Old and new vacua of 5D maximal supergravity
QQMUL-PH-20-34
Old and new vacua of 5D maximal supergravity
G. Dall’Agata , , G. Inverso , and D. Partipilo , Dipartimento di Fisica e Astronomia “Galileo Galilei”Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy INFN, Sezione di PadovaVia Marzolo 8, 35131 Padova, Italy Centre for Research in String Theory,School of Physics and Astronomy, Queen Mary University of London,327 Mile End Road, London, E1 4NS, United Kingdom
ABSTRACTWe look for critical points with U(2) residual symmetry in 5-dimensional maximallysupersymmetric gauged supergravity, by varying the embedding tensor, rather than di-rectly minimizing the scalar potential. We recover all previously known vacua and wefind four new vacua, with different gauge groups and cosmological constants. We providethe first example of a maximal supergravity model in D ≥ a r X i v : . [ h e p - t h ] J a n Introduction
Charting and analyzing vacua of supergravity theories is a fundamental task to findwhich models can be related to string theory as well as to understand supersymmetrybreaking, the possible mechanisms to generate critical points with a positive value of thecosmological constant and which supergravities lead to Anti-de Sitter (AdS) vacua withan interesting holographic dual. Among all possible theories, the maximally supersym-metric ones stand out for their fixed matter content and the limited number of possibledeformations. For these reasons there has been an active interest in their gaugings andin the analysis of the resulting scalar potentials to understand their critical points, witha special emphasis on the theories obtained by reducing string or M-theory on spheres,which give models with vacua dual to maximally supersymmetric Conformal Field The-ories (CFT).The main challenges one faces when dealing with this problem are associated to thevery complicated structure of the scalar potential, a function of 70 or 42 scalars in themaximal theory in 4 and 5 dimensions respectively, which also depends on a large num-ber of parameters (912 and 351 respectively) that fix the structure of the gaugings andtherefore of the full lagrangian, according to the rules specified in [1, 2]. Clearly such alarge space of parameters makes the search for critical points complicated and attemptsat a general classification extremely difficult. However, there has been some interestingprogress in the last few years that expanded a lot our knowledge of this particular aspectof maximal supergravity theories.There are mainly three techniques that have been used so far to find and analyzecritical points of (maximal) supergravity theories. The first one relies on using symmetriesto consistently truncate a particular theory to a subset of fields containing a limitednumber of scalars and then extremising the resulting simplified potential. Pioneeredin [3, 4], this technique allowed for the first and only analytic results for the maximaltheories from the ’80s until recent years. For what concerns maximal supergravity in 5dimensions, this technique allowed the discovery of 5 vacua [5, 6, 7, 8] of the SO(6) andSO(3,3) gauged models in addition to the maximally symmetric one in [9], though oftenonly partial results were available on the spectrum about these vacua.More recently, a new numerical approach, based on Machine Learning software li-braries was developed and employed in a series of papers [10, 11, 12, 13, 14, 15, 16, 17]where many new vacua of the maximal supergravities in 4 and 5 dimensions had beenfound. This also allowed to find precise information about the spectrum of scalar fluc-tuations, residual gauge groups and residual supersymmetry. In particular, 27 new AdSvacua were found in the SO(6) maximal supergravity in 5 dimensions, with a detailedanalysis in [15, 16].While these approaches are very interesting and gave promising results, so far they1ave only been used to produce critical points for a fixed scalar potential, which is result-ing from a single specific gauging within the large infinite family of possible deformations.This leaves open the possibility that other vacua with the same residual symmetries ap-pear in different gaugings. The approach we are going to use in this work uses instead thepower of the embedding tensor formalism in a way that allows for the search of criticalpoints independently from the choice of gauging. This approach was pioneered in a verydifferent context in [18] and used in the context of maximal 4-dimensional supergravityin [19, 20, 21, 22, 23, 24, 25], as well as in half-maximal supergravity in four and threedimensions [26, 27]. In addition to the power of investigating in a single sweep all de-formations of maximal supergravity, this approach has so far produced analytic resultsfor the critical points and their full spectrum, also providing information on the gauging,the residual gauge symmetry and supersymmetry of the vacua. Moreover, for Minkowskivacua this led to understanding the moduli space of these theories [24] as well as theiruplift to string theory [28]. Finally, since the vacua are obtained without specifying firstthe gauging, this means that we can exhaustively classify vacua with a given residualsymmetry for all possible consistent gaugings.In this work we apply this last technique by investigating critical points of maximalgauged supergravities in 5 dimensions with a residual U(2) symmetry. We recover allpreviously known vacua and we find four new ones, with different gauge groups andcosmological constants. We also provide analytic results for their full mass spectra, thuscompleting partial results for old vacua as well as fully analyzing new ones. We did notfind new AdS vacua, so that the only such vacua with U(2) symmetry are those appearingin the maximal supergravity with SO(6) gauge group, but we have new Minkowski and deSitter vacua. A particularly interesting result is that two of the vacua appear in the sametheory with SO ∗ (6) =SU(3,1) gauge group, providing the first example n D ≥ .In what follows, after some summary of the main ingredients of maximal supergravityin 5 dimensions, we will discuss in some detail our technique in section 3 and then proceedwith a detailed analysis of the U(2) invariant sector in section 4. We tried to summarizeall our results in tables that could be easily consulted and used for future reference. A comprehensive presentation of the 5-dimensional N = 8 supergravity lagrangian, su-persymmetry rules and of all the details regarding the gauging procedure can be foundin [1], which we are going to use as a basis for our analysis. In order to facilitate reading, There is a similar instance in maximal supergravity in 3 dimensions [29]
