Supersymmetric gauge theories on five-manifolds
Luis F. Alday, Pietro Benetti Genolini, Martin Fluder, Paul Richmond, James Sparks
aa r X i v : . [ h e p - t h ] A ug August 25, 2015
Supersymmetric gauge theories on five-manifolds
Luis F. Alday, Pietro Benetti Genolini, Martin Fluder,Paul Richmond and James Sparks
Mathematical Institute, University of Oxford,Andrew Wiles Building, Radcliffe Observatory Quarter,Woodstock Road, Oxford, OX2 6GG, UK
Abstract
We construct rigid supersymmetric gauge theories on Riemannian five-manifolds.We follow a holographic approach, realizing the manifold as the conformal bound-ary of a six-dimensional bulk supergravity solution. This leads to a systematicclassification of five-dimensional supersymmetric backgrounds with gravity du-als. We show that the background metric is furnished with a conformal Killingvector, which generates a transversely holomorphic foliation with a transverseHermitian structure. Moreover, we prove that any such metric defines a super-symmetric background. Finally, we construct supersymmetric Lagrangians forgauge theories coupled to arbitrary matter on such backgrounds. ontents
There has recently been considerable work on defining and studying supersymmetricgauge theories on curved backgrounds. The main reason for this interest is that thesequantum field theories possess classes of observables that may be computed exactlyusing localization methods. Such non-perturbative results allow for quantitative testsof various conjectured dualities, and have also led to the discovery of new dualities. Aprimary example is the AdS/CFT correspondence, where exact strong-coupling fieldtheory calculations may be compared to semi-classical gravity.In this paper we focus on rigid supersymmetry in d = 5 dimensions, which is currentlynot as well-developed as its lower-dimensional cousins. Supersymmetric gauge theorieswere constructed and studied on the round S in [1–4]. The product background S × S 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 geometries of [7, 8], and the productbackgrounds S × Σ [9, 10] and S × M [11–13]. In the latter two cases the spheresare round, while supersymmetry on the Riemann surface Σ or three-manifold M is1chieved via a topological twist utilizing the SU (2) R symmetry of the theory. Theseconstructions have been used to successfully test AGT-type correspondences.A systematic method for constructing rigid supersymmetric field theories on curvedbackgrounds, in any dimension d , was initiated in [14]. Here one first couples the fieldtheory to off-shell supergravity, and then takes a decoupling limit in which the gravitymultiplet becomes a non-dynamical background field. This approach was applied tofive-dimensional Poincar´e supergravity [15–17] in the series of papers [18–20]. Super-symmetry of the background requires a certain generalized Killing spinor equation tohold, whose related geometry was investigated in [18], together with an algebraic “di-latino” equation which was studied in [19]. The latter reference recasts these conditionsinto local geometric constraints on the five-manifold M . As in lower dimensions, onefinds that the background is parametrized by various arbitrary functions/tensors. Inparticular ( M , g ) is equipped with a Killing vector field ξ = ∂ ψ , with dual one-form S (d ψ + ρ ) and transverse four-dimensional metric g (4) , where locally the function S = k ξ k and tensors ρ and g (4) are ξ -invariant but otherwise freely specifiable. Theauthors of [19] furthermore show that locally all deformations of the background fieldslead to Q -exact deformations of the action, where Q is the supercharge. Despite thisgenerality, these backgrounds apparently don’t include the conformally flat S × S geometry mentioned above [19]. We shall comment further on these issues later.In [1] a twisted version of N = 1 super-Yang-Mills theory is defined on contact five-manifolds ( M , η ). Here η is a contact one-form, meaning that η ∧ d η ∧ d η is a volumeform. On a Sasaki-Einstein five-manifold [22] one can construct N = 1 super-Yang-Mills coupled to matter [23]. This is essentially because the two Killing spinors ona Sasaki-Einstein manifold satisfy the same Killing spinor equations as those on theround sphere. For the special class of toric ( U (1) -invariant) Sasaki-Einstein manifoldsof [24] the localized perturbative partition function has been computed in [25–27],with the last reference also giving a conjectured formula for the full partition function.The authors of [28] furthermore show that one can define a twisted version of N = 2super-Yang-Mills theory on any K-contact five-manifold. We also note that K-contactgeometry arises as a special case in [18].In the present paper we instead take a holographic approach, similar to [29] inlower dimensions, to construct rigid supersymmetry in five dimensions. Here M is See [21] for the construction of supersymmetric Lorentzian backgrounds within the superspaceformulation of five-dimensional conformal supergravity. There are also additional freely specifiable fields, which determine the rest of the background. F (4)gauged supergravity [30]. Some of the groundwork for this was laid in [31, 32], wheresupergravity duals of the squashed five-sphere backgrounds of [7, 8] were constructed(see also [33, 34] for holographic duals to the supersymmetric R´enyi entropy in fivedimensions). We begin with a general supersymmetric asymptotically locally AdSsolution to the Romans theory, and extract the conditions this imposes on the five-dimensional conformal boundary. Although the resulting spinor equations are quitecomplicated, we will show they are completely equivalent to a very simple geometricstructure. We find that M is equipped with a conformal Killing vector ξ = ∂ ψ whichgenerates a transversely holomorphic foliation . This is compatible with an almostcontact form η = d ψ + ρ , where up to global constraints that we describe the norm S = k ξ k and ρ are arbitrary, and the transverse metric g (4) is Hermitian . The onlyother remaining freedom is an arbitrary function α (such that Sα is ξ -invariant), whichtogether with the metric determines all the remaining background data. This structureis similar to the rigid limit of Poincar´e supergravity described above, but with theaddition of an integrable transverse complex structure and Hermitian metric. In factit is a natural hybrid of the “real” three-dimensional rigid supersymmetric geometrystudied in [35,36] and the four-dimensional supersymmetric geometry of [37,38] (wherethe 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 theform of supersymmetric asymptotically locally AdS solutions to Romans supergrav-ity, in particular extracting the Killing spinor equations on the conformal boundary M . These are then used as a starting point for a purely five-dimensional analysis insection 3. We show that the spinor equations are completely equivalent to a simple ge-ometric structure on M , and present a number of subclasses and examples, includingmany of the examples referred to above. In section 4 we construct N = 1 supersymmet-ric gauge theories formed of vector and hypermultiplets on this background geometry.Our conclusions are presented in section 5. The bosonic fields of the six-dimensional Romans supergravity theory [30] consist of themetric, a scalar field X , a two-form potential B , together with an SO (3) R ∼ SU (2) R R-symmetry gauge field A i with field strength F i = d A i − ε ijk A j ∧ A k , where i = 1 , , SU (2) R doublet ofDirac spinors ǫ I , I = 1 ,
2, satisfying the following Killing spinor and dilatino equations D M ǫ I = i4 √ X + X − )Γ M Γ ǫ I − i16 √ X − F NP (Γ M NP − δ M N Γ P ) ǫ I (2.1) − X H NP Q Γ NP Q Γ M Γ ǫ I + 116 √ X − F iNP (Γ M NP − δ M N Γ P )Γ ( σ i ) I J ǫ J , − i X − ∂ M X Γ M ǫ I + 12 √ (cid:0) X − X − (cid:1) Γ ǫ I + i24 X H MNP Γ MNP Γ ǫ I − √ X − F MN Γ MN ǫ I − i8 √ X − F iMN Γ MN Γ ( σ i ) I J ǫ J . (2.2)Here Γ M are taken to be Hermitian and generate the Clifford algebra Cliff(6 ,
