Classifying solutions to Romans supergravity with a zero B-field

Expanding on previous papers, we continue studying Euclidean Romans supergravity in six dimensions with a non-trivial Abelian R-symmetry gauge field. Using a set of differential constraints on a $SU(2)$ structure, we look for further geometric solutions to such equations when we turn off the two-form potential B. We find that the six-dimensional space is described by a two-dimensional fibration over a four-dimensional manifold with a K\"ahler metric. We then classify these types of solutions.


Introduction
In previous papers [1,2], we constructed gravity duals to five-dimensional gauge theories on non-conformally flat backgrounds, specifically, certain families of squashed five-spheres and Sasaki-Einstein manifolds. In [1], we have used the six-dimensional Romans F (4) supergravity [3], which is a consistent truncation of massive IIA supergravity on S 4 [4], and, more recently, it has been shown to be also a consistent truncation of IIB supergravity on a warped product of S 2 × Σ, where Σ is a Riemann surface [5,6]. Having constructed these supergravity solutions, we then computed the holographic free energy F = − log Z by holographically renormalizing the onshell Euclidean action. The perturbative partition function for these theories has been computed in [7] and we have explicitly shown that the large N limit of these partition functions is in precise agreement with the holographic free energies of our supergravity solutions.
In [2], we have shown that real Euclidean supersymmetric solutions to Romans F (4) gauged supergravity, with a non-trivial Abelian R-symmetry gauge field, have a canonical SU (2) structure determined by the Killing spinor. More precisely, we have shown that the supersymmetry conditions together with the equations of motion are equivalent to a set of differential constraints on this SU (2) structure. This geometric formulation then led to a number of interesting applications. We showed that this structure extended into the bulk the conformal boundary SU (2) structure studied in [8], which allowed for the construction of gravity duals to families of five-dimensional gauge theories on rigid backgrounds. As another application we extended several of the results of the previous paper.
In fact, analysing supersymmetric solutions via G-structures has been widely used, first shown in [9], then developed for various string theory settings [10]- [21]. In this paper, we take the same SU (2) structure from [2] and look for further generalizations of the solutions. In particular, we use the approach from [22] to classify the families of solutions arising from a theory with a zero two-form potential B. We find that our six-dimensional space can be seen as a five-dimensional one which is orthogonal to the Killing vector. This five-dimensional space, in turn, can be seen as a product of a one-dimensional space and a four-dimensional space with a Kähler metric.
The plan for the paper is as follows. In section 2, we summarize Euclidean Romans supergravity theory with a non-trivial Abelian R-symmetry gauge field. In section 3, we summarize the differential constraints imposed on the canonical SU (2) structure of this Romans theory to ensure supersymmetric solutions. In section 4 we take the B-field to be equal zero in the differential contraints forementioned, simplifying the problem. These new equations can now be solved in closed form. We end by classifying the families of solutions.

Euclidean Romans supergravity
The bosonic fields of the six-dimensional Romans supergravity theory [3] consist of the metric, a scalar field 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 are [1] Notice that the theory contains Chern-Simons-type couplings, that become purely imaginary in Euclidean signature. The Einstein equation is 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 µ , µ = 1, . . . , 6, are taken to be Hermitian and generate the Clifford algebra Cliff(6, 0) in an orthonormal frame. We have defined the chirality operator Γ 7 = iΓ 123456 , which satisfies (Γ 7 ) 2 = 1. The covariant derivative acting on the spinor is 4 Ω νρ µ Γ νρ denotes the Levi-Civita spin connection while σ i , i = 1, 2, 3, are the Pauli matrices.
For simplicity we shall consider Abelian solutions in which A 1 µ = A 2 µ = 0, and A 3 µ ≡ A µ , with field strength F ≡ dA. Also, as in [1], we consider a "real" class of solutions for which ǫ I satisfies the symplectic Majorana condition ε J I ǫ J = Cǫ * I ≡ ǫ c I , where C denotes the charge conjugation matrix, satisfying Γ T µ = C −1 Γ µ C. The bosonic fields are all taken to be real, with the exception of the B-field which is purely imaginary and that will later be taken to be zero. With these reality properties one can show that the Killing spinor equation (2.3) and dilatino equation (2.4) 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 1 , ǫ 2 ) = (ǫ, ǫ c ).

