A deformed conifold with a cosmological constant

We find a new regular solution of six-dimensional Einstein's equations with a positive cosmological constant. It has the same isometry group as the (deformed) conifold geometry, and the superpotential approach is used to solve the equations of motion. The space is compact and interpolates between the deformed conifold and the resolved cone with a blown-up four cycle. The deformation/resolution parameters are set by the cosmological constant.

regular and, as expected, compact. For one limiting value of the (former) radial coordinate the space asymptotes to the deformed conifold solution, while in the other limit one finds a regular resolution with the blown up S 2 × S 2 four-cycle. Remarkably, the regularity at either end requires no orbifolding of the 5d base. The sizes of the 3-sphere and the 4-cycle are related and both are determined by the cosmological constant Λ. The metric has a singular limit, where (at least) one corner of the space has a conic singularity. The singular solution preserves the U (1) ψ symmetry associated with the Reeb angle of the T 1,1 base. Although our space is compact, we will still refer to the regions with the deformed and the resolved spaces as the IR and the UV respectively. The reason for these notations will be clarified later in the paper.
The Ansatz we consider involves three independent functions of the radial coordinate, and in general requires solving a set of three second-order coupled non-linear ordinary differential equations (ODEs), similar to [13]. However, the superpotential we found greatly simplifies the task, since it leads to first-order ODEs. Out of the three equations, one can be solved analytically and the other two combine into a single second-order equation that can be treated numerically. This allows us to find a relation between the "UV" resolution and the "IR" deformation parameters. Importantly, the superpotential approach is futile for the full PT Ansatz, and this is the main reason we imposed the Z 2 symmetry. As a result, we have to exclude the 2-cycle resolution from the discussion, since it does not fit into the Z 2 -symmetric Ansatz.
There are three primary motivations for this work. First, it provides a natural extension of the well-known old results. The so-called Eguchi-Hanson-de Sitter space was found more than 30 years ago in [14,15]. It is the compact version of the Eguchi-Hanson (EH) geometry [16], which solves Einstein equations with a positive cosmological constant. At both "corners" of the space the C 2 /Z 2 singularity is resolved by blown-up two-spheres of the same size. This size is, in turn, fixed by the cosmological constant. This is very similar in spirit to the results of this paper, though we find cycles of different dimensions in the IR and the UV. This should be of no surprise, since the 2-cycle resolution/deformation is the only option in four dimensions. 2 Second, consistent Kaluza-Klein reductions of type IIB supergravity on the compact sixdimensional space may lead to interesting four-dimensional gauged supergravity theories. The truncation will be probably easier for the singular version of the solution due to the preserved U (1) ψ symmetry factor. The reductions (if exist) will share many features with the one constructed in [17,18]. 3 Third, our solution can be used to build a new KS-like background in type IIB supergravity with AdS 4 and our compact 6d geometry replacing Mink 4 and the non-compact deformed conifold respectively. Solutions with 3-form imaginary self-dual (ISD) fluxes 4 on compact Einsteinflat transversal spaces have recently attracted a great deal of attention (see [21] for the most recent developments). The 5-form tadpole cancellation on the compact 6d space necessitates the introduction of either anti-D3 brane sources or orientifolds. Supergravity solutions with antibranes placed in backgrounds that contain opposite charges dissolved in the fluxes are known to have certain singularities (see [20] for the extended list of references). For example, anti-D3 branes smeared over the tip of the warped deformed conifold induce a 3-form flux singularity, as was proven in [22]. It was furthermore argued in [23] that this singularity cannot be cured by Polchinksi-Strassler polarization [24] of D5-branes warping the shrinking 2-sphere. The situation may, however, change once the deformed conifold is made compact. As a toy model capturing some of the physics, one may consider the anti-D6 singularity in massive type IIA supergravity [25]. This is the (three times) T-dual of the anti-D3's we discussed above, but now smeared over a 3-torus rather than over the S 3 at tip of the KS geometry. According to [26], for flat Minkowskian world-volume, the singularity cannot be resolved by D8-brane polarization independently of the parameters of the fully backreacted anti-D6 solution. When the D6's have AdS 7 world-volume, however, the polarization fate depends on the values of the cosmological constant and other parameters of the fully backreacted anti-D6 solution. 5 Hence, it will be exciting to see whether our "compact deformed conifold" space leads to a 3-form flux singularity that can be smoothed out by the 5-brane polarization. To answer this question one will have to understand first whether the cosmological constant is a free parameter in the fully backreacted solution or it is rather determined by the fluxes, as it happens for completely smeared sources [28]. The method presented in [29] might appear useful to answer this question without constructing the full solution.
The paper is organized as follows. In Section 2 we present the metric Ansatz, the onedimensional effective action, the superpotential equation solution and the corresponding firstorder equations of motion. We also briefly review the known non-compact solution with zero cosmological constant. In Section 3 we write down the new compact solutions, both singular and regular. In the Appendix we give the 1-forms definitions and summarize the relation to the conventions of [9].