0) inan orthonormal frame, M = 0 , . . . ,
5. We have defined the chirality operator Γ =iΓ , which satisfies (Γ ) = 1. The covariant derivative acting on the spinor is D M ǫ I = ˆ ∇ M ǫ I + i2 A iM ( σ i ) I J ǫ J , where ˆ ∇ M = ∂ M + Ω NPM Γ NP denotes the Levi-Civitaspin connection while σ i , i = 1 , ,
3, are the Pauli matrices.Given a supersymmetric asymptotically locally AdS solution we may introduce aradial coordinate r , so that the conformal boundary is at r = ∞ and the metric admitsan expansion of the formd s = 92 d r r + r (cid:20) g µν + 1 r g (2) µν + · · · (cid:21) d x µ d x ν . (2.3)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 reparametrizationinvariant under r → Λ r , where Λ = Λ( x µ ). However, the correction terms under sucha transformation are subleading in the 1 /r expansion. This will play an important rolein the next section.For simplicity we shall mainly consider Abelian solutions in which A = A = 0, and A ≡ A , with field strength F ≡ d A . Similarly to the metric (2.3) we then write thefollowing general expansions for the remaining bosonic fields X = 1 + 1 r X + · · · ,B = rb − r d r ∧ A (0) + · · · A = a + · · · , (2.4)4here we define f ≡ d a . Some of the terms a priori present in these expansions areset to zero by the equations of motion; for example, the O (1 /r ) term in the expansionof X [32]. The Killing spinors similarly admit an expansion of the form ǫ I = √ r χ I − i χ I ! + 1 √ r ϕ I i ϕ I ! + O ( r − / ) . (2.5)Here we have used the orthonormal frame E = 3 √ rr , E µ = re µ + · · · (2.6)for the metric (2.3). Furthermore, the spin connection expands asΩ νµ = − √ δ νµ + 1 r ω νµ + · · · . (2.7)Also as in [32] we consider a “real” class of solutions for which ǫ I satisfies the sym-plectic Majorana condition ε JI ǫ J = C ǫ ∗ I ≡ ǫ cI , where C denotes the charge conjugationmatrix, satisfying Γ TM = C − Γ M C . The bosonic fields are all taken to be real, withthe 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 are simply the charge conjugates of the corresponding equations for ǫ . In this way weeffectively reduce to a single Killing spinor ǫ ≡ ǫ , with SU (2) R doublet ( ǫ, ǫ c ). Wethen note the following large r expansions of bilinears: ǫ † Γ ǫ = 4 αS + · · · , i ǫ † Γ Γ (1) ǫ = 2 SrK − √ r + · · · . (2.8)Here we have defined Γ (1) ≡ Γ M E M and S ≡ χ † χ . (2.9)We also note that the bilinear ǫ † Γ (1) ǫ is a Killing one-form in the bulk [32]. This willhence 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 thefirst two orders we obtain (cid:18) ∇ µ + i2 a µ (cid:19) χ = − √
23 i γ µ ϕ − i12 √ b νσ γ νσµ χ + i3 √ b µν γ ν χ , (2.10) (cid:18) ∇ µ + i2 a µ (cid:19) ϕ = − i6 √ b µν γ ν ϕ + 116 √ f νσ γ νσµ χ − √ f µν γ ν χ (2.11)+ 148 (d b ) νρσ γ νρσ γ µ χ − A (0) ν γ νµ χ + 112 A (0) µ χ + i2 ω νµ γ ν χ . Here we take the spinors to be Grassmann even. 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 bulkdilatino equation (2.2) implies − √ b µν γ µν ϕ − √ X χ + i8 √ f µν γ µν χ + i24 (d b ) µνσ γ µνσ χ − i18 A (0) µ γ µ χ = 0 . (2.12)As explained in [32], equation (2.10) may be rewritten in the form of a chargedconformal Killing spinor equation, with additional b -field couplings. Setting b = 0one obtains the standard charged conformal Killing spinor equation, whose solutions(twistor spinors) have been studied in the holographic context for three-manifolds andfour-manifolds in [29,38–41]. On the other hand, previous work on rigid supersymmetryin five dimensions [18–20] has used Killing spinor equations of a different form, withoutthe coupling to ϕ in (2.10). We may make closer contact with this work by noting thatsupersymmetry in the bulk also implies the algebraic relation ϕ = − αχ − i2 ( K ) ν γ ν χ . (2.13)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. In this section we begin with a Riemannian five-manifold ( M , g ), on which we’d like todefine rigid supersymmetric gauge theories. The gauge/gravity correspondence impliesthis should be possible, provided the spinor equations derived in the previous sectionhold.Let us summarize the background data. In addition to the real metric g , we havetwo generalized Killing spinors χ , ϕ . Globally these are spin c spinors, being sections ofthe spin bundle of M tensored with L − / , χ, ϕ ∈ Γ[Spin( M ) ⊗ L − / ], where L is thecomplex line bundle for which the real gauge field a is a connection. This Abelian gaugefield 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 ≡ C χ ∗ with C the five-dimensional charge conjugation matrix. Thespinors χ , ϕ then satisfy the coupled Killing spinor equations (2.10), (2.11), wherethe background b -field is taken to be a purely imaginary two-form, A (0) is a purelyimaginary one-form, while ω µν = g νσ ω σµ is real and symmetric. Furthermore, χ and ϕ α and K , which are respectively a real function and real one-form. Finally we havethe dilatino equation (2.12), which introduces the real background function X .In the remainder of this section we shall analyse the geometric constraints that theseequations impose on ( M , g ). Although the background data and equations (2.10)–(2.13) appear a priori complicated, in fact we shall see that the geometry they areequivalent to is very simple. In the analysis that follows it is convenient to assume that the spin c spinor χ is nowherezero. More generally χ could vanish along some locus Z ⊂ M , and the local geometrywe shall derive below is valid on M \ Z . If Z is non-empty one would need to imposesuitable boundary conditions, although we shall not consider this further in this paper.A nowhere zero spin c spinor equips ( M , g ) with a local SU (2) structure. Specifically,we may define the bilinears S ≡ χ † χ , K ≡ S χ † γ (1) χ ,J ≡ − i S χ † γ (2) χ , Ω ≡ − S ( χ c ) † γ (2) χ . (3.1)Here we have introduced the notation γ ( n ) ≡ n ! γ µ ··· µ n d x µ ∧ · · · ∧ d x µ n , where x µ , µ = 1 , . . . ,
5, are local coordinates on M . 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 notethat K is a real unit length one-form, while J is a real two-form with square length k J k = 2. Here the square norm of a p -form φ is defined via k φ k vol = φ ∧∗ φ , where ∗ denotes the Hodge duality operator on ( M , g ) and vol denotes the Riemannian volumeform. The complex bilinear Ω is globally a two-form valued in the line bundle L − .That χ , or equivalently the bilinears (3.1), defines a local SU (2) structure followsfrom some simple group theory. The spin group is Spin(5) ∼ = Sp (2) ⊂ U (4), with thelatter acting in the fundamental representation on the spinor space C . The stabilizerof a non-zero spinor is then Sp (1) ∼ = SU (2). When M is spin and L is trivial, so that χ ∈ Γ[Spin( M )], this defines a global SU (2) structure. However, more generally werequire only that M is spin c , and in this case the global stabilizer group is enlargedto U (2): the additional U (1) factor rotates the spinor by a phase, which may beundone by a U (1) gauge transformation. To see this in more detail we introduce a7ocal orthonormal frame e a , a = 1 , . . . ,
