A method to find \( \mathcal{N} \) = 1 AdS4 vacua in type IIB

Journal of High Energy Physics, Jan 2017

In this paper, we are looking for \( \mathcal{N} \) = 1, AdS4 sourceless vacua in type IIB. While several examples exist in type IIA, there exists only one example of such vacua in type IIB. Thanks to the framework of generalized geometry we were able to devise a semi-algorithmical method to look for sourceless vacua. We present this method, which can easily be generalized to more complex cases, and give two new vacua in type IIB.

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A method to find \( \mathcal{N} \) = 1 AdS4 vacua in type IIB

Received: October Published for SISSA by Springer INFN 0 1 Sezione di Milano-Bicocca 0 1 Open Access 0 1 c The Authors. 0 1 0 Piazza della Scienza 3 , 20100 Milano , Italy 1 Physics Department, Universita` di Milano-Bicocca In this paper, we are looking for N = 1, AdS4 sourceless vacua in type IIB. While several examples exist in type IIA, there exists only one example of such vacua in type IIB. Thanks to the framework of generalized geometry we were able to devise a semi-algorithmical method to look for sourceless vacua. We present this method, which can easily be generalized to more complex cases, and give two new vacua in type IIB. Flux compactifications; Superstring Vacua - method to find AdS4 vacua in type IIB The supersymmetry conditions Description of the method Step 0: definitions Step 1: obtaining linear constraints Step 2: eliminating the derivative in the quadratic equations Step 3: simplifying the leftover quadratic equations Step 4: adding linear constraints Step 5: going back to step 1 Final step: solving the last equations Examples of new vacua An example of Lu¨st-Tsimpis solution A new solution with constant dilaton A new solution with non constant dilaton T-dual solution Introduction. Compactification to 4-dimensional anti-de Sitter (AdS4) are of relevance to several aspects of string theory. In particular, they are central in the CFT3/AdS4 correspondence. They can also be a first step toward obtaining a de Sitter vacuum if one devise a way to break supersymetry in a controlled way. In type IIA, several AdS4 vacua have been found without [1–5] or with [6–14] sources (this is a non exhaustive list of examples). On the contrary, in type IIB, there have been far less studies. Some results have been found with sources [15–18] but only one example without sources [3] (even if the solution is singular in the compactified description). It is to remedy to this state of affairs that we looked for more sourceless vacua in type IIB. This type of vacua also presents two advantages. The first one is, as we already mentioned, their use in the CFT3/AdS4 correspondence. The second one is the validity of such vacua. Indeed, in most known examples with sources, the sources are smeared and one can ask if this assumption is well-founded. Getting rid of the sources also gets rid of this problem. In order to find sourceless vacua, we use the pure spinors formalism developed in [19– 21]. This permits to obtain linear algebraic equations for the SUSY equations. We are left, thanks to the integrability theorem [10, 22, 23], with the Bianchi identities which are quadratic and differential. Since these are not solvable in all generality, one has to devise a way to solve them. Taking inspiration from [24], where parts of the quadratic equations were in fact linear and permitted to solve the whole system of quadratic equations, we put in place a semi-algorithmical method to solve the equations. We also had to take care of the differential part which was absent from [24]. This method can be easily generalized to all type of problems with the same characteristics. Thanks to it, we were able to recover an example of the known sourceless vacuum [3] and discover two new vacua which are a priori sourceless. A more careful study shows that these solutions are singular and we give for one of these examples a possible interpretation in terms of sources. This paper is organized as follows. In section 1, we present the supersymmetry conditions in the framework of generalized geometry applied to our specific case. In section 2, we expose the method to solve the quadratic equations. Finally in section 3, we give three examples of vacua, one of them already known that we recover thanks to our method and two new ones. The supersymmetry conditions the manifold the theory lives on is of the type: ds2 = e2Ads(24) + ds(26) , with A the warp factor. As discussed in [15, 25] such solutions are only possible when the compactification manifold have SU(2) structure group. Let us recall