The Ansatz for the metric and the equations of motion
Our Ansatz for the six-dimensional metric has the same isometries as the deformed conifold space of [5], which in turn is a particular example of the Papadopoulos-Tseytlin (PT) metric Ansatz [9]: (2.1) Here the functions z(τ ), w(τ ) and y(τ ) depend only on the radial coordinate τ , and the definitions of the angular one forms g i are given in Appendix A. Apart from the SU (2) × SU (2) isometry, for y = 0 the metric enjoys an additional U (1) ψ symmetry associated with the Reeb angle ψ. For non-zero y(τ ) the U (1) ψ is broken down to Z 2 . An extra Z 2 symmetry preserved by (2.1) acts on the angles as g 1,2 → −g 1,2 with the other three 1-forms being invariant. In terms of the angles in (A.2) it is merely (θ 1 , φ 1 ) ↔ (θ 2 , φ 2 ). This symmetry reduces by one the number of functions in the most general PT Ansatz. 6 We relegated to Appendix B the relations between our functions and those of [9]. The most general regular Ricci-flat solutions of the form (2.1) are the deformed conifold metric [5] and the 4-cycle resolution of the T 1,1 /Z 2 singularity, and we will review both solutions in the next section. We are, however, interested in an Einstein-flat solution. To obtain the one-dimensional effective action for the three scalar functions one has to plug (2.1) into the Einstein-Hilbert action and integrate over the five angles. The output is: where the last terms comes from the cosmological term √ g 6 Λ in the action with: and the remaining terms can be found in [9]. Surprisingly the inclusion of the new term still allows for a simple solution of the superpotential equation for (2.2): 7 Let us stress that (2.5) is not the most general solution of the superpotential equation. Typically (2.5) should be a special case of a solution depending on (maximum) two free parameters, but we were not able to find it. The superpotential (2.5) leads to the following equations of motion: Remarkably, the equation for y(τ ) has no R in it and so has exactly the same solutions as for the Ricci-flat metric [5,9]: y = 0 and e y = tanh τ 2 . (2.7) The first solution preserves the U (1) ψ , while the second one breaks it down to Z 2 (see the comment below (2.1)). In what follows we will consider both options for Λ = 0 as well as for Λ > 0. We will see that the U (1) ψ -breaking choice of y(τ ) yields a regular (deformed) solution both for the non-compact and the compact solutions. The first two equations in (2.6) can be recast in the following form: e 3z − 2 cosh y e 3z + 3 4R 2 e 4z = 0 (2.8) We see that the superpotential method leads to a single second-order equation of motion. Solving (2.10) 7 We follow the following conventions for the superpotential equation and the first-order equations of motion: The 6-dimensional metric is then: (2.11) For r 0 = 0 this reduces to the singular conifold metric, while for a non-zero r 0 it describes a geometry with a blown up 4-cycle, which is just the product of the spheres, S 2 × S 2 , spanned by (θ 1 , φ 1 ) and (θ 2 , φ 2 ). Zooming near r = r 0 one finds that in order to avoid a singularity, the Reeb angle ψ (see the definition of g 5 in (A.2)) has to be 2π-periodic. Since for the singular conifold the period is 4π, the five-dimensional base is then T 1,1 /Z 2 . Similar solutions exist also for the Y p,q and L a,b,c Sasaki-Einstein spaces. In all these examples the blowing up of the 4-cycle resolves the conic singularity at the tip provided the Reeb angle has the right periodicity. Out of the two integration constants the first one is the deformation parameter , which measures the S 3 size at the tip, and the second constant has to be fixed to avoid a singularity at τ = 0. Before closing up this section it is worth noticing here that for Λ = 0 (or equivalently infinite R) the EOMs (2.6) are invariant under (z, w) → (z + 2λ, w − 6λ). For the singular conifold solution this rescaling can be absorbed in the radial coordinate redefinition, while for the two non-conic solutions it changes the physical size of the corresponding blown-up cycles. We will return to this issue at the end of the next section.