5, so that K = e , J = e ∧ e + e ∧ e , Ω = ( e + i e ) ∧ ( e + i e ) , (3.2)where the metric is g = P a =1 ( e a ) . The U (2) = SU (2) × Z U (1) structure group actsin the obvious way on the C spanned by e + i e , e + i e . This leaves K , J and themetric g invariant, but rotates Ω by the determinant of the U (2) transformation. Inorder for this to be undone by a gauge transformation, this identifies the line bundle as L = Λ , . The latter is the space of Hodge type (2 , e , e , e , e , and with almost complex structure I for which e + i e and e + i e are (1 , U (2) structure on M (or more precisely on M \ Z ).The one-form SK = χ † γ (1) χ arises simply from the restriction of the bulk Killingone-form ǫ † Γ (1) ǫ to the conformal boundary, and thus defines a conformal Killing one-form on ( M , g ). This is easily confirmed from the Killing spinor equation (2.10) for χ ,which implies ∇ ( µ ( SK ) ν ) = L ξ (log S ) g µν , (3.3)where we have introduced the dual vector field ξ , defined by g ( ξ, · ) = SK , and L denotes the Lie derivative.One finds that the spinor equations (2.10)–(2.13) imply the following differentialconstraints: d S = − √
23 ( SK + i i ξ b ) , d( Sα ) = − √ i ξ d a , (3.4)d( SK ) = 2 √ (cid:20) αSJ + SK ∧ K + i Sb − i2 i ξ ( ∗ b ) (cid:21) , (3.5)d( SK ) = i i ξ d b − i L ξ (log S ) b , (3.6)d( SJ ) = −√ K ∧ ( SJ ) , (3.7)d( S Ω) = − i (cid:16) a − √ αK − i √ K (cid:17) ∧ ( S Ω) . (3.8)Here ( i V φ ) a ··· a p − = V b φ ba ··· a p − defines the interior contraction of a vector V into a p -form φ . Notice that the background data X , A (0) and ω µν in (2.10)–(2.13) does notenter equations (3.4)–(3.8): they simply drop out (one only needs to use the realityproperties 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-8ations α → Λ − α , a → a , K → K − √ ,S → Λ S , K → Λ K , b → Λ b ,g → Λ g , J → Λ J , Ω → Λ Ω . (3.9)This symmetry is of course inherited from invariance under the change of radial variable r → Λ r in the bulk. If S is nowhere zero notice that one might use this symmetry toset S ≡ ξ pre-serves all of the background geometric structure, provided one rescales the fields byappropriate powers of S according to their Weyl weights in (3.9). For instance, con-tracting 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 d a = 0. One can hencelocally choose a gauge in which a is invariant under L ξ , so that the second equation in(3.4) is solved by Sα = 12 √ i ξ a . (3.10)In a similar way, one can show that also S − b and S − J are invariant under L ξ , while S − Ω is invariant under L ξ in the gauge choice for which (3.10) holds. Notice that thefirst equation in (3.4) implies that i ξ K = − √ L ξ (log S ).Without loss of generality it is convenient to henceforth impose L ξ S = 0. In termsof the bulk expansion in section 2 this means choosing the radial coordinate r to beindependent of the bulk Killing vector. This is a natural choice, which in turn impliesthat L ξ S = 0 and SK is Killing, and we shall make this convenient (partial) conformalgauge choice in the following. We may then introduce a local coordinate ψ so that ξ = ∂ ψ . (3.11)The condition L ξ S = 0 is then equivalent to S being independent of ψ . The Killing vector ξ has norm S , and the dual one-form K may be written locally as K = S (d ψ + ρ ) ≡ Sη , (3.12) An exception being the S × S geometry discussed in section 3.3. i ξ ρ = 0. Notice that η has Weyl weight zero and norm 1 /S . The local frame e , e , e , e provide a basis for D = ker η , and D inherits an almost complex structurefrom J . One then defines an endomorphism Φ of the tangent bundle of M byΦ | D = I , Φ | ξ = 0 , (3.13)where I is the almost complex structure. One easily verifies that Φ = − ξ ⊗ η , whichis a defining relation of an almost contact structure . Moreover, the five-dimensionalmetric takes the form d s M = S η + d s , (3.14)where d s is Hermitian with respect to I . Although ξ is Killing, this structure isin general not a K-contact structure, which is a stronger condition. In particular thelatter requires [42] that d η is the fundamental (1 , J associated to the transversealmost complex structure (which in general is not the case here), which in turn impliesthat η is a contact form, i.e. that η ∧ d η ∧ d η is a volume form (which in general isalso not the case here). Notice that since ξ is nowhere zero, its orbits define a foliationof M .Let us now turn to the differential constraints (3.4)–(3.8). The two equations (3.4)allow us to write b = i Sη ∧ (cid:18) K + 3 √ S (cid:19) + b ⊥ , a = 2 √ Sαη + a ⊥ , (3.15)where b ⊥ and a ⊥ are basic forms for the foliation defined by ξ ; that is, they are invariantunder, and have zero interior contraction with, ξ . Recall that in writing the gauge fieldin the form in (3.15) we have made a (partial) gauge choice, as in (3.10). This leaves aresidual gauge freedom a ⊥ → a ⊥ + d λ , where λ is a basic ( ξ -invariant) function. Theequation (3.6) is simply equivalent to b being invariant under ξ .The differential constraint (3.5) reduces tod ρ = √ S ( − i ∗ b ⊥ + 2i b ⊥ + 4 αJ ) . (3.16)Here ∗ is the Hodge dual with respect to the transverse four-dimensional metric d s ,with volume form e ∧ e ∧ e ∧ e . It is then convenient to introduce b ⊥ = b + + b − , (3.17)10ecomposing into the transversely self-dual and anti-self-dual parts. Equation (3.16)is then equivalent to b + = i (cid:18) αJ − √ S d ρ + (cid:19) , b − = − i √ S d ρ − . (3.18)The constraint (3.7) simply identifies θ ≡ J d J = −√ K − d log S , (3.19)with the
Lee form θ of the transverse four-dimensional Hermitian structure. Thatis, every four-dimensional Hermitian structure with fundamental two-form J satisfiesd J = θ ∧ J . Finally, the differential constraint (3.8) now readsdΩ = ( θ − i a ⊥ ) ∧ Ω . (3.20)This implies that the almost complex structure I is integrable , thus defining a trans-versely holomorphic foliation of M . We may introduce local coordinates ψ, z , z adapted to the foliation, where the transition functions between the z , z coordinatesare holomorphic.Notice that we may rewrite (3.20) asdΩ = − i a Chern ∧ Ω , (3.21)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 explicitexpression for the Chern connection a Chern , we begin by noting that Ω ∧ ¯Ω = 2 J ∧ J .Using local coordinates z α , α = 1 , , for the transverse space we may writeΩ = f d z ∧ d z , J = i2 g (4) α ¯ β d z α ∧ d¯ z ¯ β , (3.23)which implies that | f | = p det g (4) . Notice that globally f is a section of L − , where L ∼ = Λ , ≡ K is the canonical bundle. Writing f = | f | e i φ we then havedΩ = d log f ∧ Ω = i (cid:18)
12 d c log det g (4) + d φ (cid:19) ∧ Ω , (3.24)where d c ≡ I ◦ d. We thus recognize (up to gauge) a Chern = −
12 d c log det g (4) (3.25)11s the Chern connection on the canonical bundle.The geometric content of the differential constraints (3.4)–(3.8) may hence be sum-marized as follows. M is equipped with a transversely Hermitian structure, so thatthe metric takes the form d s M = S (d ψ + ρ ) + d s . (3.26)Here the Killing vector is ξ = ∂ ψ , which generates a transversely holomorphic foliation.The almost contact form is η = (d ψ + ρ ), and d s is a transverse Hermitian metric.One is also free to specify the functions α and S . Given this data, the remainingbackground fields a and b that enter (3.4)–(3.8) are determined via a = 2 √ Sαη + a Chern + I ( θ ) ,b = − i √ Sη ∧ ( θ − S ) + 4i αJ − i √ S (3d ρ + + d ρ − ) . (3.27)In particular the choice of a transverse Hermitian metric g (4) fixes the two-form J , andhence the Lee form θ , while the Hodge type (2 , a Chern in (3.25) are also determined up to gauge. Notice that the terms
Sαη and I ( θ ) enteringthe formula for a in (3.27) are both global one-forms on M , implying that globally a is a connection on L = Λ , .We shall furthermore show in section 3.4 that any choice of transversely Hermitianstructure on M of the above form gives a supersymmetric background. In particularthe remaining background fields X , A (0) , and ω µν appearing in the spinor equations(2.10)–(2.13) are also determined by the above geometric data. In this section we shall present some explicit examples of the above construction. Theseinclude all explicit examples appearing in the literature (within the Abelian truncationon which we are mostly focusing), including examples with six-dimensional gravityduals, plus large families of new solutions.