2e collected in this section the main formulas and properties of the tensors relevant forour work.Any gauging of 5-dimensional maximal supergravity is specified by the choice of theembedding tensor Θ
M α , which selects the generators t α of the duality algebra e associ-ated to the vector fields A Mµ gauging the corresponding group. Once this tensor is fixed,everything else in the lagrangian and supersymmetry transformations follows, accordingto the analysis in [1]. The embedding tensor lives in the product representation × of E , but it is constrained by supersymmetry and gauge-invariance to the represen-tation , which is then further constrained by consistency conditions quadratic in theembedding tensor.These parameters can also be codified in a different set of tensors that explicitlytransform under the maximal compact subgroup of E , namely USp(8). Since we areinterested in the critical points of the scalar potential, we will in fact make extensive useof the fermion shifts A ij = A ji = Ω ik Ω jl A kl = ( A ij ) ∗ (2.1)and A i,jkl = A i, [ jkl ] = Ω im Ω jn Ω kp Ω lq A m,npq = ( A i,jkl ) ∗ , (2.2)satisfying A i,jkl Ω kl = 0 = A [ i,jkl ] = Ω ij A i,jkl . (2.3)These tensors are expressed in terms of the USp(8) indices i, j, . . . = 1 , . . . ,
8, which arecarried by the fermion fields and by the supersymmetry parameters (cid:15) i (Ω is the USp(8)symplectic-invariant form). In fact A ij is in the representation and A i,jkl in the of USp(8), which are precisely the representations under which the of E breaks.They are called fermion shifts because they appear in the supersymmetry transformationsof the fermion fields and are non-vanishing only when there is a non-trivial gauging, henceshifting the ungauged expression.The relation between the fermion shifts and the embedding tensor is expressed viathe T-tensors: T klmnij = 4 A q, [ klm δ n ][ i Ω j ] q + 3 A p,q [ kl Ω mn ] Ω p [ i Ω j ] q , (2.4) T ijkl = − Ω im A m,j ) kl − Ω im (cid:16) Ω m [ k A l ] j + Ω j [ k A l ] m + 14 Ω kl A mj (cid:17) , (2.5) Z ij,kl = Ω [ i [ k A l ] j ] + A i,j ] kl , (2.6)and can be written as X MN P = Θ
M α ( t α ) N P = V mnM V N kl V Pij (cid:2) δ ik T jlmn + T ijpqmn Ω pk Ω ql (cid:3) , (2.7)where V ijM are the coset representatives of the E / USp(8) scalar manifold, satisfying V ijM Ω ij = 0, and V Mij are their inverse V ijM V Nij = δ NM .3nce we use the T-tensor, the quadratic constraints on the embedding tensor have arather simple expression: T ijkl Z kl,mn = 0 = T ijklmn Z mn,pq . (2.8)Let us now come to the center of our analysis: the scalar potential and the massmatrices. While everything can be defined in terms of the fermion shifts, for the scalarmasses we preferred to use a convenient expression which is valid only at the selectedpoint of the scalar manifold we use as a basis for our analysis. As we will see we are notgoing to lose generality by this assumption.Following a well-known general rule of gauged supergravity theories, the scalar po-tential is the square of the fermion shifts: V = 3 A ij A ij − A i,jkl A i,jkl . (2.9)We are looking for maximally-symmetric vacua, where all fields are vanishing exceptfor the scalar fields, which could have a constant vacuum expectation value and for themetric, which either describes a de Sitter, Minkowski of anti-de Sitter spacetime. Thescalar equations of motion are solved by the critical point condition U ijkl −
32 Ω [ ij U kl ] pq Ω pq + 18 ( U mnpq Ω mn Ω pq ) Ω [ ij Ω kl ] = 0 , (2.10)where the tensor U ijkl is U ijkl = 43 A mq A m, [ ijk Ω l ] q + 2 A m,npq A n,m [ ij Ω | p | k Ω l ] q . (2.11)Once we find a critical point, we derive the masses of the various fields by computingthe eigenvalues of the respective mass matrices. For what concerns the gravitini ψ iµ , themass matrix is directly proportional to the A ij shift matrix M (3 / ij = 32 A ij . (2.12)The masses of the other fermions χ ijk are then fixed by the eigenvalues of (indices ijk and pqr are fully antisymmetrized) M (1 / ijk,pqr = 8 A [ i,j [ pq Ω r ] k ] + 2 A [ i [ p Ω qj Ω k ] r ] − A l, ijk A lm ( A m,stu A n,stu ) − A n,pqr . (2.13)This mass matrix is the result of subtracting from the lagrangian mass the appropriateterm to remove the goldstinos from the spectrum for susy-breaking vacua. It is understoodthat in case of a degenerate matrix A m,stu A n,stu , we only compute the inverse for its non-degenerate part, as this is the part related to the goldstino directions, which in the originallagrangian mix the gravitinos and the spin-1/2 fields. The proof that such additional termcorrectly produces M / ijk,pqr A s,pqr = 0 follows once one takes into account the equations of4otion (2.10) and one uses repeatedly the quadratic constraints (2.8). In particular thematrix we are inverting is related to the shift of the gravitinos by means of the quadraticidentity known as supersymmetric ward identity13 A j,stu A i,stu = 18 δ ij V + 3 A ip A pj , (2.14)which also tells us that the expression is explicitly dependent on the value of the cos-mological constant at the vacuum. This expression generalizes previous similar formulaefor maximal theories in 4 dimensions, which were obtained in particular instances wherethe cosmological constant was vanishing [30] or when the squared shifts had already beendiagonalized [25]. A simple way to understand this expression can also be obtained bycomparing it with the analogous expression for N = 1 supergravity presented in [31].Also the masses of the bosonic degrees of freedom can be expressed in terms of thesame tensors. The vector mass matrix is M ( v ) M N = 13 V ijM T mnpqij T mnpqkl V Nkl , (2.15)while the squared masses of the tensor fields follow from the