that a manifold is said to be of SU(2) structure if it admits a complex one form z, a real and a holomorphic parameter. The parameters kk and k ⊥ (k2 + k2 = 1) are related to the choice of structure k ⊥ on the internal manifold. When kk = 0 and k ⊥ = 1 the structure is strict SU(2), while the general case where both kk and k zxz¯ = 2 , j ∧ j = zxz = z¯xz¯ = 0 , − = − 8 ezz¯/2(kke−ij language of Generalized Complex Geometry [26, 27]. We will give here a lightning review restricted to our specific case, for some more details, see for example [16, 24] and references The idea is to express the ten-dimensional supersymmetry variations as differential equations on a pair of polyforms defined on the internal manifold. In our case they are structure.1 When kk and k structure rather than dynamical SU(2) structure [28]. As shown in [20], for type IIB compactifications to AdS4 the ten-dimensional super(d − H∧)(e2A−φΦ−) = −2μeA−φ Re Φ+ , (d − H∧)(eA−φ Re Φ+) = 0 , (d − H∧)(e3A−φ Im Φ+) = −3e2A−φ Im(μ¯Φ−) − 8 (d − H∧)(e3A−φz ∧ e−ij ) = 2|μ|e2A−φ(ωI − zRzI ωR) (d − H∧) e2A−φ(ωI − zRzI ωI ) = 0 with R and I denoting the real and imaginary part. this case here. The ten-dimensional fluxes are defined in terms of F by It is convenient to introduce the rescaled forms but for simplicity of notation, we will drop the ˆ symbols in the rest of the paper. Plugging the explicit form of (1.3) and (1.4), into the SUSY variations (1.5a)–(1.5c), kk = 0 We will choose the first case namely a strict SU(2) structure. In this case, the equations (1.5a)–(1.5c) become: In this section, we present the semi-algorithmical method used to find new sourceless vacua parts of the equations are linear, i.e. of the type (2.5), and parts of the equations are quadratic/differential, i.e. of the type (2.6). Step 0: definitions Let ei be a 6D vielbein on the internal manifold. Define: z1 = eA(e1 + ie2) z2 = eA(e3 + ie4) z3 = eA(e5 + ie6) E1i = z = z1 j = (z2 ∧ z2 + z3 ∧ z3) = e2A(e34 + e56) − e46 + i(e36 + e45) define a SU(2) structure on the internal manifold. Moreover define dei = − 12 f ijkejk dA = dAiei F1 = F1iei We are looking for a sourceless solution in type IIB with a strict SU(2) structure internal manifold. That is to say that we have to solve for (1.12a)–(1.12c) and for the in order to constrain more the system and be sure to obtain a well-defined manifold at the end of the day. We also define the following set of variables: Ti = {f ijk, dAi, dφi, F1i, F3i, F5i, H3i} the variables in order to obtain fully linear (2.5) and fully quadratic (2.6) equations. We can claim we have a solution when we find a set of Ti that solves the aforementioned equations. Indeed, all the equations of motion are solved in this case (see for example [19] and references therein). Step 1: obtaining linear constraints The equations (1.12a)–(1.12c) are linear in the Ti’s and of the form: We can easily solve for them and thus eliminate some of the Ti’s. The rest of the equations are quadratic in the Ti’s and are of the form: E2i = X αijk(dT )jk + X βi jkTjTk = 0 linear in the (dT )ij’s so we can “solve” for them to simplify the system and obtain two sets of equations of the type: E3i = E4i = X αijk(dT )jk + X βi jkTjTk = 0 Maybe it can be better explained with an example. Assume the system E2i is composed keeping one of the two equations unchanged, and replace (dT )12 in the other one to obtain with all the (dT )ij’s and the other, E4i with only Ti’s. Step 3: simplifying the leftover quadratic equations We can still simplify a bit the system of equations E4i (2.8). Indeed, in general, all the equations are not independent and there exists a simple trick to easily get a minimal system. and by solving it, one obtains a minimal system in the (T T )ij’s. One just has to go back to the Ti’s to have simplified E4i. Moreover while solving for the (T T )ij’s, we can make it so that (T T )0j appears as much as possible. It will help us get simpler equations for step 4. Step 4: adding linear constraints The goal of this step is to obtain a linear constraint from the set of quadratic constraints to simplify the original problem. This is inspired by [24] where some linear conditions were hidden in the quadratic constraints and permitted to fully solve these equations. Having simplified the system in steps 2 and 3, some equations may immediately give such a linear constraint: If one is not in one of the case above, one has to make an assumption. The system can often give an hint on what is a sensible assumption or not. Indeed, some equations are simpler than other and help make a choice. But how can one find these simpler equations in a system which can be quite