General families
We begin by noting some special families of backgrounds: • Setting ρ = 0 and S ≡ M = R × M or M = S × M ,where M is any Hermitian four-manifold. Notice this four-manifold geometry12s the same as the rigid supersymmetric geometry one finds in four dimensions[29, 37]. The first reference here follows a similar holographic approach to thepresent paper, while the second takes a rigid limit of “new minimal” supergravityin four dimensions. • If d θ = 0 then the transverse Hermitian metric is locally conformally K¨ahler. – If furthermore θ = 0 then the transverse four-metric is K¨ahler. – If θ = 0 and d ρ is a positive constant multiple of J then the five-metricis locally conformally Sasakian. Supersymmetric gauge theories on Sasaki-Einstein manifolds, for which furthermore S ≡ g is a positively curvedEinstein metric, were defined in [23], and further studied in [25–28]. • We may take any circle bundle over a product of Riemann surfaces S ֒ → Σ × Σ .The Hermitian metric may be taken to be simply a product of two metrics onthe Riemann surfaces, while ρ is the connection one-form for the fibration. Onecan generalize this further by allowing S orbibundles over a product of orbifoldRiemann surfaces. – If we only fibre over Σ , this leads to direct product M × Σ solutions, where M is a Seifert fibred three-manifold. Notice this three-manifold geometryis the same as the rigid supersymmetric geometry in three dimensions [35].Maximally supersymmetric Yang-Mills theory has been studied on similarbackgrounds in [9–13], including the direct products S × Σ and M × S .Here the spheres are equipped with round metrics and the associated canon-ical spinors, while the spinors on Σ and M are constructed by topologicallytwisting with the SU (2) R symmetry. • Finally, if d ρ has Hodge type (1 ,
1) the transversely holomorphic foliation admitsa complexification [43], i.e. adding a radial direction to ξ we naturally have acomplex six-manifold M , with a transversely holomorphic foliation. Notice thatSasakian geometry and the direct product S × M are special cases. Whenthe orbits of ξ all close, M fibres over a Hermitian four-orbifold M , and theassociated U (1) orbibundle is the unit circle in a Hermitian holomorphic lineorbibundle over M . The corresponding complex M is then simply the totalspace of the associated C ∗ bundle over M .13 quashed Sasaki-Einstein We have already noted that a Sasakian five-manifold is a particular case of a super-symmetric background. Recall that Sasakian metrics take the formd s = η + d s , (3.28)where η defines a contact structure on M , with Reeb Killing vector field ξ , and d s isa transverse K¨ahler metric. Moreover d η = d ρ = 2 J . If the transverse K¨ahler metric g (4) is Einstein, then the metric (3.28) is said to be a squashed Sasaki-Einstein metric. For a given choice of transverse K¨ahler-Einstein metric, we obtain a two-parameterfamily of backgrounds, parametrized by the constants c , c : S ≡ , α = c , K = − √ θ ≡ ,a = c η , b = i(4 c − √ J . (3.29)The K¨ahler-Einstein metric g (4) satisfies the Einstein equation Ric (4) = 2(2 √ c − c ) g (4) . Notice that we have presented the solution (3.29) in a different gauge choiceto (3.10). We may impose the latter gauge choice by simply transforming a → a +(2 √ c − c )d ψ , although the form of a in (3.29) makes it clear that we may take a tobe a global one-form on M for this particular class of solutions.When g is taken to be the standard metric on CP , the above geometry is a squashedfive-sphere. This corresponds to the conformal boundary of the 1/4 BPS bulk Romanssupergravity solutions constructed in [32]. Black hole boundary
In this section we consider the conformal boundary of the 1/2 BPS topological blackhole solutions constructed in [33]. We begin with the following product metric on S × H , where H is hyperbolic four-space:d s = d τ + 1 q + 1 d q + q (d ϑ + sin ϑ d ϕ + cos ϑ d ϕ ) . (3.30)Here τ is a periodic coordinate on S , q is a radial coordinate with q ∈ [0 , ∞ ), ϑ ∈ [0 , π ]while ϕ , ϕ have period 2 π . The metric in brackets is simply the round metric on aunit radius S . For this solution b vanishes identically, while a is gauge-equivalent tozero. The Killing spinors for this background [33] in general depend on four integration In the mathematical literature [42] these are called η -Sasaki-Einstein metrics. S = p q + 1 , α = − √ p q + 1 ,K = − √ qq + 1 d q = − √ q + 1) , (3.31)while in a gauge in which a = 0 the U (2) structure is given by K = 1 p q + 1 (cid:2) d τ + q (cos ϑ d ϕ − sin ϑ d ϕ ) (cid:3) ,J = q ϑ d ϑ ∧ (d ϕ + d ϕ ) + q ( q + 1) d q ∧ (cid:20) d τ + sin ϑ d ϕ − cos ϑ d ϕ (cid:21) , Ω = − q e i( ϕ − τ − ϕ ) p q + 1 (cid:20) sin 2 ϑ ( q d τ − id q ) ∧ (d ϕ + d ϕ ) + q sin 2 ϑ d ϕ ∧ d ϕ +2i q d ϑ ∧ (d τ + sin ϑ d ϕ − cos ϑ d ϕ ) − q ∧ d ϑ (cid:21) . (3.32)The supersymmetric Killing vector is ξ = g ( SK , · ) = ∂ τ + ∂ ϕ − ∂ ϕ . (3.33)Furthermore, notice that rescaling J by 1 / ( q + 1) leads to a closed two-form, henceshowing that the Hermitian metric transverse to ξ is conformal to a K¨ahler metric.Moreover, one can also check that the almost contact form η = K /S is a contact formin this case, i.e. that η ∧ d η ∧ d η is a volume form. Conformally flat S × S In this section we consider the conformally flat metric on S × S , which we may writeas d s = d τ + d s S , (3.34)where d s S = d β + sin β (d ϑ + sin ϑ d ϕ + cos ϑ d ϕ ) . (3.35)Here τ is a periodic coordinate on S , while the metric in brackets in (3.35) is simplythe round metric on a unit radius S , as in the previous black hole boundary example. This is different to the gauge choice (3.10), where instead a = − τ for this solution. β ∈ [0 , π ]. The metric (3.34) of course arises as the conformalboundary 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 