SU (2) structure
Consider a Dirac spinor ǫ in six dimensions, such that (ǫ 1 , ǫ 2 ) = (ǫ, ǫ c ) solves (2.3) and (2.4) above. We may construct the following scalar bilinears We have chosen a basis for the gamma matrices in which they are purely imaginary and anti-symmetric, with charge conjugation matrix C = −iΓ 7 . A short computation reveals that The integrability condition for this equation immediately implies F = dA = 0 unless f ≡ 0 (notice that X is nowhere zero). We will henceforth restrict our analysis to the case f ≡ 0, which is necessary for a non-trivial R-symmetry gauge field. 1 We may then write where −Γ 7 ǫ ± = ±ǫ ± , and furthermore the condition f ≡ 0 allows us to introduce as done in [22]. Here η 1 , η 2 are two orthogonal unit norm chiral spinors, so that η † 1 η 1 = η † 2 η 2 = 1 and η † 2 η 1 = 0. These each define a canonical SU (3) structure, and together determine a canonical SU (2) structure. Concretely, in six dimensions such a structure is specified by two one-forms K 1 , K 2 and a triplet of two-forms J i , i = 1, 2, 3, given by Here we have introduced the notation Γ (n) ≡ 1 n! Γ µ 1 ···µn dx µ 1 ∧ · · · ∧ dx µn , where x µ are local coordinates. We also define The canonical SU (2) structure is thus determined by (K 1 , K 2 , J, Ω). We note that K 1 and K 2 are orthonormal one-forms, and both are orthogonal to J and Ω, with J ∧ Ω = 0 and 2J ∧ J = Ω ∧Ω.
The SU (2) structure (S, ϑ, K 1 , K 2 , J, Ω) that arises naturally from a supersymmetric solution is thus related to the canonical SU (2) structure by the square norm S and angle ϑ, via (3.8).
It was shown in [2] that the differential contraints on this SU (2) structure are given by where fields were divided into components parallel and perpendicular to K 1 , so that K 1 F ⊥ = 0. It is also shown in [2] that, when supplemented by solving these equations is equivalent to finding supergravity solutions to Euclidean Romans theory.

Classifying solutions with a zero B-field
Briefly, the idea is to take Euclidean Romans supergravity theory and set the twoform potential B to be zero. This leads us to a simpler set of equations of motion, dilatino and Killing spinor equations.
We analyzed possible solutions to Romans supergravity in previous papers [1,2], and a particular case where B = 0 was discussed in [23]. One can expect, however, to find more possible solutions to this case. By following the procedure adopted in [22], here we classify these families of solutions.

Taking B = 0
Taking the two-form potential B = 0 in (3.11) and (3.12) implies and Solving these equations is sufficient to ensure we have a supersymmetric solution to the Euclidean equations of motion of Romans theory in this particular case. It is also worth recalling from [2] that the one-form K 1 can be written as where ∂ ψ is the supersymmetric Killing vector that preserves all the structure, and also that part of F perpendicular to K 1 can be written as In order to analyse these equations, we will need to break the derivatives (and the remaining of each equation) further into components. We start by defining a radial coordinate in the K 2 direction, this will given by ρ = XS , such that dρ = d(XS) . (4.23) The exterior derivative can be written as i.e., in the K 1 , K 2 and M 4 directions. Another term that requires attention is the one-form σ. For instance, one may consider where σ 4 is the σ component in the direction that is both perpendicular to K 1 and K 2 . Its derivative is then given by Notice however that, if we reparametrise ψ (that enters in the definition of the oneform K 1 ), one can make a gauge choice of shifting it in such a way that This way, in writing down dψ + σ, we have σ ρ dρ being cancelled, and we can simply and relabel as both terms are in the d 4 direction. This way, we are free to make a choice where σ ≡ σ 4 . Notice however that we still have to consider dσ = d 4 σ 4 + dρ ∧ (∂ ρ σ 4 ).