eigenvalues of the matrix M ( t ) M N = V ijM Z ij mn Z mn kl V Nkl . (2.16)These mass matrices are clearly redundant, because the sum of vector and tensor fieldspresent in the theory is fixed, given that the tensor fields appear by dualization of thevector fields. This means that both M ( v ) and M ( t ) are degenerate and contain zeros inthe directions where the fields have been dualized.All the above expressions have general validity and should be evaluated at the criticalpoints satisfying (2.10). For the scalar fields, on the other hand, following [19] we providean expression that is valid only when the critical point is the base-point of the manifold,i.e. when all scalars are vanishing. While this could seem a restriction, as we will explainin the next section, it allows us to obtain the full spectrum for any critical point in anyarbitrary gauging. This is given in terms of the embedding tensor, the e generators,the e structure constants f αβγ and the e Cartan–Killing metric η αβ : M αβ = 165 (cid:0) Θ M σ ( t α t β ) M N Θ N γ ( δ γσ + 5 η σγ ) + Θ M σ ( t α ) M N Θ N γ f βγσ +Θ M σ ( t β ) M N Θ N γ f αγσ + Θ M σ Θ M γ f αγδ f βδσ (cid:1) . (2.17)The matrix is non-zero only in the non-compact directions, i.e. along the generators t α ∈ e \ usp (8). Moreover all goldstone fields appear with a zero eigenvalue. The procedure used to find and analyze the scalar potential has been developed in thecase of maximal supergravities in [19], developing on an old idea presented in a very5ifferent context [18]. The main point is that the scalar potential is a function of thescalar fields via the coset representatives V ijM and the embedding tensor Θ M α V ( φ ) = V ( V ( φ ) , Θ) . (3.1)As explained above, vacua of the theory follow as solutions of the minimization condition(2.10). This is generally a rather complicated expression of the scalar fields (at bestratios of polynomials and exponentials of the scalar fields). This is the reason why thetask of finding solutions to such complicated system of equations has always been verychallenging and researchers usually focussed on restricted sets of scalar fields in order tosimplify the task, which anyway is often performed only numerically.The alternative proposed in [19] maps the problem to a coupled set of second and firstorder algebraic conditions on the gauging parameters. This is possible because the scalarmanifold is homogeneous and therefore each point on the manifold can be mapped to anyother by an E transformation and at the same time the scalar potential is invariant under the simultaneous action of these trasformations on both the coset representativesand on the embedding tensor. This implies that we can always map any critical pointof the scalar potential to the “origin” at φ = 0. At such point, the scalar potential is asimple quadratic function of the embedding tensor V = 215 Θ M α Θ M β ( δ αβ + 5 η αβ ) (3.2)and the minimization conditions become quadratic conditions on the embedding tensor,which should be solved together with the quadratic constraints (2.8). The result is thatrather than fixing the gauging and then performing a scan of all possible critical pointsof the scalar potential and then scan among all possible gaugings, one can simply solve aset of quadratic conditions on the embedding tensor and then read the resulting valuesof Θ that fix at the same time the gauge group, the value of the cosmological constantand the masses at the critical point. Clearly any choice of point on the scalar manifoldis equivalent, but choosing φ = 0 has the advantage that it is a fixed point under theaction of the maximal compact subgroup of the isometries, namely USp(8), and thereforewe can consider modifications of the embedding tensor related only to the non-compacttransformations, so that that there is a one-to-one correspondence between the parametersin Θ related to the scalar fields and the independent directions on the scalar manifold.We advise the reader to consult [19] for more details.As we mentioned in the introduction, all our results are fully analytic. The reason weare able to produce such results is related to the procedure we used to solve the quadraticconditions coming from the minimization of the scalar potential and from the quadraticconstraints. While in fact we reduced our problem to a set of quadratic equations, we stillhave generically a very large number of parameters and quadratic equations. This implies6hat not always one can see a straightforward analytic solution, because the equationsare coupled and they could become very high in order in terms of a single variable.We mainly used two techniques. The first one is based on a simplification of the setof quadratic equations by employing a choice of a more convenient Gr¨obner basis for thepolynomial generating the same solutions. This has been done with the aid of the com-puter algebra system for polynomial computations SINGULAR [32]. Unfortunately whenthe number of variables is very large, this can be extremely costly in time and thereforeone has to resort to a different way of reducing the set of equations. We found a veryeffective procedure by borrowing an algorithm developed in the context of cryptographywhere the solution of quadratic equations on finite fields is a common problem. In par-ticular we used the so-called XL algorithm [33], or extended linearization. The idea israther simple. Rather than solving directly the given set of quadratric equations, oneproduces sets of linear equations in the monomials appearing in the equations and in allequations obtained by multiplying the original set of equations by the variables and bytheir products up to a fixed order. This produces sets of linear equations that can besolved rapidly and, once interpreted in terms of the original variables, they may reduceto equations in a single variable or in simpler sets of polynomial equations (like equalitybetween different monomials). This allows to fix and eliminate some of the variables fromthe problem and then face a simpler set of equations, which could be solved directly orfurther simplified by another iteration of the same procedure, or by a more convenientchoice of Gr¨obner basis.