oneso as to present simple expressions for the remaining background data. We find S = e − τ , α = 0 , K = 3 √ τ . (3.36)The U (2) structure is given by K = sin β d β − cos β d τ ,J = sin β sin( ϕ + ϕ ) (cid:26) cot( ϕ + ϕ ) (d ϑ ∧ d τ − cot β d β ∧ d ϑ ) − sin ϑ d ϑ ∧ d ϕ − cos ϑ d ϑ ∧ d ϕ + sin ϑ cos ϑ h (cot β d β + d τ ) ∧ (d ϕ − d ϕ ) − cot( ϕ + ϕ ) d ϕ ∧ d ϕ i(cid:27) , Ω = i sin β sin( ϕ + ϕ ) h cot β d β ∧ d ϑ − d ϑ ∧ d τ + sin ϑ cos ϑ d ϕ ∧ d ϕ i + sin β sin ϑ h sin ϑ + i cos ϑ cos( ϕ + ϕ ) i(cid:16) cot β d β ∧ d ϕ − cot ϑ d ϑ ∧ d ϕ +d τ ∧ d ϕ (cid:17) + sin β cos ϑ h cos ϑ − i sin ϑ cos( ϕ + ϕ ) i(cid:16) cot β d β ∧ d ϕ + tan ϑ d ϑ ∧ d ϕ + d τ ∧ d ϕ (cid:17) . (3.37)Notice that in this example we obtain a conformal Killing vector from the Killingspinor bilinear, but not a Killing vector. As described at the end of section 3.1, wemay 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 correspondingWeyl-transformed metric is then (locally) flat, with the Weyl-transformed J and Ω bothclosed and hence defining a transverse hyperK¨ahler structure. Nevertheless, the factthat the metric (3.34) leads to a conformal Killing vector explains why this backgroundis missing from the rigid supersymmetric geometry in [18, 19]: in the latter referencesthe corresponding bilinear is necessarily a Killing vector. This also suggests that theconjecture made in [19] is likely to be correct: that is, to obtain the S × S backgroundfrom a rigid limit of supergravity, one should begin with conformal supergravity in fivedimensions, rather than Poincar´e supergravity. See [44] for a related discussion on this point. quashed S We consider the squashed five-sphere metricd s = 1 s (d τ + C ) + d σ + 14 sin σ (d ϑ + sin ϑ d ϕ )+ 14 cos σ sin σ (d β + cos ϑ d ϕ ) , (3.38)where s ∈ (0 ,
1] is the squashing parameter and C ≡ −
12 sin σ (d β + cos ϑ d ϕ ) . (3.39)The coordinates σ, β, ϑ, ϕ are coordinates on the base CP , with β having period 4 π , ϕ having period 2 π , while σ ∈ [0 , π ], ϑ ∈ [0 , π ], and d C is the K¨ahler two-form on CP . For the “toric” family discussed in [31, 32] we find S = cos σb + sin σb , (3.40)where b = 1 + √ − s , b = 1 − √ − s . (3.41)The other background fields are, in an appropriate gauge ( i.e. not that in (3.10)), α = b ( b + b )( b − b + ( b − b ) cos 2 σ )4 √ b cos σ + b sin σ ) ,a = b − b b (d τ + C ) ,b = − i( b − b )2 √ b b ( b + b ) d C ,K = √ b − b ) sin 2 σb cos σ + b sin σ d σ = −√ b cos σ + b sin σ ) . (3.42)17he U (2) structure is K = 14 b b ( b + b ) (cid:0) b cos σ + b sin σ (cid:1) (cid:20) ( b + b )( b − b + ( b + b ) cos 2 σ )d τ −
12 sin σ (cid:16) ( b − b ) cos 2 σ + b − b b − b (cid:17) (d β + cos ϑ d ϕ ) (cid:21) ,J = sin σ b b ( b + b ) (cid:0) b cos σ + b sin σ (cid:1) (cid:20) σ (cid:16) b + b ) d σ ∧ d τ − b d σ ∧ (d β + cos ϑ d ϕ ) (cid:17) + 2 sin ϑ sin σ ( b cos σ + b sin σ )d ϑ ∧ d ϕ (cid:21) , Ω = sin σ e i( τ − β ) b b ( b + b ) (cid:0) b cos σ + b sin σ (cid:1) (cid:20) − sin 2 σ (cid:16) i sin ϑ ( b d ϕ ∧ d β +2( b + b ) d τ ∧ d ϕ ) − b + b ) d ϑ ∧ d τ + b d ϑ ∧ (d β + cos ϑ d ϕ ) (cid:17) − b cos σ + b sin σ ) (sin ϑ d σ ∧ d ϕ + i d ϑ ∧ d σ ) (cid:21) . (3.43)The supersymmetric Killing vector is ξ = b ∂ τ + 2( b + b ) ∂ β . (3.44)One also computes η ∧ d η ∧ d η = b b ( b + b ) (cid:0) b cos σ + b sin σ (cid:1) (cid:0) ( b − b ) cos 2 σ + b − b b − b (cid:1) × (cid:0)(cid:0) b − b (cid:1) cos 2 σ + b − b b + b (cid:1) vol , (3.45)where vol denotes the Riemannian volume form and η = K /S is the almost contactform. The right hand side of (3.45) can have non-trivial zeros, and we thus see that ingeneral η does not define a contact structure. These backgrounds arise as the conformalboundary of the 3/4 BPS solutions of Romans supergravity constructed in [31, 32]. In this section we will show that any choice of transversely Hermitian structure on M 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 thegeometry also determines the fields X , A (0) and ω µν , in such a way that the originalspinor equations (2.10)–(2.13) are satisfied.18e first examine the Killing spinor equation (2.10) for χ . In order to proceed it isconvenient to choose a set of projection conditions (see for example [45]) γ χ = γ χ = i χ , γ χ = χ . (3.46)These allow one to substitute for the fields b and K in terms of the geometry, via(3.27) and (3.19), into the right hand side of equation (2.10). In doing this calculationit is also convenient to write Ω = J + i J , J = J so that J = e + e , J = e − e , J = e + e . (3.47)Notice that J i , i = 1 , , γ m χ = J mn γ n χ , where m, n = 1 , . . . ,
4, and ( β − ) mn γ mn χ =0 for any transverse anti-self-dual two-form β − .In this way it is straightforward to show that the µ = 5 (the ψ direction) componentof (2.10) simply imposes ∂ ψ χ = 0. Thus χ is independent of ψ . Taking instead µ = m , m = 1 , , ,
4, one finds (2.10) is equivalent to ∇ (4) m χ = 14 θ n γ mn χ − i2 ( a ⊥ ) m χ + 12 ( ∂ m log S ) χ , (3.48)where ∇ (4) denotes the Levi-Civita spin connection for the transverse four-dimensionalmetric. Recall that the latter metric is Hermitian. It is then more natural to expressequation (3.48) in terms of an appropriate Hermitian connection, which preserves boththe metric and the two-form J . The Chern connection is such a connection, defined by ∇ Chern m χ = ∂ m χ + 14 ( ω Chern m ) pq γ pq χ , where ( ω Chern m ) pq ≡ ( ω (4) m ) pq + 12 J nm (d J ) npq . (3.49)This coincides with the Levi-Civita connection if and only if d J = 0 (equivalently θ = 0), so that the metric is K¨ahler.Next, let us notice that under the Weyl transformation (3.9) we have χ → Λ / χ , sothat it is also natural to introduce ˜ χ ≡ S − / χ , (3.50) Without loss of generality we take the four-dimensional frame e , . . . , e to be independent of theKilling vector ξ = ∂ ψ .