Conditions for a supersymmetric M 6
Notice that once we take the B-field to be zero, we immediately get B 1 = 0, this gives us 3 √ 2S sin 2ϑ d(XS) + X −2 K 2 = 0 . (4.31) As we defined, d(XS) = dρ (now confirming that K 2 is in fact in the dρ direction), indeed, In (4.21), notice that the first term is zero, and it simply reduces to Next, equation (4.14) reads This is equivalent to so that we can say cos 2ϑ = X 4 λ(ρ). We then get an equivalent to equation (2.36) in [22], namely Notice that from this equation also follows that d 4 σ has no components proportional to Ω (but it still could have an anti-self-dual part). Similarly, equation (4.17) reads Here again we have used gauge freedom to remove the part of A ⊥ proportional to dρ, so that A ⊥ = A 4 . These equations imply that the geometry at constant ρ (and ψ) is (conformally) Kähler, with Kähler metricĝ 4 associated toĴ andΩ. Moreover, since the derivative ofΩ in the ρ direction is proportional toΩ, this shows that the associated complex structureÎ is independent of ρ, ∂ ρÎ = 0. Since whereP is the canonical Ricci one-form potential, we identify Note that this can be rewritten as Next, we turn to equation (4.18). Multiplying it by X 2 and substituting from (4.22), this reads and This implies that the one-form in brackets is a (1, 0)−form, and hence Next we turn to equation (4.18). One finds that the component of this equation in the dρ direction is precisely equivalent to equation (4.51). The remainder of equation (4.18) is equivalent to Finally we turn to the scalar equation (4.20). After a computation, one remarkably finds precisely equation (4.52) plus a (generically) non-zero function times ∂ ρ (ρλ(ρ)). One concludes that where c is an integration constant. One can check that equation (4.53) is also true for the four-parameter family of BPS black hole solutions discussed in [23], a highly non-trivial check. We have thus solved these equations for one of the functions in the problem. Notice that now one can write ρ = cX 4 sec 2ϑ . We thus really have only one free function in the problem, and we can take it to be X.
We have now analysed all the content of all the equations, apart from equations (4.15) and (4.21). After quite a lengthy calculation, and using many of the equations above, one can show that the dρ component of equation (4.15) is precisely equivalent to (4.52). Notice that the anti-self-dual part of F 4 enters, which is related to (d 4 σ) − via (4.34), but this combines with d 4 σ, and in the end only the self-dual part of the it remains, and it is proportional to J, as Ω∧d 4 σ = 0. The remainder of equation (4.15) is easier to compute, and one finds an equivalent to (4.51). Thus (4.15) is implied by all the other equations, and hence imposes nothing new.
It thus remains only to impose the equation of motion (4.21). The two components read where from (4.34), we have together with the scalar equation We conclude by noting that a few equations are redundant. First (4.51) is precisely the dρ component of d(4.46). Here we have the second equation which may be combined with (4.55) to obtain an Einstein-like equation (involving the Ricci form of the Kähler metric). To see this, recall thatP = 1 2Îḋ 2 log √ detĝ. But since alsoΩ ∧Ω = 4v ol = 2Ĵ ∧Ĵ is automatically true, it follows by taking ∂ ρ that is an identity. Using this, one can check that equations (4.41) and (4.43) in fact imply equation (4.52). The latter is hence implied by the other equations, and it is also redundant.

Summary
We can now put together the necessary and sufficient conditions to have a supersymmetric solution. The metric is given by where now g SU (2) is a conformally Kähler manifold. We can rewrite it as whereĝ is a one-parameter family of Kähler metrics depending only on ρ, for which the complex strucutureÎ is independent of ρ. The vector ∂ ψ is Killing, and preserves all of the structure. The functions X and ϑ are related by where c is a non-zero constant, so that we may substitute The evolution equations for the Kähler structure are and ∂ ρ tan 2ϑΩ = 3 c sin 2ϑΩ . (4.65) Notice that Ω ∧ d 4 σ = 0 is consistent with equation (4.64) andĴ must remain type (1, 1) as the complex structure is independent of ρ. From (4.46), we have the Einstein-like equation together with Finally, we must impose the B-field equation of motion components and The norm here is with respect to g 4 (rather thanĝ 4 ). Notice that remarkably the supersymmetry equations above are almost exactly the same (essentially up to numerical factors) to the equations in [22].

