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Supersymmetric gauge theories on five-manifolds

We construct rigid supersymmetric gauge theories on Riemannian five-manifolds. We follow a holographic approach, realizing the manifold as the conformal boundary of a six-dimensional bulk supergravity solution. This leads to a systematic classification of five-dimensional supersymmetric backgrounds with gravity duals. We show that the background metric is furnished with a conformal Killing vector, which generates a transversely holomorphic foliation with a transverse Hermitian structure. Moreover, we prove that any such metric defines a supersymmetric background. Finally, we construct supersymmetric Lagrangians for gauge theories coupled to arbitrary matter on such backgrounds.


Introduction
There has recently been considerable work on defining and studying supersymmetric gauge theories on curved backgrounds. The main reason for this interest is that these quantum field theories possess classes of observables that may be computed exactly using localization methods. Such non-perturbative results allow for quantitative tests of various conjectured dualities, and have also led to the discovery of new dualities. A primary example is the AdS/CFT correspondence, where exact strong-coupling field theory calculations may be compared to semi-classical gravity.
In this paper we focus on rigid supersymmetry in d = 5 dimensions, which is currently not as well-developed as its lower-dimensional cousins. Supersymmetric gauge theories were constructed and studied on the round S 5 in [1][2][3][4]. The product background S 1 × S 4 studied in [5,6] leads to the superconformal index. As in lower dimensions, the first constructions of non-conformally flat backgrounds were produced via various ad hoc methods. These include the squashed S 5 geometries of [7,8], and the product backgrounds S 3 × Σ 2 [9,10] and S 2 × M 3 [11][12][13]. In the latter two cases the spheres are round, while supersymmetry on the Riemann surface Σ 2 or three-manifold M 3 is gauged supergravity [29]. Some of the groundwork for this was laid in [30,31], where supergravity duals of the squashed five-sphere backgrounds of [7,8] were constructed (see also [32,33] for holographic duals to the supersymmetric Rényi entropy in five dimensions). We begin with a general supersymmetric asymptotically locally AdS solution to the Romans theory, and extract the conditions this imposes on the fivedimensional conformal boundary. Although the resulting spinor equations are quite complicated, we will show they are completely equivalent to a very simple geometric structure. We find that M 5 is equipped with a conformal Killing vector ξ = ∂ ψ which generates a transversely holomorphic foliation. This is compatible with an almost contact form η = dψ + ρ, where up to global constraints that we describe the norm S = ξ and ρ are arbitrary, and the transverse metric g (4) is Hermitian. The only other remaining freedom is an arbitrary function α (such that Sα is ξ-invariant), which together with the metric determines all the remaining background data. This structure is similar to the rigid limit of Poincaré supergravity described above, but with the addition of an integrable transverse complex structure and Hermitian metric. In fact it is a natural hybrid of the "real" three-dimensional rigid supersymmetric geometry studied in [34,35] and the four-dimensional supersymmetric geometry of [36,37] (where the four-manifold is complex with a compatible Hermitian metric).
The outline of the rest of the paper is as follows. In section 2 we summarize the form of supersymmetric asymptotically locally AdS solutions to Romans supergravity, in particular extracting the Killing spinor equations on the conformal boundary M 5 . These are then used as a starting point for a purely five-dimensional analysis in section 3. We show that the spinor equations are completely equivalent to a simple geometric structure on M 5 , and present a number of subclasses and examples, including many of the examples referred to above. In section 4 we construct N = 1 supersymmetric gauge theories formed of vector and hypermultiplets on this background geometry.
Our conclusions are presented in section 5.