19o that ˜ χ is Weyl invariant. In this notation (3.48) becomes ∇ Chern m ˜ χ + i2 a Chern ˜ χ = 0 , (3.51)where recall that a Chern = a ⊥ − I ( θ ) is the Chern connection for the canonical bundle K ≡ Λ , , given explicitly by (3.25). It is then a standard fact, and is straightforward toshow, that any Hermitian space admits a canonical solution ˜ χ to (3.51). Specifically,any Hermitian space admits a canonical spin c structure, with twisted spin bundlesSpin c = Spin ⊗ K − / . In four dimensions this is isomorphic toSpin c ∼ = (cid:0) Λ , ⊕ Λ , (cid:1) ⊕ Λ , , (3.52)where Λ p,q denotes the bundle of forms of Hodge type ( p, q ). In the case at hand, theseare defined transversely to the foliation generated by the Killing vector ξ . Under (3.52)the Killing spinor ˜ χ = S − / χ is a section of the trivial line bundle Λ , . Moreover,the Chern connection restricted to this summand is flat, with the induced connection − a Chern on the twist factor K − / effectively cancelling that coming from the spinbundle. Concretely, in terms of local complex coordinates z α , α = 1 ,
2, we have( ω Chern ) βα = ( ∂g (4) ) α ¯ γ ( g (4) ) ¯ γβ , and using the projection conditions (3.46) one can showthis precisely cancels the contribution from (3.25). The spin c spinor ˜ χ is simply aconstant length section of this flat line bundle. Put simply, the rescaled Killing spinor˜ χ = S − / 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 holdsprovided A (0) = − ∗ d ∗ b − i √ b ∧ b ! , (3.53)and X = − α − h K , K i − i6 √ S h η, A (0) i − h d a ⊥ , J i − √ h K , d log S i . (3.54)Here we have introduced the notation φ ∧ ∗ φ = p ! h φ , φ i vol for the inner productbetween two p -forms φ , φ . Notice that the expression (3.53) for the imaginary one-form A (0) coincides with that in [32], which was derived by solving the bulk equationsof motion near the conformal boundary, in terms of the boundary data. Notice thatunder the Weyl scaling (3.9) we have A (0) → (cid:18) A (0) + 92 i d log Λ b (cid:19) , X → X . (3.55)20he fact that X has Weyl weight − A (0) in (3.55) naively appears to contra-dict (2.4), for which A (0) has Weyl weight −
1. However, this is where the commentabove equation (2.4) is relevant: the reparametrization r → Λ r does not preserve the subleading terms in the metric (2.3). It is therefore not a strict symmetry of the sys-tem 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 Weylsymmetry (3.9), while the higher order term A (0) arising in the expansion of the B -fielddoes not. One could restore the full Weyl symmetry by adding a cross term 9 d rr C µ d x µ into the metric (2.3), so that C → C − d log Λ , (3.56)under r → Λ r preserves the form of the metric. Then C is a new background field on M , and one finds A (0) = − ∗ " (d + 2 C ∧ ) ∗ b − i √ b ∧ b . (3.57)This now has Weyl weight −
1, as expected, and the anomalous variation in (3.55) arisessimply because we have made the gauge choice C = 0 in our original expansion. Ingeneral notice that a field of Weyl weight w will couple to a Weyl covariant derivativeD µ ≡ ∂ µ + w C µ , and w = 2 for ∗ b .It remains to show that the background geometry implies the ϕ Killing spinor equa-tion (2.11). At this point notice that everything is fixed uniquely in terms of thefree functions α and S , and the transversely Hermitian structure on M , apart fromthe higher order spin connection term ω µν which appears in (2.11). After a lengthycomputation, in our orthonormal frame one finds the expression ω = − √ α − √ h K , K i − √ X − √ h d a ⊥ , J i − h K , d log S i ,ω m = (cid:20) − i3 √ i K b ⊥ + i d log S (cid:18) αJ + 1 √ S d ρ − (cid:19)(cid:21) m = ω m ,ω mn = √
23 ( K ) m ( K ) n − ∇ (4)( m ( K ) n ) − (cid:0) Sα d ρ − + 1 √ a −⊥ (cid:1) mp J pn + (cid:18) √ α + √ X − √ h K , K i + 14 √ h d α ⊥ , J i (cid:19) δ mn . (3.58)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 pn is symmetric for any transverse21nti-self-dual two-form β − . Thus (2.11) is satisfied provided ω µν is given by (3.58). Weconclude this subsection by noting the following formula ω µµ = 2 √ α + √ X − h K , √ K + d log S i + 12 √ h d a ⊥ , J i − ∇ (4) m K m . (3.59)This trace will appear in the supersymmetric Lagrangians constructed in section 4. A supersymmetric asymptotically locally AdS solution to six-dimensional Romans su-pergravity leads to the coupled spinor equations (2.10)–(2.13) on the conformal bound-ary M . These are a rather complicated looking set of equations for the spin c spinors χ , ϕ , depending on the large number of background fields g, X , a, A (0) , b and ω µν on M , with ϕ and χ related to each other by the further background fields α and K via(2.13). However, we have shown these equations are completely equivalent to a verysimple geometric structure:(i) The five-manifold M is equipped with a transversely holomorphic foliation, withthe one-dimensional leaves generated by the (conformal) Killing vector field ξ = ∂ ψ . This structure is a natural odd-dimensional cousin of a complex manifold, andmeans we may cover M locally with coordinates ψ, z , z , where the transitionfunctions between the z , z coordinates are holomorphic (more formally we havean open cover { U i } and submersions f i : U i → C with one-dimensional fibres,such that on overlaps U i ∩ U j we have f j = g ji ◦ f i where g ji are biholomorphismsof open sets in C ).(ii) This foliation is compatible with an almost contact form η = d ψ + ρ . Choosea particular ρ = ρ , 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 + ν , where ν is a global basic one-form . That is, ν is a global one-form on M satisfying L ξ ν = 0 = i ξ ν .(iii) One can choose an arbitrary transverse Hermitian metric d s , 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). 22n interesting special case is when all the leaves of the foliation are closed, so that ξ generates a U (1) action on M and ψ is a periodic coordinate. In this case M fibresover a complex Hermitian orbifold M = M /U (1), where η is a global angular formfor the U (1) orbibundle. Different choices of ν in (ii) above are then simply differentconnections on this bundle, with (iii) giving different Hermitian metrics on M .We have shown that any choice of the data (i)–(iv) determines a supersymmet-ric background, solving the spinor equations (2.10)–(2.13), and conversely any suchsolution determines a choice of the above geometric data. Furthermore, solving (2.10)–(2.13) is equivalent to finding a supersymmetric asymptotically locally AdS solutionto Romans supergravity, to the first few orders in an expansion around the confor-mal boundary M . Of course whether or not this extends to a complete non-singularsupergravity solution, as some of the explicit examples in section 3.3 do, is anothermatter. In this section we construct N = 1 supersymmetric gauge theories formed of vectorand hypermultiplets on the background geometry described in section 3.We adopt the same notation as [2], in particular using ξ and η to denote five-dimensional Killing spinors. The γ µ are 4 × ,