In this work we decided to scan gauged maximal supergravity in 5 dimensions for vacuawith a residual U(2) symmetry. Asking for a residual U(2) invariance of the vacuum(with respect to a gauged or global symmetry) imposes restrictions on the allowed co-efficients of the embedding tensor and consequently of the fermion shift tensors, whichshould be singlets with respect to this residual symmetry. To perform a full analysis,we therefore looked at all the inequivalent embeddings of SU(2) in USp(8) and then sin-gled out all possible inequivalent charge assignments for the remaining U(1)s, if any. Wethen performed the branching of the and representations of USp(8) specifyingthe fermion shifts with respect to the chosen embedding and classified all inequivalentcases. When the commutant of the residual symmetry group in USp(8) was non-trivial weused the commuting symmetries to further reduce the number of inequivalent variablesby removing those that could be generated by the action of the commutant. Once thenon-vanishing components of the fermion shifts had been identified we then proceededto solve the set of quadratic algebraic conditions coming from the scalar equations ofmotion (2.10) and the quadratic constraints (2.8) and then collected all solutions, which7ay still be related by duality transformations. Finally, we analyzed their properties andcomputed their mass spectrum as we will discuss momentarily. In the summary tables wecollected all inequivalent vacua and reported the most general mass spectra for each ofthem. Unfortunately, two of the branchings still present a very large number of singlets( ≥
48) and even combining all the techniques mentioned above we have not been able tofully scan and solve their equations for all the allowed parameters, though for all solutionswe recovered the same vacua we found in other branchings.As a first step we list the branchings we analyzed by the inequivalent decompositionsof the -dimensional representation of USp(8) under SU(2) and then give one of thebranching routes leading to this decomposition. For each case we also give a table withthe subcases based on possible different choices of the U(1) factor, when present. We alsolist the number of singlets in the fermion shifts, which are going to be the variables to befixed by the quadratic conditions in order to find vacua. We find 13 different branchings of the fundamental representation of USp(8) under SU(2),which we therefore analyze separately. The labels on the various factors are self explana-tory: we use letters from the beginning of the alphabet to keep track of the various factorsin the decompositions and we use S and diag to specify the symmetric and diagonal em-bedding of the group. Case 1: 8 → The branching path is USp(8) → SU(2) S . (4.1)This case leaves no singlets to discuss, so no vacua are possible for this choice. Case 2: 8 → + The branching path isUSp(8) → SU(2) A × USp(6) → SU(2) A × SU(2) S → SU(2) diag (4.2)There is only one singlet in A i,jkl . Case 3: 8 → + + The branching path isUSp(8) → SU(2) A × USp(6) → A × SU(2) S (4.3)There are 3 singlets in A ij and no singlets in A i,jkl .8 ase 4: 8 → + The branching path isUSp(8) → USp(4) → USp(4) diag → SU (2) S (4.4)We have one singlet in A ij and 3 singlets in A i,jkl . Case 5: 8 → + + The branching path isUSp(8) → USp(4) A × USp(4) B → [SU(2) × U(1)] A × SU(2) S → SU(2) diag (4.5)We find just one singlet in A ij and 8 in A i,jkl . Case 6: 8 → + + + The branching path isUSp(8) → USp(4) × USp(4) → [SU(2) A × SU(2) B ] × SU(2) S → SU(2) B + S (4.6)There are 3 singlets in A ij and 5 singlets in A i,jkl . Case 7: 8 → + 4 · The branching path isUSp(8) → USp(4) A × USp(4) B → [SU(2) S ] A × [SU(2) × U(1)] B → SU(2) S × U(1) A × U(1) B (4.7)In this case we have two inequivalent choices of U(1) ⊂ U(1) A × U(1) B , which we listin the table 1. charges
36 315 decomposition choice singlets singlets7 → + ± ± ( q A , q B )7a → + 2 · ± q A → + ± + 2 · q A + q B Case 8: 8 → · + The branching path isUSp(8) → SU(2) A × USp(6) → SU(2) A × [SU(3) × U(1)] B → SU(2) A × SO(3) B → SU(2) diag (4.8)The decomposition contains 3 singlets for A ij and 6 singlets for A i,jkl .9 ase 9: 8 → · + 2 · The branching path isUSp(8) → SU(4) × U(1) B → SU(3) × U(1) A × U(1) B → SU(2) s × U(1) A × U(1) B . (4.9)This case has already 21 SU(2) singlets overall, therefore we distinguish various subcasesaccording to the choices of a U(1) factor, which we report in the table 2. charges
36 315 decomposition choice singlets singlets9 → + − − + − + − ( q A , q B ) 2 19a → ± + ± q A → ± + ± q B , q A + q B → · + ± q A − q B → ± + 2 · q A +3 q B Case 10: 8 → · The branching path isUSp(8) → SU(2) A × SU(2) B × SU(2) C → SU(2) C × U(1) A × U(1) B (4.10)This case has 51 singlets of SU(2) and therefore we classify various subcases according toa remaining U(2) symmetry. We collect all different branchings in table 3. charges