Rigid supersymmetry from holography
The bosonic fields of the six-dimensional Romans supergravity theory [29] consist of the metric, a scalar field X, a two-form potential B, together with an SO(3) R ∼ SU(2) R Rsymmetry gauge field A i with field strength F i = dA i − 1 2 ε ijk A j ∧ A k , where i = 1, 2, 3. Here we are working in a gauge in which the Stueckelberg one-form is zero, and we set the gauge coupling constant to 1. The Euclidean signature equations of motion for this theory may be found in [31], although we will not require their explicit form here.
A solution is supersymmetric provided there exists a non-trivial SU(2) R doublet of Dirac spinors ǫ I , I = 1, 2, satisfying the following Killing spinor and dilatino equations Here Γ M are taken to be Hermitian and generate the Clifford algebra Cliff(6, 0) in an orthonormal frame, M = 0, . . . , 5. We have defined the chirality operator Γ 7 = iΓ 012345 , which satisfies (Γ 7 ) 2 = 1. The covariant derivative acting on the spinor is Given a supersymmetric asymptotically locally AdS solution we may introduce a radial coordinate r, so that the conformal boundary is at r = ∞ and the metric admits an expansion of the form ds 2 = 9 2 dr 2 r 2 + r 2 g µν + 1 Here x µ , µ = 1, . . . , 5, are coordinates on the conformal boundary, which has metric g = (g µν ). Notice that the particular form of the metric in (2.3) is not reparametrization invariant under r → Λr, where Λ = Λ(x µ ). However, the correction terms under such a transformation are subleading in the 1/r expansion. This will play an important role in the next section.
For simplicity we shall mainly consider Abelian solutions in which A 1 = A 2 = 0, and Similarly to the metric (2.3) we then write the following general expansions for the remaining bosonic fields where we define f ≡ da. Some of the terms a priori present in these expansions are set to zero by the equations of motion; for example, the O(1/r) term in the expansion of X [31]. The Killing spinors similarly admit an expansion of the form Here we have used the orthonormal frame for the metric (2.3). Furthermore, the spin connection expands as Also as in [31] we consider a "real" class of solutions for which ǫ I satisfies the symplectic Majorana condition ε J I ǫ J = C 6 ǫ * I ≡ ǫ c I , where C 6 denotes the charge conjugation matrix, satisfying Γ T M = C −1 6 Γ M C 6 . The bosonic fields are all taken to be real, with the exception of the B-field which is purely imaginary. With these reality properties, one can show that the Killing spinor equation (2.1) and dilatino equation (2.2) for ǫ 2 are simply the charge conjugates of the corresponding equations for ǫ 1 . In this way we effectively reduce to a single Killing spinor ǫ ≡ ǫ 1 , with SU(2) R doublet (ǫ, ǫ c ). We then note the following large r expansions of bilinears: 2 We also note that the bilinear ǫ † Γ (1) ǫ is a Killing one-form in the bulk [31]. This will hence restrict to a conformal Killing vector on the boundary at r = ∞.
Substituting the expansions (2.5) into the bulk Killing spinor equation (2.1), at the first two orders we obtain Here we take the spinors to be Grassmann even.
Here γ µ generate the Clifford algebra Cliff(5, 0) in an orthonormal frame, while ∇ denotes the Levi-Civita spin connection for the boundary metric g. Similarly, the bulk As explained in [31], equation (2.10) may be rewritten in the form of a charged conformal Killing spinor equation, with additional b-field couplings. Setting b = 0 one obtains the standard charged conformal Killing spinor equation, whose solutions (twistor spinors) have been studied in the holographic context for three-manifolds and four-manifolds in [28,[37][38][39][40]. On the other hand, previous work on rigid supersymmetry in five dimensions [18][19][20] has used Killing spinor equations of a different form, without the coupling to ϕ in (2.10). We may make closer contact with this work by noting that supersymmetry in the bulk also implies the algebraic relation This follows from the bilinear expansions (2.8).
In the remainder of the paper we shall take equations (2.10), (2.11), (2.12), and (2.13) as our starting point for a purely five-dimensional analysis.

Background geometry
In this section we begin with a Riemannian five-manifold (M 5 , g), on which we'd like to define rigid supersymmetric gauge theories. The gauge/gravity correspondence implies this should be possible, provided the spinor equations derived in the previous section hold.
Let us summarize the background data. In addition to the real metric g, we have two generalized Killing spinors χ, ϕ. Globally these are spin c spinors, being sections of the spin bundle of M 5 tensored with L −1/2 , χ, ϕ ∈ Γ[Spin(M 5 ) ⊗ L −1/2 ], where L is the complex line bundle for which the real gauge field a is a connection. This Abelian gauge field is a background field for U(1) R ⊂ SU(2) R , with (χ, χ c ), (ϕ, ϕ c ) forming SU(2) R doublets, where χ c ≡ Cχ * with C the five-dimensional charge conjugation matrix. The spinors χ, ϕ then satisfy the coupled Killing spinor equations (2.10), (2.11), where the background b-field is taken to be a purely imaginary two-form, A (0) is a purely imaginary one-form, while ω µν = g νσ ω σ µ is real and symmetric. Furthermore, χ and ϕ are related algebraically by (2.13), which introduces the additional background fields α and K 2 , which are respectively a real function and real one-form. Finally we have the dilatino equation (2.12), which introduces the real background function X 2 .
In the remainder of this section we shall analyse the geometric constraints that these equations impose on (M 5 , g). Although the background data and equations (2.10)-(2.13) appear a priori complicated, in fact we shall see that the geometry they are equivalent to is very simple.