0) in an orthonormal frame. A complete set of 4 × , γ µ , γ µν ) and we choose γ µνρστ = − ǫ µνρστ with ǫ = +1. The five-dimensional charge conjugation matrix, C = ( C αβ ), is unitary and anti-symmetric inthe spinor indices α, β = 1 , , , ∼ = Sp (2). The matrices C γ µ are anti-symmetric in spinor indices whereas C γ µν are symmetric. Spinor bilinears are denoted( ηγ µ ··· µ n ξ ) = η α ( C γ µ ··· µ n ) αβ ξ β . Finally, the Fierz identity for Grassmann odd spinorsin five dimensions is γ A η α ( ξγ B λ ) = −
14 ( ηξ ) γ A γ B λ α −
14 ( ηγ µ ξ ) γ A γ µ γ B λ α + 18 ( ηγ µν ξ ) γ A γ µν γ B λ α , (4.1)where γ A , γ B denote arbitrary elements of Cliff(5 , An off-shell N = 1 vector multiplet in five dimensions consists of a gauge field A µ , areal scalar σ , a gaugino λ I , and a triplet of auxiliary scalars D IJ , all transforming in the23djoint representation of the gauge group G . Here I, J = 1 , SU (2) R symmetryindices. The gaugino is a symplectic-Majorana spinor which satisfies ( λ αI ) ∗ = ε IJ C αβ λ βJ whilst the auxiliary scalars satisfy ( D IJ ) † = ε IK ε JL D KL , where recall that ε IJ is theLevi-Civita symbol.We introduce the following covariant derivatives: F µν = ∂ µ A ν − ∂ ν A µ − i[ A µ , A ν ] ,D µ σ = ∂ µ σ − i[ A µ , σ ] ,D µ λ I = ∇ µ λ I − i[ A µ , λ I ] ,D µ D IJ = ∂ µ D IJ − i[ A µ , D IJ ] , (4.2)where ∇ is the Levi-Civita spin connection. In general we may consider turning on an SU (2) R background gauge field a iµ , i = 1 , ,
3, or equivalently we may introduce V µIJ ≡ − i2 a iµ ( σ i ) IJ , (4.3)where σ i , i = 1 , ,
3, denote the Pauli matrices. In section 2 recall that for simplicity werestricted to an Abelian background gauge field, with a = a = 0, a = a , but in thissection we will relax this assumption. There is also a background two-form b -field andwe 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 fieldactive we modify the covariant derivatives toD µ σ = D µ σ − C µ σ , D µ λ I = D µ λ I − C µ λ I − V µI J λ J , D µ D IJ = D µ D IJ − C µ D IJ − V µ ( I K D J ) K , (4.4)so that they are covariant with respect to both Weyl and R-symmetry transforma-tions. These correspond to Weyl weights w = ( − , , − , −
2) for the gauge multiplet( σ, A µ , λ I , D IJ ).Given this background data we consider the following (conformal) supersymmetryvariations: δ ξ σ = i ε IJ ξ I λ J ,δ ξ A µ = i ε IJ ξ I γ µ λ J ,δ ξ λ I = − γ µν ξ I F µν + γ µ ξ I D µ σ − D IJ ξ J + i3 √ γ µν ξ I b µν σ − √ ξ I σ ,δ ξ D IJ = − ξ ( I γ µ D µ λ J ) + 2[ σ, ξ ( I λ J ) ] + 2 √
23 ˜ ξ ( I λ J ) − √ ξ ( I γ µν λ J ) b µν . (4.5)24his has Grassmann odd supersymmetry parameters ξ I , ˜ ξ I . We find that these trans-formations close onto[ δ ξ , δ η ] σ = − i v ν D ν σ − √ ̺σ , (4.6)[ δ ξ , δ η ] A µ = − i v ν F νµ + D µ Υ , [ δ ξ , δ η ] λ I = − i v ν D ν λ I + i[Υ , λ I ] − √ (cid:20) ̺λ I + R JI λ J −
14 Θ αβ γ αβ λ I (cid:21) , [ δ ξ , δ η ] D IJ = − i v ν D ν D IJ + i[Υ , D IJ ] − √ (cid:2) ̺D IJ + R KI D JK + R KJ D IK (cid:3) , where we have defined v µ = 2 ε IJ ξ I γ µ η J , Υ = − ε IJ ξ I η J σ ,̺ = − ε IJ ( ξ I ˜ η J − η I ˜ ξ J ) ,R IJ = − ξ I ˜ η J + ξ J ˜ η I − η I ˜ ξ J − η J ˜ ξ I ) , Θ αβ = − ε IJ ( ˜ ξ I γ αβ η J − ˜ η I γ αβ ξ J ) − ε IJ ( ξ I η J ) b αβ + i4 ε µνραβ b µν v ρ , (4.7)and R JI = ε JK R IK , provided that the spinors ( ξ, ˜ ξ ) and ( η, ˜ η ) satisfy the SU (2) R -covariantization of the ( χ, ϕ ) spinor equations (2.10)–(2.12). More preciselyD µ ξ I = − √ γ µ ˜ ξ I − i12 √ b νρ γ µνρ ξ I + i3 √ b µν γ ν ξ I , D µ ˜ ξ I = − i6 √ b µν γ ν ˜ ξ I −
116 D ν b ρσ γ µνρσ ξ I + 116 D µ b νρ γ νρ ξ I −
18 D ν b µρ γ νρ ξ I + i8 √ V νρI J γ µνρ ξ J − √ V µνI J γ ν ξ J − A (0) ν γ µν ξ I + 112 A (0) µ ξ I + i2 ω µν γ ν ξ I , − √ b µν γ µν ˜ ξ I − √ X ξ I + i8 D µ b νρ γ µνρ ξ I − i18 A (0) µ γ µ ξ I − √ V µνI J γ µν ξ J , (4.8)with V µνIJ ≡ ∂ [ µ V ν ] IJ − V [ µK ( I V ν ] K J ) . Recall that b has Weyl weight w = 1, whilethe spinors have weight w = ± / ω µν = ω νµ , which is the same conditionused in deriving the differential constraints (3.4)–(3.8). Also as for that computationthe closure of the supersymmetry algebra is insensitive to the explicit form of ω µν , A (0) or X . Let us also notice that the supersymmetry variations (4.5) reduce to those ofthe round S in [2] (in particular b ≡ S , and ˜ ξ here I = √ i ˜ ξ there I ).25e now consider the on-shell hypermultiplet which consists of two complex scalars q I and a spinor ψ , all transforming in an arbitrary representation of the gauge group.A system of r hypermultiplets is described by q AI , ψ A with A = 1 , . . . , r . The fieldssatisfy the reality conditions ( q AI ) ∗ = Ω AB ε IJ q BJ and ( ψ Aα ) ∗ = Ω AB C αβ ψ Bβ with Ω AB being the invariant tensor of Sp ( r ). The supersymmetry variations for the system of r hypermultiplets coupled to the vector multiplet are δ ξ q AI = − ξ I ψ A ,δ ξ ψ A = ε IJ γ µ ξ I D µ q AJ + i ε IJ ξ I σq AJ − √ ε IJ ˜ ξ I q AJ . (4.9)The commutator of two supersymmetry transformations leads to[ δ ξ , δ η ] q AI = − i v µ D µ q AI + iΥ q AI − √ (cid:20) ̺q AI + R I J q AJ (cid:21) , [ δ ξ , δ η ] ψ A = − i v µ D µ ψ A + iΥ ψ A − √ (cid:20) ̺ψ A −
14 Θ αβ γ αβ ψ A (cid:21) + 12 v ρ Γ ρ (cid:18) i γ µ D µ ψ A + σψ A + ε IJ λ I q AJ − √ γ µν ψ A b µν (cid:19) − ε KL ( ξ K η L ) (cid:18) i γ µ D µ ψ A + σψ A + ε IJ λ I q AJ − √ γ µν ψ A b µν (cid:19) , (4.10)where D µ q AI = ∂ µ q AI − i A µ q AI − C µ q AI − V µI J q AJ , D µ ψ A = ∇ µ ψ A − i A µ ψ A − C µ ψ A . (4.11)Closure of the algebra occurs only on-shell and this identifies the fermionic equation ofmotion as E ψ ≡ i γ µ D µ ψ A + σψ A + ε IJ λ I q AJ − √ γ µν ψ A b µν = 0 . (4.12)Acting on E ψ with the supersymmetry transformations gives the bosonic equation ofmotion: ε IJ (cid:18) D µ D µ q AJ + σ q AJ − X q AJ + 1 √ ω µµ q AJ − ψ A λ J ) (cid:19) + i D IJ q AJ = 0 . (4.13) The action for a vector multiplet in five dimensions is determined by the prepotential F ( V ), which is a real and gauge invariant function of the vector superfield V . Gauge26nvariance limits the prepotential to being at most cubic in V [46] and classically ittakes the form F ( V ) = Tr (cid:20) g V + k V (cid:21) . (4.14)Here g is the dimensionful gauge coupling constant and k is a real constant whichis subject to a quantization condition dependent on the gauge group [47]. Writingthe components of the vector superfield as V a T a = ( σ a T a , A aµ T a , λ aI T a , D aIJ T a ) where T a are generators of the gauge group in the adjoint representation we find the cubicprepotential term in our curved backgrounds to be L cubic = d abc (cid:20) ǫ µνρστ A aµ F bνρ F cστ + i8 ε IJ ( λ aI γ µν λ bJ ) F cµν + i4 D a,IJ ( λ bI λ cJ ) (cid:21) + d abc σ a (cid:20) F bµν F c,µν −