36 315 decomposition choice singlets singlets10 → ± ± ( q A , q B ) 2 310a → · [ ± ] q A → · [ ] + 2 ± q A + q B → [ ± ] + [ ± ] 2 q A + q B ase 11: 8 → · + 2 · The branching path isUSp(8) → SU(2) × USp(6) → SU(2) × [SU(3) × U(1)] → SU(2) × [SU(2) × U(1) × U(1)] → SU(2) diag × U(1) A × U(1) B (4.11)This case has 39 singlets of SU(2) and therefore we classify various subcases according toa remaining U(2) symmetry. Results are collected in table 4. charges
36 315 decomposition choice singlets singlets11 → + + − − + − + − ( q A , q B ) 2 511a → ± + + 1 ± q A → ± + + 1 ± q B → ± + + 1 ± q A + q B → · + 1 ± q A − q B → ± + + 2 · q A +2 q B Case 12: 8 → · + 4 · The branching path isUSp(8) → USp(4) A × USp(4) B → [SU(2) × U(1)] A × [SU(2) × SU(2)] B → SU(2) A × U(1) A × U(1) B × U(1) C (4.12)This case has 64 singlets of SU(2) and therefore we classify various subcases according toa remaining U(2) symmetry, which we list in table 5.11 charges
36 315 decomposition choice singlets singlets12 → ± + ± + ± ( q A , q B , q C ) 3 512a → ± + 4 · q A
11 2112b → · + ± + 2 · q B → ± + 2 · ± q A + q B + q C → ± + ± + 2 · q A + q B → · [ + ± ] q B + q C → ± + ± + 2 · q A + 2 q B Case 13: 8 → + 6 · The branching path isUSp(8) → SU(2) × USp(6) → SU(2) × [SU(2) A × USp(4)] → SU(2) × [SU(2) A × SU(2) B × SU(2) C ] → SU(2) × U(1) A × U(1) B × U(1) C (4.13)This case has 124 singlets of SU(2) and therefore we classify various subcases accordingto a remaining U(2) symmetry. Note that cases 13e and 13f have only a subset of thesinglets present in the other cases, so it is enough to solve cases 13a–13d. Results arepresented in table 6.The branchings 13a and 13d present more than 48 singlets and this hampered thesimplification of the problem with any of the techniques used in this work in a reasonableamount of time. Anyway, all solutions we have been able to find for these branchingswere already present in one of the other branchings.12 charges
36 315 decomposition choice singlets singlets13 → + ± + ± + ± ( q A , q B , q C ) 3 913a* → + ± + 4 · q A
11 3713b → + 2 · [ + ± ] q A + q B → + ± + ± + 2 · q A + q B → + 3 · ± q A + q B + q C → + 2 · ± + ± q A + q B + 2 q C → + ± + 2 · ± q A + 2 q B + 2 q C The search for vacua has been carried out by solving the sets of quadratic equations forthe singlets in the tables above. Once we found solutions, we checked for each candidatevacuum the rank of the embedding tensor, the signature of the resulting Cartan–Killingmatrix and the full mass spectrum. Overall we found 5 different Anti-de Sitter vacua, 5Minkowski vacua and 2 de Sitter vacua. The vacua with negative cosmological constantare all pertaining to the same gauging, namely the maximal SO(6) theory of [9], and wereall already known [6, 7, 8]. Among the Minkowski vacua there are the Cremmer–Scherk–Schwarz gaugings [34, 35] with various mass parameters and a supersymmetric vacuumfor the SO ∗ (6) theory discovered in [36], but we also find three new vacua with a non-abelian gauge group (like those in [24] for the analogous analysis of maximal supergravityin 4 dimensions). Finally, we also find 2 de Sitter vacua, resulting from gauging ofthe semisimple groups SO(3,3) [5] and SO ∗ (6), the latter being new. All the vacuaare reported in the table 7, together with the number of supersymmetry they preserve,the original gauging, the residual gauge group and the reference where they were firstdiscovered. In the appendix we provide for each vacuum one instance of fermion shiftvalues reproducing the critical point mentioned in the table.13acuum susy G gauge G res ref. branchingA1 8 SO(6) SO(6) [9, 8] 4,9,10,12A2 0 SO(6) SO(5) [6, 7, 8] 4, 10, 12 cef A3 0 SO(6) SU(3) [6, 7, 8] 9 a , 12 ef A4 2 SO(6) SU(2) × U(1) [8] 12 cef
A5 0 SO(6) SU(2) × U(1) × U(1) [8] 12 be M1 0,2,4,6 U(1) (cid:110) R U(1) [35] 5,6,7,1011,12,13M2 2 SO ∗ (6)=SU(3,1) SU(3) × U(1) [36] 8,9,1112 abcef , 13 b M3 4 SO ∗ (4) (cid:110) R U(2) here 12 abcef
11, 13 bc M4 0 [SO(3,1) × SO(2,1)] (cid:110) R U(2) here 10 b M5 4 SO*(4) (cid:110) R SO(3) here 12 be D1 0 SO(3,3) SO(3) [5] 9 b ,10 ab D2 0 SO ∗ (6)=SU(3,1) SU(2) here 9 b Table 7: Summary of vacua found in this work.Given the nature of the gaugings generating such vacua, we can also see how someof these could be obtained from string theory reductions. All AdS vacua appear in theSO(6) theory, which is a consistent truncation of type IIB supergravity compactified on S [39]. A subset of the CSS gaugings and their vacua M1 are known to be the result ofa twisted torus reduction [34], while the most general gauging and vacuum in this classmay admit an uplift through a generalised Scherk–Schwarz Ansatz analogous to the onesdescribed for four-dimensional CSS gaugings in [28].It is interesting to notice that for the first time in a maximal theory in D ≥ ∗ (6) =SU(3,1) gauging, that contains at the same time aMinkowski and a