Differential constraints
In the analysis that follows it is convenient to assume that the spin c spinor χ is nowhere zero. More generally χ could vanish along some locus Z ⊂ M 5 , and the local geometry we shall derive below is valid on M 5 \ Z. If Z is non-empty one would need to impose suitable boundary conditions, although we shall not consider this further in this paper.
A nowhere zero spin c spinor equips (M 5 , g) with a local SU(2) structure. Specifically, we may define the bilinears Here we have introduced the notation γ (n) ≡ 1 n! γ µ 1 ···µn dx µ 1 ∧ · · · ∧ dx µn , where x µ , µ = 1, . . . , 5, are local coordinates on M 5 . Since χ is nowhere zero the scalar function S is strictly positive, and it makes sense to normalize the bilinears as in (3.1). We note that K 1 is a real unit length one-form, while J is a real two-form with square length J 2 = 2. Here the square norm of a p-form φ is defined via φ 2 vol 5 = φ∧ * φ, where * denotes the Hodge duality operator on (M 5 , g) and vol 5 denotes the Riemannian volume form. The complex bilinear Ω is globally a two-form valued in the line bundle L −1 .
That χ, or equivalently the bilinears (3.1), defines a local SU(2) structure follows from some simple group theory. The spin group is Spin(5) ∼ = Sp(2) ⊂ U(4), with the latter acting in the fundamental representation on the spinor space C 4 . The stabilizer of a non-zero spinor is then Sp(1) ∼ = SU (2). When M 5 is spin and L is trivial, so that χ ∈ Γ[Spin(M 5 )], this defines a global SU(2) structure. However, more generally we require only that M 5 is spin c , and in this case the global stabilizer group is enlarged to U(2): the additional U(1) factor rotates the spinor by a phase, which may be undone by a U(1) gauge transformation. To see this in more detail we introduce a local orthonormal frame e a , a = 1, . . . , 5, so that where the metric is g = 5 a=1 (e a ) 2 . The U(2) = SU(2) × Z 2 U(1) structure group acts in the obvious way on the C 2 spanned by e 1 + ie 2 , e 3 + ie 4 . This leaves K 1 , J and the metric g invariant, but rotates Ω by the determinant of the U(2) transformation. In order for this to be undone by a gauge transformation, this identifies the line bundle as L = Λ 2,0 . The latter is the space of Hodge type (2, 0)-forms for the four-dimensional vector bundle spanned by e 1 , e 2 , e 3 , e 4 , and with almost complex structure I for which e 1 + ie 2 and e 3 + ie 4 are (1, 0)-forms. Thus our rigid supersymmetric geometry will in general be equipped with a global U(2) structure on M 5 (or more precisely on M 5 \ Z).
The one-form SK 1 = χ † γ (1) χ arises simply from the restriction of the bulk Killing one-form ǫ † Γ (1) ǫ to the conformal boundary, and thus defines a conformal Killing oneform on (M 5 , g). This is easily confirmed from the Killing spinor equation (2.10) for χ, which implies where we have introduced the dual vector field ξ, defined by g(ξ, ·) = SK 1 , and L denotes the Lie derivative.
One finds that the spinor equations (2.10)-(2.13) imply the following differential constraints: Notice that the background data X 2 , A (0) and ω µν in (2.10)-(2.13) does not enter equations (3.4)-(3.8): they simply drop out (one only needs to use the reality properties we specified, together with the fact that ω µν = ω νµ is symmetric).
It is straightforward to verify that (3.4)-(3.8) are invariant under the Weyl transfor- This symmetry is of course inherited from invariance under the change of radial variable r → Λr in the bulk. If S is nowhere zero notice that one might use this symmetry to set S ≡ 1.
Using equations (3.4)-(3.8) one can show that the conformal Killing vector ξ preserves all of the background geometric structure, provided one rescales the fields by appropriate powers of S according to their Weyl weights in (3.9). For instance, contracting ξ into the second equation in (3.4) shows that L ξ (Sα) = 0. On the other hand, taking the exterior derivative of the same equation one finds L ξ da = 0. One can hence locally choose a gauge in which a is invariant under L ξ , so that the second equation in (3.4

) is solved by
In a similar way, one can show that also S −1 b and S −2 J are invariant under L ξ , while S −2 Ω is invariant under L ξ in the gauge choice for which (3.10) holds. Notice that the . Without loss of generality it is convenient to henceforth impose L ξ S = 0. 3 In terms of the bulk expansion in section 2 this means choosing the radial coordinate r to be independent of the bulk Killing vector. This is a natural choice, which in turn implies that L ξ S = 0 and SK 1 is Killing, and we shall make this convenient (partial) conformal gauge choice in the following. We may then introduce a local coordinate ψ so that (3.11) The condition L ξ S = 0 is then equivalent to S being independent of ψ.