New compact solutions, singular and regular
In this section we will consider solutions with finite R. For y = 0 the z(τ ) equation (2.8) can be solved analytically: 8 Upon the redefinition the metric takes the following form: with α ∈ [0, 2π]. Near α = 0 the T 1,1 part of the metric shrinks and the space looks like the singular conifold geometry. On the other hand, the four cycle has a finite size at α = π.
Expanding near this point we find that the g 5 part has no conical deficit provided ψ is 2 3 πperiodic, implying that we have to quotient T 1,1 by Z 6 in order to end up with a regular space at α = π. The geometry will, however, be still singular at α = 0.
We will now study the main subject of this paper: the U (1) ψ -breaking solution with e y = tanh τ 2 and a non-zero cosmological constant. The regular solution for small τ is: At leading order it coincides with the deformed conifold solution (2.12) with C IR ∼ 4/3 . The constant C IR is a free IR parameter that has to be properly adjusted by the large-τ boundary conditions, 9 since for a generic C IR the solution will be singular for large τ . Note that according to (2.6) e z and e −w are both monotonic increasing functions. This means that the 4-cycle spanned by g 1,2,3,4 acquires a non-zero size for large τ . At the same time, the equations of motion imply that the ψ 1-cycle shrinks there. In other words, for large τ the geometry is that of (2.11) with a non-zero r 0 . Since the periodicity of ψ is already fixed in τ = 0 to be 4π, the space is regular if and only if the g 5 part of the metric (2.1) looks asymptotically as e −τ dτ 2 + g 2 5 . 10 This is, in turn, possible only if (e z ) behaves at infinity as e −τ . Such a solution indeed exists and its asymptotic expansion for large τ is: We finally conclude that the IR integration constant C IR has to be chosen such that in the UV the function e z(τ ) approaches the value 3R 2 . This will guaranty that the S 1 ψ shrinks there smoothly.
As the equation (2.8) does not allow for an analytic solution for the U (1) ψ -breaking choice of y, see (2.7), we have to use numerics to solve this equation. The output for the e z(τ ) function is shown on Figure 1. For R = 1 the right "UV" solution is obtained for C IR = 1.5(0). Moreover, matching the numerical solution to the subleading terms of (3.5) we find that C UV = 1.9 (6).
To summarize, we have found a new regular solution of Einstein's equations with a non-zero cosmological constant having the isometries of the deformed conifold. The space is schematically presented on Figure 2. At the minimal value of the (former) radial coordinate the geometry looks 9 We will refer to the small and the large τ regions as the IR and the UV even though the space is now compact.
The main reason for that is (2.7), which is the same as for the non-compact solution, where large τ corresponds to the UV region. 10 Upon the definition r = e τ 2 one gets 4dr 2 + r 2 g 2 s implying that ψ is indeed 4π-periodic. In fact, the constant C IR , the τ = 0 deformation parameter, behaves as C IR ∼ R 2 . This immediately follows from (3.4) either by using the dimensional analysis of (2.1) or by noting that for finite R the scaling symmetry mentioned at the end of the previous section modifies to: (z, w, R) → z + 2λ, w − 6λ, e λ R . (3.6) Together with the numerical result for R = 1 it implies that Similar analysis reveals also that C UV does not scale with R or, in other words, the R = 1 result, C UV = 1.9(6), holds actually for any R.
It is worth to emphasize again that (2.5) is supposedly not the most general solution of the superpotential equation. It is reasonable to believe that there exists a solution for which the sizes of the blown-up cycles depend on additional free parameters and not only on R. Playing with these parameters it should be (presumably) possible to obtain the singular metric (3.3) as a special limit of the regular solution. It remains to be seen whether such a general solution does exist, and if yes, whether it follows from a certain superpotential. helpful suggestions and corrections. This work was supported in part by the ERC Starting Grant 240210 String-QCD-BH and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by Theiss Research).

A Angular one-forms
In this appendix we summarize the definitions of the metric 1-forms in terms of the angular coordinates θ 1,2 , φ 1,2 and ψ.

B Relation to other conventions in the literature
Here we present the connection between the functions of the Ansatz (2.1) and the metric functions used in [9]: Notice that the relation between a and g is required by the Z 2 symmetry we discussed below (2.1).