complicated? The answer is to look at the eigenvalues of are usually sufficiently simple to make sensible assumptions (see section 3.1 for an explicit Step 5: going back to step 1 We are now going back to step 1 with the additional linear constraints obtained in step 4. We are forced to do all the work again for the following reason. Assume you had for simplify sufficiently the quadratic constraints to spot the linear constraints hidden in them to be able to discover even more linear constraints. Thus we are going from step 1 to step 4 to step 1 again until one of the three following things happen: • The system has no solution: it means that one of the assumptions made in step 4 is wrong and should be discarded or that there is no solution within the ansatz one was • E4i (2.8) is empty then one can go to the final step. • E4i (2.8) is not empty but is sufficiently simple to be able to find a non linear solution of it. Then one can go to the final step. Final step: solving the last equations Ideally at this point both E3i and E4i defined in step 2 are empty but this is often not the case. Nevertheless, they are usually sufficiently simple to be solved by traditional methods. To sum up, the above steps take care of the linear parts of the equations and of some of the quadratic constraints by assuming some linear constraints. What is left are the differential and quadratic parts. An explicit example of this step will be given in section 3.1. Examples of new vacua In this section we give some examples of solutions found by the above method. One of them is already known as a Lu¨st-Tsimpis solution [3]. The other two, as far as the author knows, are two new vacua in type IIB. An example of Lu¨st-Tsimpis solution We will give an example of a Lu¨st-Tsimpis solution [3]. In this section we will also give a detailed account of how the method works in this particular case. First of all we assume that there is no vector or tensor in the torsion classes as they do in [3]. These are linear constraints in our variables Ti’s so can already be put in step 1. x3 being coordinates. This requirement is also linear and can be put in step 1. Finally, we will require that all the variables are functions of only x2 and x3. Part of this requirement (dT )ij = 0 for j 6= 2, 3 and appears in step 2. We run the algorithm from step 1 to step 3 and take a look at the resulting system = | | Then we rerun the algorithm from step 1. In step 4, we obtain only one equation in E4i namely: (f 425)2 + 4(dA2)2 − We rerun the algorithm from step 1 and find that E4i is empty. So we go to the final step and take a look at E3i (2.7). There are 4 equations in it (the projections on e2 and e3 5|μ|2 = 0. We are in the case where there is no obvious of the two following expressions): d(f 434) = 2(f 434)2 − f 434f 636 e d(f 636) = 5|μ|2 − 2(f 434)2 + 2f 434f 636 + (f 636)2 e3 x0) with x0 an integration constant. With this, E3i and E4i are empty which means we have successfully solved all the relevant equations. Let’s now give explicitly the results. We have: − f366e56 de2 = d(dx2) = 0 de5 = de3 = d(dx3) = 0 de6 = −f366e36 The fluxes, dilaton and warp factor being: F1 = 0 F3 = |μ|e−2A √5(−e135 + e146) − e234 − e256 H = |μ|e2A √5(e134 + e156) + e235 − e246 with f 636(x3) = and so of all the equations of motion. Moreover, there are no vectors and no tensors in the torsion classes. In order to understand more this solution, it is useful to give a coordinate expression of the metric or at least identify each part of the space. In that regard, one can take the following change of variables: e1 = e4 = √ 2f (x3)dx6 e2 = dx2 e5 = √ e3 = dx3 e6 = with f (x3) = √ explicitly the squashed Sasaki-Einstein of Lu¨st-Tsimpis [3], we now give the correspondence dtLT = e2 αLT = 6 |μ|2 sin(θLT)(e34 + e56) + cos(θLT)(e35 βLT = 6 |μ|2 cos(θLT)(e34 + e56) − sin(θLT)(e35 − e46) − e46) A new solution with constant dilaton Applying the method to more complex cases, we were able to identify two new vacua. Here we present the first one which has the particularity to have a constant dilaton. We will make an ansatz on the solution to make the method converge more rapidly (this ansatz de1 = 4dA3(x3) − dφ3(x3) e13 + 2|μ|(e36 + e45) de2 = 4dA3(x3) − dφ3(x3) e23 de3 = d(dx3) = 0 de4 = −f344(x3)e34 de5 = − dφ3(x3) + f344(x3) e35 − f263(x3)e23 − 2dA3(x3) − dφ3(x3) − f344(x3) e36 + 4dA3(x3) − 2dφ3(x3) − 2f344(x3) e45 with dA = dA3(x3)e3 and dφ = dφ3(x3)e3. Moreover, in order to verify the Bianchi system E3i (2.7) in this case): (dA3)′ = 5|μ|2 + 6(dA3)2 + dA3f344 + (dφ3)′ = 10dA3dφ3 − 2(dφ3)2 + dφ3f344 + (f263)2 (f263)′ = 4dA3f263 Unfortunately, the author hasn’t been able to solve these