Geometric structure
The Killing vector ξ has norm S, and the dual one-form K 1 may be written locally as where i ξ ρ = 0. Notice that η has Weyl weight zero and norm 1/S. The local frame e 1 , e 2 , e 3 , e 4 provide a basis for D = ker η, and D inherits an almost complex structure from J. One then defines an endomorphism Φ of the tangent bundle of M 5 by where I is the almost complex structure. One easily verifies that Φ 2 = −1+ξ ⊗η, which is a defining relation of an almost contact structure. Moreover, the five-dimensional metric takes the form where ds 2 4 is Hermitian with respect to I. Although ξ is Killing, this structure is in general not a K-contact structure, which is a stronger condition. In particular the latter requires [41] that dη is the fundamental (1, 1)-form J associated to the transverse almost complex structure (which in general is not the case here), which in turn implies that η is a contact form, i.e. that η ∧ dη ∧ dη is a volume form (which in general is also not the case here). Notice that since ξ is nowhere zero, its orbits define a foliation of M 5 .
Let us now turn to the differential constraints (3.4)-(3.8). The two equations (3.4) allow us to write where b ⊥ and a ⊥ are basic forms for the foliation defined by ξ; that is, they are invariant under, and have zero interior contraction with, ξ. Recall that in writing the gauge field in the form in (3.15) we have made a (partial) gauge choice, as in (3.10). This leaves a residual gauge freedom a ⊥ → a ⊥ + dλ, where λ is a basic (ξ-invariant) function. The equation (3.6) is simply equivalent to b being invariant under ξ.
The differential constraint (3.5) reduces to Here * 4 is the Hodge dual with respect to the transverse four-dimensional metric ds 2 4 , with volume form e 1 ∧ e 2 ∧ e 3 ∧ e 4 . It is then convenient to introduce decomposing into the transversely self-dual and anti-self-dual parts. Equation (3.16) is then equivalent to The constraint (3.7) simply identifies with the Lee form θ of the transverse four-dimensional Hermitian structure. That is, every four-dimensional Hermitian structure with fundamental two-form J satisfies dJ = θ ∧ J. Finally, the differential constraint (3.8) now reads This implies that the almost complex structure I is integrable, thus defining a CR structure on M 5 . It also means that ξ defines a transversely holomorphic foliation of M 5 , and we may introduce local coordinates ψ, z 1 , z 2 adapted to the foliation, where the transition functions between the z 1 , z 2 coordinates are holomorphic.
Notice that we may rewrite (3.20) as where we have defined a Chern ≡ a ⊥ − I(θ) , (3.22) and I(θ) ≡ −i θ # J, where θ # is the vector field dual to θ. To obtain an explicit expression for the Chern connection a Chern , we begin by noting that Ω ∧Ω = 2J ∧ J.
Using local coordinates z α , α = 1, 2, for the transverse space we may write which implies that |f | = det g (4) . Notice that globally f is a section of L −1 , where L ∼ = Λ 2,0 ≡ K is the canonical bundle. Writing f = |f |e iφ we then have where d c ≡ I • d. We thus recognize (up to gauge) as the Chern connection on the canonical bundle.
The geometric content of the differential constraints (3.4)-(3.8) may hence be summarized as follows. M 5 is equipped with a transversely Hermitian structure, so that the metric takes the form Here the Killing vector is ξ = ∂ ψ , which generates a transversely holomorphic foliation. The almost contact form is η = (dψ + ρ), and ds 2 4 is a transverse Hermitian metric. One is also free to specify the functions α and S. Given this data, the remaining background fields a and b that enter (3.4)-(3.8) are determined via In particular the choice of a transverse Hermitian metric g (4) fixes the two-form J, and hence the Lee form θ, while the Hodge type (2, 0)-form Ω and Chern connection a Chern in (3.25) are also determined up to gauge. Notice that the terms Sαη and I(θ) entering the formula for a in (3.27) are both global one-forms on M 5 , implying that globally a is a connection on L = Λ 2,0 .
We shall furthermore show in section 3.4 that any choice of transversely Hermitian structure on M 5 of the above form gives a supersymmetric background. In particular the remaining background fields X 2 , A (0) , and ω µν appearing in the spinor equations (2.10)-(2.13) are also determined by the above geometric data.

General families
We begin by noting some special families of backgrounds: • Setting ρ = 0 and S ≡ 1 gives a product metric where M 4 is any Hermitian four-manifold. Notice this four-manifold geometry is the same as the rigid supersymmetric geometry one finds in four dimensions [28,36]. The first reference here follows a similar holographic approach to the present paper, while the second takes a rigid limit of "new minimal" supergravity in four dimensions.
-If θ = 0 and dρ is a positive constant multiple of J then the five-metric is locally conformally Sasakian. Supersymmetric gauge theories on Sasaki-Einstein manifolds, for which furthermore S ≡ 1 and g is a positively curved Einstein metric, were defined in [23], and further studied in [21,[25][26][27].
• We may take any circle bundle over a product of Riemann surfaces S 1 ֒→ Σ 1 ×Σ 2 .
The Hermitian metric may be taken to be simply a product of two metrics on the Riemann surfaces, while ρ is the connection one-form for the fibration. One can generalize this further by allowing S 1 orbibundles over a product of orbifold Riemann surfaces.
-If we only fibre over Σ 1 , this leads to direct product M 3 ×Σ 2 solutions, where M 3 is a Seifert fibred three-manifold. Notice this three-manifold geometry is the same as the rigid supersymmetric geometry in three dimensions [34]. Maximally supersymmetric Yang-Mills theory has been studied on similar backgrounds in [9][10][11][12][13], including the direct products S 3 × Σ 2 and M 3 × S 2 .
Here the spheres are equipped with round metrics and the associated canonical spinors, while the spinors on Σ 2 and M 3 are constructed by topologically twisting with the SU(2) R symmetry.
• Finally, if dρ has Hodge type (1, 1) the transversely holomorphic foliation admits a complexification [42], i.e. adding a radial direction to ξ we naturally have a complex six-manifold M 6 , with a transversely holomorphic foliation. Notice that Sasakian geometry and the direct product