de Sitter vacuum. Our claim that they reside in the same model followsboth from the analysis of the embedding tensors that generate them, and the directidentification of a truncated scalar potential for the SU(3,1) theory where both vacua areeasily found. From the embedding tensors we find which generators of e are involved in14heir corresponding model and analyzing the commutants we find in both cases that therepresentation decomposes in the representations + + of the gauge group. Thiscorresponds to the correct branching under SU(3,1) and since the adjoint is unique in thebranching, we argue that the gaugings are the same. Moreover, if we directly decomposethe representation of e from the branching above for the we see that there isa unique singlet with respect to SU(3,1) and therefore there is a unique possible form ofembedding tensor leading to this gauging up to duality transformations.Figure 1: Scalar potential for the two common scalars invariant under the residualsymmetries of the vacua (M2) and (D2). We see a Minkowski vacuum at the center ofthe picture, surrounded by a family of de Sitter vacua, with a massless modulus.Actually, for this specific model we can provide a truncated scalar potential, where wemake explicit the dependence on the two scalar fields that are singlets of both symmetrygroups. Furthermore, both vacua arise as different solutions of the 9 b case and thecommutator of the residual U(2) group with the non-compact generators of e leaves onlytwo generators g and g , for which we can provide a truncated scalar potential whereboth vacua can be found. We construct the coset representative L ( x, y ) = exp ( g x + g y ) , (4.14)which induces the scalar potential V = − (cid:0) −
16 cosh(2 x ) cosh(2 y ) + 4 cosh (2 x ) cosh (2 y ) (cid:1) , (4.15)where x and y are canonically normalized scalar fields. The scalar potential has two vacua,a Minkowski one at x = y = 0 and a line of unstable de Sitter vacua at cosh(2 x ) cosh(2 y ) =2. At any point in the family of de Sitter vacua we see that the masses of the twofluctuations are indeed zero and m / Λ = −
24. These coincide with one of the moduli15nd one of the unstable directions of the full scalar spectrum about the de Sitter vacuum(see table 14).A similar discussion could apply to the vacua (M3) and (M5). They both have thesame gauge group, though in this case they do not belong to the same model. In fact,there are 4 U(2) invariant scalar fields in both models, but the scalar potentials show onlya single vacua in each of the potentials constructed from (M3) and (M5) by introducingthe appropriate coset representatives. For instance, using canonically normalized fields,the potential of (M3) is V = x − √ x − √ x x (cid:16) x √ − (cid:17) (cid:20) − m m x √ x (cid:0) x − (cid:1) (cid:16) x √ + 2 x √ + 3 (cid:17) (cid:0) x − (cid:1) + m x √ (cid:18)(cid:16) x √ + 2 x √ + 3 (cid:17) (cid:0) x + 1 (cid:1) (cid:0) x (cid:1) + 4 (cid:16) x √ − (cid:17) x x (cid:19)(cid:16)(cid:16) x √ + 2 x √ + 3 (cid:17) (cid:0) x + 1 (cid:1) (cid:0) x (cid:1) − (cid:16) x √ + 6 x √ + 1 (cid:17) x x (cid:17) + m (cid:18)(cid:16) x √ + 2 x √ + 3 (cid:17) (cid:0) x + 1 (cid:1) (cid:0) x (cid:1) − (cid:16) x √ − (cid:17) x x (cid:19)(cid:16)(cid:16) x √ + 2 x √ + 3 (cid:17) (cid:0) x + 1 (cid:1) (cid:0) x (cid:1) + 4 (cid:16) x √ + 6 x √ + 1 (cid:17) x x (cid:17)(cid:105) . (4.16)This shows a single critical point at x i = 1, where the scalars x , , are moduli, while thescalar x is massive with mass m . Actually x is a modulus that simply rescales themass parameters. While the gauge group is the same, the two vacua indeed pertain totwo different gaugings. This is possible because the decomposition of the of E under SO*(4) shows 6 singlets and therefore one could find inequivalent embeddings ofthe same gauge group. In this final section we present the mass spectra of all the vacua listed in the previous table.The masses for backgrounds with non-vanishing cosmological constant are normalized interms of the (A)dS radius squared L = | /V | , so that supersymmetric gravitinos havea normalized squared mass of 9 / m / (cid:2) (cid:3) × L m vec [0] × L m tens [1] × L m / (cid:2) (cid:3) × , (cid:2) (cid:3) × L m scal [ − × , [ − × , [0] × Table 8: Masses for the AdS vacuum A1 L m / (cid:2) (cid:3) × L m vec [0] × , (cid:2) (cid:3) × L m tens (cid:2) (cid:3) × , [6] L m / [0] × , (cid:2) (cid:3) × , (cid:2) (cid:3) × L m scal (cid:2) − (cid:3) × , [ − × , [0] × , [8] × Table 9: Masses for the AdS vacuum A2 L m / (cid:2) (cid:3) × , (cid:2) (cid:3) × L m vec [0] × , (cid:2) (cid:3) × , [8] × L m tens (cid:2) (cid:3) × , (cid:2) (cid:3) × L m / [0] × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × L m scal (cid:2) − (cid:3) × , (cid:2) − (cid:3) × , [0] × , [8] × Table 10: Masses for the AdS vacuum A3The spectrum of the vacuum (A4) is particularly interesting in the context