Here F 4 = dC 3 and C 3 is a 3-form.
The eleven-dimensional space-time manifold decomposes into is a Lorentzian five-dimensional manifold. Some useful details on the mathematics of CY 3 as well as the reduction on CY 3 can be found for example in [17][18][19][20].
We shall briefly present the basics of the reduction relevant to our discussion. One where q i are the Kähler moduli taken to depend on the coordinates of M 5 and are the basis of h (1,1) harmonic forms, ξ A represent the three complex coordinates of CY 3 .
Note that the Kähler form and the volume of CY 3 are given by The Kähler moduli space metric is given by where we have used the notation Next one has to evaluate the eleven-dimensional Ricci curvature in terms of the Kähler moduli taking into consideration that for a CY 3 , we have In addition, we use the Kaluza-Klein ansatz for the three-form then the reduction of the action (2.12) after a rescaling of the five-dimensional metric and redefining scalars gives 2 where G ij (h) is given by (2.4) and F i 2 = dA i . Note that G ij (h) is obtained from (2.16) by simply replacing the M i with h i .
The action of the eleven-dimensional supergravities constructed by Hull [14] can be written in the form where α 2 = −1 for the theories with signatures (1, 10), (5,6) and (9,2) , and α 2 = 1 for the mirror theories with signatures (10, 1), (6,5) and (2,9). In the reduction of the theories with 2 We have ignored a kinetic term for the scalar field related to the volume of the Calabi-Yau and belongs to the hypermultiplet sector.

Four-Dimensional Supergravity
Starting with the action (2.22) of the N = 2 supergravity theory in five dimensions with (t, s) signature, we reduce the theory on a space-like and time-like circle. The Kaluza-Klein reduction ansatz is given bŷ All the fields are taken to be independent of the compact dimension labelled by index 0, and the vector A 0 has a vanishing component along the compact dimension. The non-vanishing components of the spin connection are given bŷ where ǫ = 1 corresponds to a reduction on a time-like circle, and ǫ = −1 on a space-like circle. Note that all the indices on the right hand side of (3.2) are four dimensional, ω a,bc are the spin connections of the four-dimensional theory with basis e a and F 0 = dA 0 .
The reduction of the action in (2.22) results in the following Lagrangian Cyyy.
In what follows we shall demonstrate that the action (3.3) describes a four-dimensional N = 2 supergravity theory with various signatures coupled to vector multiplets with the Lagrangian with the prepotential The n complex scalar fields z i of N = 2 vector multiplets are coordinates of a projective special (para)-Kähler manifold. In the symplectic formulation of the theory [21], one introduces the symplectic vectors satisfying the symplectic constraint Here X I = ReX I + i ǫ ImX I , i ǫ satisfiesī ǫ = −i ǫ and i 2 ǫ = τ , where τ = 1 for the case when the scalar fields geometry is given by a projective special para-Kähler manifold and τ = −1 when it is given by a projective special Kähler manifold, F I = ∂F ∂X I and F IJ = ∂ 2 F ∂X I ∂X J . The constraint (3.9) can be solved by setting X I = e K(z,z)/2 X I (z) (3.10) where K(z,z) is the Kähler potential. Then we have The metric of the special (para)-Kähler manifold is given by and locally its U(1) connection A is given A convenient choice of inhomogeneous coordinates z i are the special coordinates defined by (3.14) Defining then for theories with cubic prepotentials given in (3.7), we obtain for the scalars kinetic term The gauge field coupling matrix is given bȳ which for theories with cubic prepotential gives (3.18) Using the above information we obtain After making the identification τ = −α 2 ǫ, and definingF iab = τ 2 ǫ abcd F i cd , (3.6) is equivalent to (3.3).
The Killing spinors equations of the five-dimensional supergravity theories with signature (2, 3), (4, 1) and (0, 5) are given bŷ Those can be shown to reduce to

Summary
In this work we have constructed N = 2 four and five-dimensional supergravity theories in various space-time signatures. The five-dimensional theories were constructed by employing the integrability conditions of the Killing spinor equations as well as by via the reduction of the eleven-dimensional supergravities constructed by Hull [14] on a CY 3 . Among the fivedimensional theories constructed, we obtained the Euclidean five-dimensional supergravity recently constructed in [9] and its mirror theory. The four-dimensional supergravity theories were then obtained as reductions of the five-dimensional theories on a time-like and space-like circles. One of the new four-dimensional supergravity theories obtained are the Lorentzian theories with signature (1, 3) with projective special Kähler geometry and with the wrong sign of the gauge coupling terms. Solutions of these (1, 3) theories with space-like Killing vectors were considered in [22]. There, these theories were labelled as fake theories. In the present work, however, they were shown to be genuine theories with higher dimensional origins. Also, in four dimensions a new theory with signature (2, 2) is obtained where the scalar manifold is described by a projective special para-Kähler manifold. A future direction is finding solutions to all these theories. The Killing spinor equations constructed should provide a starting point for a systematic analysis of their supersymmetric solutions. Also of interest is the reduction of the four-dimensional theories down to three dimensions and the investigations of the resulting c-maps along the lines of [4]. We hope to address these questions in forthcoming publications.