Squashed Sasaki-Einstein
We have already noted that a Sasakian five-manifold is a particular case of a supersymmetric background. Recall that Sasakian metrics take the form where η defines a contact structure on M 5 , with Reeb Killing vector field ξ, and ds 2 4 is a transverse Kähler metric. Moreover dη = dρ = 2J. If the transverse Kähler metric g (4) is Einstein, then the metric (3.28) is said to be a squashed Sasaki-Einstein metric. 4 For a given choice of transverse Kähler-Einstein metric, we obtain a two-parameter family of backgrounds, parametrized by the constants c 1 , c 2 : The Kähler-Einstein metric g (4) satisfies the Einstein equation Ric (4) = 2(2 √ 2c 1 − c 2 )g (4) . Notice that we have presented the solution (3.29) in a different gauge choice to (3.10). We may impose the latter gauge choice by simply transforming a → a + (2 √ 2c 1 − c 2 )dψ, although the form of a in (3.29) makes it clear that we may take a to be a global one-form on M 5 for this particular class of solutions.
When g 4 is taken to be the standard metric on CP 2 , the above geometry is a squashed five-sphere. This corresponds to the conformal boundary of the 1/4 BPS bulk Romans supergravity solutions constructed in [31].

Black hole boundary
In this section we consider the conformal boundary of the 1/2 BPS topological black hole solutions constructed in [32]. We begin with the following product metric on S 1 × H 4 , where H 4 is hyperbolic four-space: Here τ is a periodic coordinate on S 1 , q is a radial coordinate with q ∈ [0, ∞), ϑ ∈ [0, π 2 ] while ϕ 1 , ϕ 2 have period 2π. The metric in brackets is simply the round metric on a unit radius S 3 . For this solution b vanishes identically, while a is gauge-equivalent to zero. The Killing spinors for this background [32] in general depend on four integration 4 In the mathematical literature [41] these are called η-Sasaki-Einstein metrics.
constants (being 1/2 BPS), but for simplicity here we present only the "toric" solution discussed in [32]. The remaining fields are then while in a gauge 5 in which a = 0 the U(2) structure is given by The supersymmetric Killing vector is Furthermore, notice that rescaling J by 1/(q 2 + 1) leads to a closed two-form, hence showing that the Hermitian metric transverse to ξ is conformal to a Kähler metric. Moreover, one can also check that the almost contact form η = K 1 /S is a contact form in this case, i.e. that η ∧ dη ∧ dη is a volume form.

Conformally flat S 1 × S 4
In this section we consider the conformally flat metric on S 1 × S 4 , which we may write as where ds 2 S 4 = dβ 2 + sin 2 β(dϑ 2 + sin 2 ϑdϕ 2 1 + cos 2 ϑdϕ 2 2 ) . Here τ is a periodic coordinate on S 1 , while the metric in brackets in (3.35) is simply the round metric on a unit radius S 3 , as in the previous black hole boundary example.
The polar coordinate β ∈ [0, π]. The metric (3.34) of course arises as the conformal boundary of Euclidean AdS in global coordinates, and as such the background fields a = 0 = b. There are many Killing spinors in this case, and here we simply choose one so as to present simple expressions for the remaining background data. We find The U(2) structure is given by Notice that in this example we obtain a conformal Killing vector from the Killing spinor bilinear, but not a Killing vector. As described at the end of section 3.1, we may always make a Weyl transformation of the background to obtain a Killing vector.
In the case at hand this corresponds to the Weyl factor Λ = e τ , and the corresponding Weyl-transformed metric is then (locally) flat, with the Weyl-transformed J and Ω both closed and hence defining a transverse hyperKähler structure. Nevertheless, the fact that the metric (3.34) leads to a conformal Killing vector explains why this background is missing from the rigid supersymmetric geometry in [18,19]: in the latter references the corresponding bilinear is necessarily a Killing vector. This also suggests that the conjecture made in [19] is likely to be correct: that is, to obtain the S 1 ×S 4 background from a rigid limit of supergravity, one should begin with conformal supergravity in five dimensions, rather than Poincaré supergravity.