Complex M 6 (Setting d 4 ϑ = 0)
In [22] the equations of this form were solved in closed form, with the additional assumption of d 4 ϑ = 0, leading to new solutions. It is then natural, due to the similarity of the system, to make the same assumption here.
In order to have a six-dimensional complex manifold with Hermitian metric, we require the three-form given by Ω (3) = Ω ∧ (K 1 + iK 2 ) to have a derivative in the form with v = 0. This restriction will imply that dσ ≡ d 4 σ, and implies that d 4 X = d 4 ϑ = d 4 S = 0. From this, one can deduce that Next, we may look at (4.66), which readŝ The Ricci scalar of the Kähler metricĝ 4 isR =Ĵ ijR ij , so that using (4.52), we computeR Since the right hand side is a function only of ρ, we deduce that d 4R = 0, andĝ 4 is a constant scalar curvature Kähler metric (for fixed ρ).
We may similarly computeR ijR ij =R ijR ij from equation (4.72). Using again (4.52) to computeĴ d 4 σ, and (4.69) to compute (d 4 σ) − 2 , the right hand side is again a function only of ρ, and we deduce that It follows that at fixed ρ, the Ricci tensorR ij has two pairs of constant eigenvalues. If these eigenvalues are the same, this is a Kähelr-Einstein metric, while if they are distinct and M 4 is compact, then the Goldberg conjecture implies that M 4 is locally a product of two Riemann surfaces of (distinct) constant curvature.
We shall consider both cases separately.

Kähler-Einstein base solutions
For a Kähler -Einstein metricR ∝Ĵ , with the constant of proportionality depending only on ρ. Thus d 4 σ is also proportional toĴ. One checks that there are no solutions with d 4 σ = 0, so, without loss of generality, we set where ∂ ρJ = 0. Thus the rescaled Kähler metricJ is independent of ρ, and the Kähler-Einstein condition readsR where κ ∈ R is a constant. Solving first equation (4.64), we find where a in an integration constant. Substitutuing this into (4.72), and X in (4.62), one can find ϑ, given by Notice that at this point, all the functions have been completely determined. Next, solving (4.65), we can write a and c in terms of κ (whereΩ = F (ρ)Ω) It follows that X ≡ 1. Finally, notice that (d 4 σ) − = 0, and one can check that the right hand side of the equation (4.69) is in fact zero. At this point we have solved all the equations. The final solution is therefore given by The six-dimensional metric is where dσ =J, andg 4 is a constant (in ρ) Kähler-Einstein metric withR = κJ. The gauge field A has dA = −dR, so that A is a connection on the canonical bundle of M 4 . The ρ coordinate in the metric (4.81) is somewhat peculiar. A better system of coordinates is set by making the change (4.82) The metric then becomes ds 2 6 = 9 κ 2 + 2r 2 dr 2 + r 2 (dψ + σ) 2 + 3 2κĝ 4 . This is simply the hyperbolic cone over a regular Sasaki-Einstein manifold. Notice that the full gauge field F = 0. Thus, this solution was known.

Product of two Riemann surfaces base solutions
Analogous to the Kähler-Einstein solutions, we can consider the metric wherê with F 1 (ρ) and F 2 (ρ) depending only on ρ. Then we also have d 4 σ given by with the factors c 1 and c 2 being constants. Here the two-formsJ 1 andJ 2 are such that ∂ ρJ1 = ∂ ρJ2 = 0. Again the rescaled Kähler metric is independent of ρ and we can writeR = k 1J1 + k 2J2 , (4.86) where k i ∈ R are constants. Solving equation (4.64), we find where a and b are integration constants. We can now substitute this into (4.72), with X given by (4.62). Notice that there are two equations for ϑ. We find cos 2ϑ = 4 a c (4.89) Now solving equation (4.65) order by order, one also finds constraints to c, a and c 2 , namely We then check equation (4.69) and confirm that it holds. The final solution is a function of the parameters left, i.e., k 1 , k 2 and c 1 , and it is given by