equations in all generality. But f344 = 4dA3 and f263 = 0): The fluxes, dilaton and warp factor being: de1 = 4dA3(x3)e13 + 2|μ|(e36 + e45) de2 = 4dA3(x3)e23 de3 = d(dx3) = 0 de4 = −4dA3(x3)e34 de5 = −4dA3(x3)e35 − 4dA3e45 F1 = 0 F3 = e2A 4dA3(x3)e125 + |μ|(−3e234 + 2e256) H = e2A 4dA3(x3)e124 + |μ|(3e235 + 2e246) dA3(x3) = | | tan 5|μ|(x3 − x0) A(x3) = − 10 We then put its expression in coordinates by the following change of variables: e2 = f (x3) 5 dx2 e3 = dx3 = dA3(x3)f (x3) 10 dx′3 e4 = f (x3) 5 dx4 e5 = f (x3) 5 dx5 e6 = 4dA3(x3)f (x3) 25 e˜1 + dA3(x3)f (x3) 10 1 with de˜1 = dx′3 ∧ dx6 + dx4 ∧ dx5 and f (x3) = 5|μ|2 +10dA3(x3)2 2 has not been able to obtain an explicit change of variables to go from x3 to x′3. . Unfortunately, the author A new solution with non constant dilaton The solution Another solution arose from the method described, one with non constant dilaton. Once again to make the method converge more rapidly one takes an ansatz (this ansatz has been After some iterations of the algorithm, one obtains: The fluxes, dilaton and warp factor being: de1 = dA3(x3)e13 + 2|μ|(e36 + e45) de2 = dA3(x3)e23 de3 = d(dx3) = 0 de4 = − dA3(x3)e35 − dA3(x3)e36 + − dA3(x3) F1 = e−φ |μ|e1 − 2dA3(x3)e6 H = 0 dA = dA3(x3)e3 = | | tan 2|μ|(x3 − x0) e3 A = − 4 log cos 2|μ|(x3 − x0) Once again a coordinate expression is useful. Do the following change of variables: e2 = sin(X) 4 dx2 e3 = e4 = cos(X)dx4 e5 = − sin(2|μ|x4) sin(X) 4 dx1 + cos(2|μ|x4) sin(X) 4 dx5 metric in the (x1, x2, X, x4, x5, x6) system of coordinates: gij =  √sin(X) sin(X) 2 =X→0 X 2 = |μ|3x˜3. This shows that this system can be mapped to a D5-D7 intersecting system which are delocalized in the {1, 2, 5, 6} directions. For example D5 along x1, x4 and D7 along x2, x4, x5, x6. Indeed, we are in the case of a system similar to (10) of [29] with only one transverse direcds2 = Then define x˜ = μX , the metric around 0 becomes: | | goes to 0. This shows that, a priori, this space is singular. Around 0, this metric doesn’t have the form of the D-brane metric so one has to better understand this singularity. In ds2 = √ ds(24) + (dx12 + dx22 + dx25 + dx26) + + √ One can see that there exists several isometric directions for this solution (at first sight dx1, direction. The resulting solution in IIA is: de1 = dA3(x3)e13 + 2|μ|(e36 + e45) de2 = dA3(x3)e23 de3 = d(dx3) = 0 de4 = − dA3(x3)e35 − dA3(x3)e36 + − dA3(x3) The fluxes, dilaton and warp factor being: F0 = F4 = H = 0 − 2dA3e15 + 2dA3e26 dA = dA3(x3)e3 = | | tan 2|μ|(x3 − x0) e3 A = − 4 log cos 2|μ|(x3 − x0) Note that the space the solution lives on is the same in both IIA and IIB. But in IIA, contrary to IIB, we have, the following SU(3) structure: z1 = eA(ie1 − e2) z2 = eA(e3 + ie4) z3 = eA(ie5 − e6) z = z1 j = J = z ∧ z¯ + j − = − 8 e−iJ (z2 ∧ z2 + z3 ∧ z3) = e2A(e34 + e56) − e35 + e46 + i(−e36 − e45) Conclusion and outlooks. In this paper, we managed to identify two new vacua in type IIB which are explicit. It is a step forward in identifying the web of vacua in type II. We also have been able to discover a new IIA solution by applying T-duality along an isometric direction on one of the solutions. One caveat should be pointed out: these solutions are indeed sourceless if the space is smooth which is not guaranteed by the analysis. Indeed, one could find localized sources (or partially localized sources as we did for the second example) but it is not in the scope of this paper. To obtain these new vacua, we devised a semi-algorithmical method which can be applied to lots of other similar situations. Indeed, one can apply it to type IIA to discover new vacua (and we should be able to easily recover the one we found here), or to type IIB with dynamic SU(2) structure instead of the strict SU(2) structure we restricted to in this paper. More generaly, one can apply it to all problems with a linear part and a quadratic/differential part of the type (2.5), (2.6). In that respect, one can see this paper as a proof of concept for the method. There is also lots of room for improvement for the method depending on which problems one applies it to. Indeed, in this paper we restricted to having only one parameter tions of variables to be non zero. Then one can modify step 3 and step 4 to take that into account and be provided with even more linear constraints. Another improvement concerns the automatization. In step 4, it is quite common to have constraints of the type instead of just choosing one. 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Gautier Solard. A method to find \( \mathcal{N} \) = 1 AdS4 vacua in type IIB, Journal of High Energy Physics, 2017, 42, DOI: 10.1007/JHEP01(2017)042