Squashed S 5
We consider the squashed five-sphere metric where s ∈ (0, 1] is the squashing parameter and C ≡ − 1 2 sin 2 σ(dβ + cos ϑdϕ) . (3.39) The coordinates σ, β, ϑ, ϕ are coordinates on the base CP 2 , with β having period 4π, ϕ having period 2π, while σ ∈ [0, π 2 ], ϑ ∈ [0, π], and 1 2 dC is the Kähler two-form on CP 2 . For the "toric" family discussed in [30,31] we find The other background fields are, in an appropriate gauge (i.e. not that in (3.10)), The U(2) structure is The supersymmetric Killing vector is One also computes where vol 5 denotes the Riemannian volume form and η = K 1 /S is the almost contact form. The right hand side of (3.45) can have non-trivial zeros, and we thus see that in general η does not define a contact structure. These backgrounds arise as the conformal boundary of the 3/4 BPS solutions of Romans supergravity constructed in [30,31].

From geometry to supersymmetry
In this section we will show that any choice of transversely Hermitian structure on M 5 defines a supersymmetric background. The background U(1) R gauge field a and the b-field are given in terms of the geometry by (3.27). It then remains to show that the geometry also determines the fields X 2 , A (0) and ω µν , in such a way that the original spinor equations (2.10)-(2.13) are satisfied.
We first examine the Killing spinor equation (2.10) for χ. In order to proceed it is convenient to choose a set of projection conditions (see for example [43]) These allow one to substitute for the fields b and K 2 in terms of the geometry, via (3.27) and (3.19), into the right hand side of equation (2.10). In doing this calculation it is also convenient to write Ω = J 2 + iJ 1 , J = J 3 so that In this way it is straightforward to show that the µ = 5 (the ψ direction) component of (2.10) simply imposes ∂ ψ χ = 0. 6 Thus χ is independent of ψ. Taking instead µ = m, m = 1, 2, 3, 4, one finds (2.10) is equivalent to where ∇ (4) denotes the Levi-Civita spin connection for the transverse four-dimensional metric. Recall that the latter metric is Hermitian. It is then more natural to express equation (3.48) in terms of an appropriate Hermitian connection, which preserves both the metric and the two-form J. The Chern connection is such a connection, defined by This coincides with the Levi-Civita connection if and only if dJ = 0 (equivalently θ = 0), so that the metric is Kähler.
Next, let us notice that under the Weyl transformation (3.9) we have χ → Λ 1/2 χ, so that it is also natural to introduceχ where recall that a Chern = a ⊥ − I(θ) is the Chern connection for the canonical bundle K ≡ Λ 2,0 , given explicitly by (3.25). It is then a standard fact, and is straightforward to show, that any Hermitian space admits a canonical solutionχ to (3.51). Specifically, any Hermitian space admits a canonical spin c structure, with twisted spin bundles Spin c = Spin ⊗ K −1/2 . In four dimensions this is isomorphic to where Λ p,q denotes the bundle of forms of Hodge type (p, q). In the case at hand, these are defined transversely to the foliation generated by the Killing vector ξ. Under (3.52) the Killing spinorχ = S −1/2 χ is a section of the trivial line bundle Λ 0,0 . Moreover, the Chern connection restricted to this summand is flat, with the induced connection − 1 2 a Chern on the twist factor K −1/2 effectively cancelling that coming from the spin bundle. Concretely, in terms of local complex coordinates z α , α = 1, 2, we have (ω Chern ) β α = (∂g (4) ) αγ (g (4) )γ β , and using the projection conditions (3.46) one can show this precisely cancels the contribution from (3.25). The spin c spinorχ is simply a constant length section of this flat line bundle. Put simply, the rescaled Killing spinor χ = S −1/2 χ is constant.
Next we turn to the dilatino equation (2.12). Substituting for ϕ in terms of χ, using (2.13), after a somewhat lengthy computation one finds the dilatino equation holds provided and Here we have introduced the notation φ 1 ∧ * φ 2 = 1 p! φ 1 , φ 2 vol 5 for the inner product between two p-forms φ 1 , φ 2 . Notice that the expression (3.53) for the imaginary oneform A (0) coincides with that in [31], which was derived by solving the bulk equations of motion near the conformal boundary, in terms of the boundary data. Notice that under the Weyl scaling (3.9) we have The fact that X 2 has Weyl weight −2 is clearly consistent with the bulk expansion (2.4), but the "anomalous" transformation of A (0) in (3.55) naively appears to contradict (2.4), for which A (0) has Weyl weight −1. However, this is where the comment above equation (2.4) is relevant: the reparametrization r → Λr does not preserve the subleading terms in the metric (2.3). It is therefore not a strict symmetry of the system we have defined. However, the leading order terms in the expansions (2.3), (2.4) are invariant. This explains why the differential constraints (3.4)-(3.8) have the Weyl symmetry (3.9), while the higher order term A (0) arising in the expansion of the B-field does not. One could restore the full Weyl symmetry by adding a cross term 9 dr r C µ dx µ into the metric (2.3), so that under r → Λr preserves the form of the metric. Then C is a new background field on M 5 , and one finds This now has Weyl weight −1, as expected, and the anomalous variation in (3.55) arises simply because we have made the gauge choice C = 0 in our original expansion. In general notice that a field of Weyl weight w will couple to a Weyl covariant derivative D µ ≡ ∂ µ + w C µ , and w = 2 for * b.
It remains to show that the background geometry implies the ϕ Killing spinor equation (2.11). At this point notice that everything is fixed uniquely in terms of the free functions α and S, and the transversely Hermitian structure on M 5 , apart from the higher order spin connection term ω µν which appears in (2.11). After a lengthy computation, in our orthonormal frame one finds the expression This is manifestly real and symmetric, apart from the last term in the penultimate line.
However, it is straightforward to show that (β − ) mp J p n is symmetric for any transverse anti-self-dual two-form β − . Thus (2.11) is satisfied provided ω µν is given by (3.58). We conclude this subsection by noting the following formula This trace will appear in the supersymmetric Lagrangians constructed in section 4. (ii) This foliation is compatible with an almost contact form η = dψ + ρ. Choose a particular ρ = ρ 0 , which notice is defined only locally in the foliation patches, gluing together to give the global η. Then for fixed foliation any other choice of ρ is related to this by ρ = ρ 0 + ν, where ν is a global basic one-form. That is, ν is a global one-form on M 5 satisfying L ξ ν = 0 = i ξ ν.