of theAdS/CFT correspondence, as it fixes the anomalous dimensions of the operators of thecorresponding N=1 deformation of super-Yang–Mills in 4 dimensions [37]. L m / (cid:2) (cid:3) , [4] , (cid:2) (cid:3) × L m vec [0] , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , [6] × L m tens (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , L m / (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , [1] × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , [4] × , [0] × , (cid:2) ± √ (cid:3) × L m scal [0] × , [ − , (cid:2) − (cid:3) × , (cid:2) − (cid:3) × , [ − × , (cid:2) − (cid:3) × , [3] × , [4 ± √ × Table 11: Masses for the AdS vacuum A417 m / (cid:2) (cid:3) × , (cid:2) (cid:3) × L m vec [0] × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × L m tens (cid:2) (cid:3) × , [4] × , (cid:2) (cid:3) × L m / [0] × , (cid:104) ± (cid:113) (cid:105) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , [ ± √ × L m scal (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) − (cid:3) × , [ − × , (cid:2) − (cid:3) × , (cid:2) − (cid:3) × , [0] × Table 12: Masses for the AdS vacuum A5.The full spectra of the de Sitter vacua (D1) and (D2) are new and show that suchvacua are unstable with very large instabilities, of the order of the cosmological constant,or larger. L m / [0] × L m vec [0] × , [8] × L m tens [2] × L m / [0] × , [8] × L m scal [ − × , [ − × ,[0] × , [10] × , [16] × Table 13: Masses for the dS vacuum D1. L m / (cid:2) (cid:3) × , (cid:2) (cid:3) × L m vec [0] × , [24] × , [96] × L m tens [32] × , [56] × L m / [0] × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × , (cid:2) (cid:3) × L m scal [ − × , [0] × ,[4(29 ± √ × ,[40] × , [112] × , [120] × Table 14: Masses for the dS vacuum D2.For what concerns the Minkowski vacua, since there is no intrinsic scale associated tothe vacuum, we parametrized all masses in terms of the ones of the gravitini. We easilyreproduce the expected spectrum for the CSS vacua, while the results for all the othervacua are new.By also looking at the fermion shifts collected in the appendix, is interesting to noticethat all the vacua we found show spectra that do not depend on additional parametersexcept for a few masses (or the cosmological constant, if different from zero). Thismeans that for all the gaugings considered the vacua appear in a unique theory with thatgauge group and there are no continuous families of models with the same gauge groupcontaining such vacua. This differs from what was discovered in the 4-dimensional case1820, 38], where it was found that one can have infinite families of gaugings with the samegauge group and vacua whose existence and whose value of the cosmological constantmay depend on the parameter specifying the family of gaugings. m / [ m ] × , [ m ] × , [ m ] × , [ m ] × m vec [0] × , [( m ± m ) ] × , [( m ± m ) ] × , [( m ± m ) ] × , [( m ± m ) ] × m tens [0] × , [( m ± m ) ] × , [( m ± m ) ] × m / [0] × , [ m i ] × , [( m ± m ± m ) ] × , [( m ± m ± m ) ] × , [( m ± m ± m ) ] × , [( m ± m ± m ) ] × m scal [0] × , [( m ± m ) ] × , [( m ± m ) ] × , [( m ± m ± m ± m ) ] × Table 15: Masses for the CSS vacuum M1. m / [0] , [ m ] , [ m ] , [ m ] m vec [0] , [( m ± m ) ] , [( m ± m ) ] , [( m ± m ) ] ,m tens [ m ] , [ m ] , [ m ] ,m / [0] , [ m ] , [ m ] , [ m ] , [( m ± m ) ] , [( m ± m ) ] , [( m ± m ) ] , [( m ± m ± m ) ] m scal [0] , [ m ] , [ m ] , [ m ] , [( m ± m ± m ) ] , Table 16: Masses for the Minkowski vacuum M2. m / [0] × , [ m ] × , [ m ] × m vec [0] × , [ m ] × , [ m ] × , [( m ± m ) ] × m tens [0] × , [ m ] × , [ m ] × m / [0] × , [ m ] × , [ m ] × , [( m ± m ) ] × m scal [0] × , [ m ] × , [ m ] × , [( m ± m ) ] Table 17: Masses for the Minkowski vacuum M3.19 / [ m ] × , [3 m ] × m vec [0] × , [4 m ] × , [8 m ] × m tens [0] × , [4 m ] × m / [0] × , [ m ] × , [3 m ] × , [7 m ] × , [9 m ] × m scal [0] × , [4 m ] × , [8 m ] × , [12 m ] × Table 18: Masses for the Minkowski vacuum M4. m / [0] × , [ m ] × , [ m ] × m vec [0] × , [ m ] × , [ m ] × , [ m + m ± m ] × m tens [0] × , [ m ] × , [ m ] × m / [0] × , [ m ] × , [ m ] × , [ m + m ± m ] × m scal [0] × , [ m ] × , [ m ] × , [ m + m ± m ] × Table 19: Masses for the Minkowski vacuum M5.The other interesting fact that emerges from the spectra is that also in 5 dimensions,like in 4, Minkowski vacua have moduli. In fact, once we remove the scalars that are eatenby the massive vectors in the usual Higgs mechanism, we see that the vacuum (M2) hastwo additional massless fields, the vacuum (M3) has 6 additional moduli, the vacuum(M4) 7 and the vacuum (M5) again 6. Like in the 4-dimensional case [24], it may beworth investigating if these gaugings can be connnected to each other by infinite distancelimits along their moduli spaces. Quite possibly, the most general such limits may alsogenerate novel gaugings with new Minkowski vacua and residual symmetries other thanU(2).