12 D µ σ b D µ σ c − D bIJ D c,IJ − i2 √ σ b F cµν b µν + 13 σ b σ c √ ω µµ + 23 X − b µν b µν ! (4.15)+ i2 ε IJ ( λ bI γ µ D µ λ cJ ) − ε IJ λ bI [ λ J , σ ] c + 18 √ ε IJ ( λ bI γ µν λ cJ ) b µν . Here d abc ∝ kπ Tr (cid:0) T ( a T b T c ) (cid:1) is a symmetric invariant tensor of the gauge group. Itvanishes for all simple gauge groups except U (1) or SU ( N ) with N ≥
3. The La-grangian L cubic is invariant under the superconformal transformations (4.5) providedthe supersymmetry parameters satisfy (4.8), and in addition A (0) is given by A (0) = − ∗ (d + 2 C ∧ ) ∗ b − i √ b ∧ b ! , (4.16)which matches precisely the expression (3.57) in section 3.The quadratic term in the prepotential includes Yang-Mills kinetic terms and is notconformally invariant. We therefore expect to break conformality by using the relation˜ ξ I = − α I J ξ J − i2 ( K ) µ γ µ ξ I , (4.17)which is the SU (2) R -covariantization of (2.13). The Lagrangian describing the quadraticpiece can be found from L cubic by identifying one of the vector superfields with a con-stant supersymmetry preserving Abelian vector multiplet [7]. That is L YM = 12 g V (1) Tr[ V ] , (4.18) For example, taking the gauge group to be G = SO ( N ) so that the Lie algebra generators satisfy T ta = − T a then Tr (cid:0) T ( a T b T c ) (cid:1) = Tr (cid:0) T ( a T b T c ) (cid:1) t = − Tr (cid:0) T ( a T b T c ) (cid:1) . V (1) = ( σ (1) , A (1) µ , λ (1) I , D (1) IJ ). We choose σ (1) = 1 and λ (1) I = 0. Then V (1) issupersymmetry preserving if the fermion variation δ ξ λ (1) I = − γ µν ξ I F (1) µν − D (1) IJ ξ J + i3 √ γ µν ξ I b µν − √ α IJ ξ J − √
23 ( K ) µ γ µ ξ I , = 0 , (4.19)holds for non-trivial spinor parameters ξ I and some choice of D (1) IJ , A (1) µ such that F (1) = d A (1) . Here we have substituted for ˜ ξ I using (4.17). To progress, note thatthere are two natural one-forms in our geometry namely K and K . If we concentrateon K which, with S = 1, satisfies (3.5)d K = 2 √ (cid:20) αJ + K ∧ K + i b − i2 i ξ ( ∗ b ) (cid:21) , (4.20)then upon SU (2) R -covariantizing and multiplying by − γ µν ξ I we find0 = − γ µν ξ I (d K ) µν − i √ b µν ! − √ γ µ ξ I ( K ) µ − √ α IJ ξ J . (4.21)To derive the previous equation we have used the projection conditions satisfied by thebackground geometry: ( K ) µ γ µ χ = χ and J µν γ µν χ = 4i χ , along with ( K ) µ ( K ) µ =0 = ( K ) µ C µ and − i( K ) µ b µν = ( K ) ν + √ C ν . Comparing this to (4.19) gives theconstant vector multiplet as V (1) = ( σ (1) , A (1) µ , λ (1) I , D (1) IJ ) = (1 , ( K ) µ , , √ α IJ ) , (4.22)and the corresponding Yang-Mills Lagrangian is L YM = 1 g Tr (cid:20) F µν F µν −
12 D µ σ D µ σ − D IJ D IJ + i2 ǫ IJ ( λ I γ µ D µ λ J ) − ε IJ λ I [ λ J , σ ]+ 18 ǫ µνρστ F µν F ρσ ( K ) τ − i √ σ F µν b µν + 12 σ F µν (d K ) µν (4.23) − √ σD IJ α IJ + σ √ ω µµ + 23 X − b µν b µν − i2 √ K ) µν b µν ! + i8 ε IJ ( λ I γ µν λ J )(d K ) µν + 18 √ ε IJ ( λ I γ µν λ J ) b µν − √ λ I λ J ) α IJ . The second candidate one-form is K but taking F (1) = d K does not lead to (4.19).28he superconformal Lagrangian for the vector coupled hypermultiplets exists irre-spective of the gauge group and is straightforward to construct: we simply integratethe equations of motion (4.12) and (4.13) found from closing the superalgebra to find L hm = Ω AB (cid:20) − ε IJ D µ q AI D µ q BJ + 12 ε IJ q AI σ q BJ + i2 q AI D IJ q BJ − ε IJ q AI ( ψ B λ J ) + ε IJ q AI q BJ (cid:18) √ ω µµ − X (cid:19) + i( ψ A γ µ D µ ψ B ) + ψ A σψ B − √ ψ A γ µν ψ B ) b µν (cid:21) . (4.24) In this paper we have constructed rigid supersymmetric gauge theories with matter on ageneral class of five-manifold backgrounds. By construction these are the most generalbackgrounds that arise as conformal boundaries of six-dimensional Romans supergrav-ity solutions. We find that ( M , g ) is equipped with a conformal Killing vector whichgenerates a transversely holomorphic foliation. In particular the transverse metric g (4) is an arbitrary Hermitian metric with respect to the transverse complex structure. Thisis a natural hybrid/generalization of the rigid supersymmetric geometries in three andfour dimensions constructed in [35, 37, 38], and includes many previous constructionsas special cases.It is interesting to compare the geometry we find to the rigid limit of Poincar´e su-pergravity [18, 19] and the twisting of [28]. In the former case the backgrounds naivelyappear to be more general, as there is no almost complex structure singled out, norintegrability condition. However, they don’t include the S × S geometry relevantfor the supersymmetric index, which as we showed in section 3.3 is included in ourbackgrounds. In fact the singling out of the almost complex structure associated to J = J , where recall that Ω = J + i J , in our geometry is almost certainly related tothe 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 becausethe known solutions to Romans supergravity discussed previously also have this prop-erty. Nevertheless, the supersymmetry variations and Lagrangians we constructed insection 4 are valid for an arbitrary background SU (2) R gauge field, and it should be rel-atively straightforward to analyse the geometric constraints in this more general case.Indeed, this is certainly necessary, and presumably sufficient, to reproduce the partiallytopologically twisted backgrounds S × M of [11–13], since the SU (2) spin connection29f M is twisted by SU (2) R . On the other hand recall that the twisting in [28] requiresthat M be a K-contact manifold. This shares many features with our geometry, withone 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 somesense precisely the opposite, namely transversely holomorphic. These intersect pre-cisely for Sasakian manifolds. It is interesting that these various approaches generallyseem to lead to different supersymmetric geometries, with varying degrees of overlap.Given the geometry we find and the results of [48], it is natural to conjecture thatthe partition function and other BPS observables depend only on the transverselyholomorphic foliation, i.e. for fixed such foliation they are independent of the choiceof the remaining background data (functions S , α , the one-form ν defined in section3.5, and the transverse Hermitian metric g (4) ). It will be interesting to verify thatthis is indeed the case, and to compute these quantities using localization methods.Notice that locally a transversely holomorphic foliation always looks like R × C , whichperhaps also explains why in [19] the authors found that locally all deformations of theirbackgrounds were Q -exact. Finally, our construction allows one to address holographicduals of these questions, which we plan to return to in future work. Acknowledgments
The work of L. F. A., M. F. and P. R. is supported by ERC STG grant 306260. L. F. A.is a Wolfson Royal Society Research Merit Award holder. J. F. S. is supported by theRoyal Society. P. B. G. is supported by EPSRC and a Scatcherd European Scholarship.
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