Summary
(iii) One can choose an arbitrary transverse Hermitian metric ds 2 4 , invariant under ξ and compatible with the foliation.
(iv) Finally, one is free to choose the ξ-invariant real functions α and S (with S nowhere zero).
An interesting special case is when all the leaves of the foliation are closed, so that ξ generates a U(1) action on M 5 and ψ is a periodic coordinate. In this case

Supersymmetry algebra
An off-shell N = 1 vector multiplet in five dimensions consists of a gauge field A µ , a real scalar σ, a gaugino λ I , and a triplet of auxiliary scalars D IJ , all transforming in the adjoint representation of the gauge group G. Here I, J = 1, 2 are SU(2) R symmetry indices. The gaugino is a symplectic-Majorana spinor which satisfies (λ α I ) * = ε IJ C αβ λ β J whilst the auxiliary scalars satisfy (D IJ ) † = ε IK ε JL D KL , where recall that ε IJ is the Levi-Civita symbol.
We introduce the following covariant derivatives: where ∇ is the Levi-Civita spin connection. In general we may consider turning on an SU(2) R background gauge field a i µ , i = 1, 2, 3, or equivalently we may introduce where σ i , i = 1, 2, 3, denote the Pauli matrices. In section 2 recall that for simplicity we restricted to an Abelian background gauge field, with a 1 = a 2 = 0, a 3 = a, but in this section we will relax this assumption. There is also a background two-form b-field and we choose to introduce the gauge field C µ associated with restoring Weyl invariance -see the earlier discussion around equation (3.56). With this background gauge field active we modify the covariant derivatives to so that they are covariant with respect to both Weyl and R-symmetry transformations. These correspond to Weyl weights w = (−1, 0, − 3 2 , −2) for the gauge multiplet (σ, A µ , λ I , D IJ ).
Given this background data we consider the following (conformal) supersymmetry variations: This has Grassmann odd supersymmetry parameters ξ I ,ξ I . We find that these transformations close onto and R J I = ε JK R IK , provided that the spinors (ξ,ξ) and (η,η) satisfy the SU(2) Rcovariantization of the (χ, ϕ) spinor equations (2.10)-(2.12). More precisely Recall that b has Weyl weight w = 1, while the spinors have weight w = ±1/2.
It is crucial for the closure of the algebra that ω µν = ω νµ , which is the same condition used in deriving the differential constraints (3.4)-(3.8). Also as for that computation the closure of the supersymmetry algebra is insensitive to the explicit form of ω µν , A (0) or X 2 . Let us also notice that the supersymmetry variations (4.5) reduce to those of the round S 5 in [2] (in particular b ≡ 0 for the round S 5 , andξ here invariance limits the prepotential to being at most cubic in V [44] and classically it takes the form (4.14) Here g is the dimensionful gauge coupling constant and k is a real constant which is subject to a quantization condition dependent on the gauge group [45]. Writing the components of the vector superfield as V a T a = (σ a T a , A a µ T a , λ a I T a , D a IJ T a ) where T a are generators of the gauge group in the adjoint representation we find the cubic prepotential term in our curved backgrounds to be Here d abc ∝ k π 2 Tr T (a T b T c) is a symmetric invariant tensor of the gauge group. It vanishes 7 for all simple gauge groups except U(1) or SU(N) with N ≥ 3. The Lagrangian L cubic is invariant under the superconformal transformations (4.5) provided the supersymmetry parameters satisfy (4.8), and in addition A (0) is given by which matches precisely the expression (3.57) in section 3.
The quadratic term in the prepotential includes Yang-Mills kinetic terms and is not conformally invariant. We therefore expect to break conformality by using the relatioñ which is the SU(2) R -covariantization of (2.13). The Lagrangian describing the quadratic piece can be found from L cubic by identifying one of the vector superfields with a constant supersymmetry preserving Abelian vector multiplet [7]. That is 7 For example, taking the gauge group to be G = SO(N ) so that the Lie algebra generators satisfy IJ ). We choose σ (1) = 1 and λ (1) I = 0. Then V (1) is supersymmetry preserving if the fermion variation holds for non-trivial spinor parameters ξ I and some choice of D µ such that F (1) = dA (1) . Here we have substituted forξ I using (4.17). To progress, note that there are two natural one-forms in our geometry namely K 1 and K 2 . If we concentrate on K 1 which, with S = 1, satisfies (3.5) then upon SU(2) R -covariantizing and multiplying by − 1 2 γ µν ξ I we find To derive the previous equation we have used the projection conditions satisfied by the background geometry: (K 1 ) µ γ µ χ = χ and J µν γ µν χ = 4iχ, along with (K 1 ) µ (K 2 ) µ = 0 = (K 1 ) µ C µ and −i(K 1 ) µ b µν = (K 2 ) ν + 3 √ 2 C ν . Comparing this to (4.19) gives the constant vector multiplet as IJ ) = (1, (K 1 ) µ , 0, 2 √ 2iα IJ ) , (4.22) and the corresponding Yang-Mills Lagrangian is The second candidate one-form is K 2 but taking F (1) = dK 2 does not lead to (4.19).
The superconformal Lagrangian for the vector coupled hypermultiplets exists irrespective of the gauge group and is straightforward to construct: we simply integrate the equations of motion (4.12) and (4.13) found from closing the superalgebra to find (4.24)