Acknowledgments
We would like to thank T. Fischbacher, C. Krishnan and especially M. Trigiante foruseful correspondence. This work is supported in part by the MIUR-PRIN contract2017CC72MK003 “Supersymmetry breaking with fields, strings and branes” . This projecthas received funding from the European Union’s Horizon 2020 research and innovationprogramme under the Marie Sk(cid:32)lodowska-Curie grant agreement No 842991.20
Fermion shifts at the vacuum
In this appendix we provide an instance of the value of the fermion shifts generating thevacua of table 7. For all examples we have chosen a basis where eitherΩ = ⊗ i σ (A.1)or Ω = i σ ⊗ . (A.2) A1 . In the basis with Ω as in (A.1), the maximal AdS supersymmetric vacuum iseasily obtained by setting A = A = − A = − A = g, A i,jkl = 0 . (A.3) A2 . In the basis with Ω as in (A.1), the SO(5) non-supersymmetric AdS vacuumfollows from choosing A = A = − A = − A = g, (A.4)and A = A = A = A = A = A = A = A = g , (A.5) A = A = A = A = A = A = A = A = A = A = A = A = A = A = A = A = 316 g, (A.6) A = A = A = A = A = A = A = A = g , (A.7) A = A = A = A = A = A = A = A = 516 g. (A.8) A3 . In the basis with Ω as in (A.2), the SU(3) invariant AdS vacuum follows from A = A = A = 79 i m , A = − i m , (A.9)and A = A = A = A = A = A = A = A = A = A = A = A = i g, (A.10) A = A = A = A = A = A = 29 i g, (A.11) A = A = A = A = A = A = 12 √ g, (A.12)21 = A = √ g. (A.13) A4 . In the basis with Ω as in (A.1), the N = 2 AdS vacuum with U(2) residualsymmetry follows from A = − A = 712 g, A = − i g , A = i g, (A.14)and A = A = A = A = − g, (A.15) A = A = A = A = A = A = A = A = i g, (A.16) A = A = A = A = A = A = A = A = g , (A.17) A = A = A = A = g , (A.18) A = A = A = A = g , (A.19) A = A = − i g , (A.20) A = A = A = A = i g , (A.21) A = A = A = A = g , (A.22) A = A = A = A = i g . (A.23) A5 . In the basis with Ω as in (A.1), the N = 0 AdS vacuum with SU(2) × U(1) residual symmetry follows from A = − A = 13 (cid:114) g, A = − i g , A = − A = i g , (A.24)and A = A = A = A = g √ , (A.25) A = A = A = A = 112 (cid:114) − √ g, (A.26) A = A = A = A = 112 (cid:114) √ g, (A.27) A = A = A = A = A = A = A = A = g √ , (A.28)22 = A = A = A = i (cid:16) √ (cid:17) g, (A.29) A = A = A = A = i (cid:16) − √ (cid:17) g, (A.30) A = A = A = A = i g . (A.31) M1 . The general CSS Minkowski vacuum in the basis with Ω as in (A.1), followsfrom A = A = m , A = A = m , A = A = m , A = A = m , (A.32)and A = A = − m , (A.33) A = A = A = A = m , (A.34) A = A = − m , (A.35) A = A = A = A = m , (A.36) A = A = − m , (A.37) A = A = A = A = m , (A.38) A = A = − m , (A.39) A = A = A = A = m . (A.40)Obviously the vacua that appear in the context of our analysis have some of the masseseither set to zero or proportional to each other, in order to respect the correct U(2)residual symmetry, but they are always subcases of the one presented here. M2 . The Minkowski vacuum from the SU(3,1) gauging appears in the basis with Ωas in (A.1) by choosing A = i m , A = i m , A = A = m A = A = − i m , (A.42) A = A = A = A = i m , (A.43)23 = A = − i m , (A.44) A = A = A = A = i m , (A.45) A = A = − m , (A.46) A = A = A = A = m . (A.47) M3 . The first new Minkowski vacuum we found appears in the basis with Ω as in(A.1) by choosing A = − i m , A = A = m A = A = i m , (A.49) A = A = A = A = − i m , (A.50) A = A = − m , (A.51) A = A = A = A = m . (A.52) M4 . The new non-supersymmetric Minkowski vacuum appears in the basis with Ωas in (A.1) by choosing A = − A = m , A = − A = m √ A = A = A = A = A = A = A = A = m , (A.54) A = A = A = A = A = A = A = A = m √ , (A.55) A = A = A = A = A = A = A = A = m √ , (A.56) A = A = A = A = A = A = A = A = m , (A.57) A = A = A = A = m √ A = A = A = A = m . (A.59) M5 . The new N = 4 Minkowski vacuum appears in the basis with Ω as in (A.1) bychoosing A = − i m , A = A = m A = A = i m , (A.61) A = A = A = A = − i m , (A.62) A = A = − m − m m , (A.63) A = A = − m
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