Discussion
In this paper we have constructed rigid supersymmetric gauge theories with matter on a general class of five-manifold backgrounds. By construction these are the most general backgrounds that arise as conformal boundaries of six-dimensional Romans supergravity solutions. We find that (M 5 , g) is equipped with a conformal Killing vector which generates a transversely holomorphic foliation. In particular the transverse metric g (4) is an arbitrary Hermitian metric with respect to the transverse complex structure. This is a natural hybrid/generalization of the rigid supersymmetric geometries in three and four dimensions constructed in [34,36,37], and includes many previous constructions as special cases.
It is interesting to compare the geometry we find to the rigid limit of Poincaré supergravity [18,19] and the twisting of [21]. In the former case the backgrounds naively appear to be more general, as there is no almost complex structure singled out, nor integrability condition. However, they don't include the S 1 × S 4 geometry relevant for the supersymmetric index, which as we showed in section 3.3 is included in our backgrounds. In fact the singling out of the almost complex structure associated to J = J 3 , where recall that Ω = J 2 + iJ 1 , in our geometry is almost certainly related to the fact that in section 3 we focused on the case where we turn on only an Abelian U(1) R ⊂ SU(2) R . This was motivated in part for simplicity, and in part because the known solutions to Romans supergravity discussed previously also have this property. Nevertheless, the supersymmetry variations and Lagrangians we constructed in section 4 are valid for an arbitrary background SU(2) R gauge field, and it should be relatively straightforward to analyse the geometric constraints in this more general case. Indeed, this is certainly necessary, and presumably sufficient, to reproduce the partially topologically twisted backgrounds S 2 × M 3 of [11][12][13], since the SU(2) spin connection of M 3 is twisted by SU(2) R . On the other hand recall that the twisting in [21] requires that M 5 be a K-contact manifold. This shares many features with our geometry, with one important difference: for a K-contact manifold the transverse two-form J is closed, so the corresponding foliation is transversely symplectic; however, our case is in some sense precisely the opposite, namely transversely holomorphic. These intersect precisely for Sasakian manifolds. It is interesting that these various approaches generally seem to lead to different supersymmetric geometries, with varying degrees of overlap.
Given the geometry we find and the results of [46], it is natural to conjecture that the partition function and other BPS observables depend only on the transversely holomorphic foliation, i.e. for fixed such foliation they are independent of the choice of the remaining background data (functions S, α, the one-form ν defined in section 3.5, and the transverse Hermitian metric g (4) ). It will be interesting to verify that this is indeed the case, and to compute these quantities using localization methods. Notice that locally a transversely holomorphic foliation always looks like R × C 2 , which perhaps also explains why in [19] the authors found that locally all deformations of their backgrounds were Q-exact. Finally, our construction allows one to address holographic duals of these questions, which we plan to return to in future work.