Revisiting eight-manifold flux compactifications of M-theory using geometric algebra techniques
aa r X i v : . [ h e p - t h ] O c t (c) 2018 Romanian Journal of Physics (for accepted papers only) REVISITING EIGHT-MANIFOLD FLUX COMPACTIFICATIONS OFM-THEORY USING GEOMETRIC ALGEBRA TECHNIQUES
ELENA-MIRELA BABALIC , CALIN IULIU LAZAROIU Department of Theoretical Physics,“Horia Hulubei” National Institute for Physics and Nuclear Engineering,Reactorului 30, RO-077125, POB-MG6, Magurele-Bucharest, Romania
Email : [email protected], [email protected]
Compiled October 17, 2018
Motivated by open problems in F-theory, we reconsider warped compactifica-tions of M-theory on 8-manifolds to
AdS spaces in the presence of a non-trivial fieldstrength of the M-theory 3-form, studying the most general conditions under which suchbackgrounds preserve N = 2 supersymmetry in three dimensions. In contrast with pre-vious studies, we allow for the most general pair of Majorana generalized Killing pinorson the internal 8-manifold, without imposing any chirality conditions on those pinors.We also show how such pinors can be lifted to the 9-dimensional metric cone overthe compactification 8-manifold. We describe the translation of the generalized Killingpinor equations for such backgrounds to a system of differential and algebraic con-straints on certain form-valued pinor bilinears and develop techniques through whichsuch equations can be analyzed efficiently. Key words : string theory compactifications, M-theory, supergravity, supersym-metry, differential geometry.
PACS : 11.25.Mj, 11.25.Yb, 04.65.+e, 11.30.Pb, 02.40.-k
1. INTRODUCTION
Consider eleven-dimensional supergravity on a background ˜ M endowed witha spinnable Lorentzian metric ˜ g of ‘mostly plus’ signature. The fields of the theoryare the three-form potential ˜ C with four-form field strength ˜ G , the gravitino ˜Ψ M and the metric. As in [1, 2], we consider compactifications down to an AdS spaceof cosmological constant Λ = − κ , where κ is a positive real parameter — thisincludes the Minkowski case as the limit κ → . Thus ˜ M = N × M , where N is anoriented 3-manifold diffeomorphic with R and carrying the AdS metric while M is an oriented Riemannian eight-manifold with metric denoted by g . The metric ˜ g isa warped product with the warp factor ∆ . For the field strength ˜ G , we use the ansatz: ˜ G = e G with G = vol ∧ f + F , where f = f m e m ∈ Ω ( M ) , F = F mnpq e mnpq ∈ Ω ( M ) and vol is the volumeform of N . RJP v.1.1 r2012c Romanian Academy Publishing House ISSN: 1221-146X Elena-Mirela Babalic, Calin Iuliu Lazaroiu (c) 2018 RJP
For the eleven-dimensional supersymmetry generator ˜ η , we use the ansatz: ˜ η = e ∆2 η with η = ψ ⊗ ξ , where ξ is a real pinor of spin / on the internal space M and ψ is a real pinor onthe AdS space N . As in [1, 2] (and in contradistinction with [3]) we do not requirethat ξ has definite chirality ∗ . Mathematically, ξ is a section of the pinor bundleof M , which is a real vector bundle of rank defined on M , carrying a fiberwiserepresentation of the Clifford algebra Cl(8 , . The corresponding morphism γ :( ∧ T ∗ M, ⋄ ) → (End( S ) , ◦ ) of bundles of algebras is an isomorphism. As in [4], wehave set γ m def . = γ ( e m ) and γ (9) def . = γ ◦ . . . ◦ γ . Assuming that ψ is a Killing pinor onthe AdS space, the supersymmetry condition δ ˜ η ˜Ψ M = 0 amounts to the following constrained generalized Killing (CGK) pinor equations [5] for ξ : D m ξ = 0 , Qξ = 0 , (1)where D m is a linear connection on S and Q ∈ Γ( M, End( S )) is a globally-definedendomorphism of the vector bundle S , given explicitly by: D m = ∇ Sm + A m , A m = 14 f p γ mp γ (9) + 124 F mpqr γ pqr + κγ m γ (9) , (2) Q = 12 γ m ∂ m ∆ − F mpqr γ mpqr − f p γ p γ (9) − κγ (9) . (3)The space of solutions to (1) is a finite-dimensional R -linear subspace K ( D, Q ) of thespace Γ( M, S ) of smooth globally-defined sections of S . The problem of interest isto find those metrics and fluxes on M for which some fixed amount of supersymme-try is preserved in three dimensions, i.e. for which the space K ( D, Q ) has some givennon-vanishing dimension, which we denote by s . The case s = 1 (which correspondsto N = 1 supersymmetry in three dimensions) was studied in [1, 2] and reconsid-ered in [5] by using geometric algebra techniques. The case s = 2 (which leads to N = 2 supersymmetry in three dimensions) was studied in [3], but considering onlyMajorana- Weyl solutions of (1), i.e. solutions ξ which also satisfy the supplementaryconstraint γ (9) ξ = ± ξ . Here, we consider the case when no such chirality constraintis imposed on the solutions of (1).
2. THE GEOMETRIC ALGEBRA APPROACH
Since the procedure used is explained in detail in [5,6] (being also summarizedin [4]) in what follows we give only a brief overview. ∗ The geometric product.
Following an idea originally due to Chevalley and Riesz[7, 8], we identify
Cl( T ∗ M ) with the exterior bundle ∧ T ∗ M , thus realizing the Clif-ford product as the geometric product , which is the fiberwise associative, unital andbilinear binary composition ⋄ : ∧ T ∗ M × M ∧ T ∗ M → ∧ T ∗ M given on sections bythe expansion: ω ⋄ η = [ d ] X k =0 ( − k (2 k )! ω ∧ k η + [ d − ] X k =0 ( − k +1 (2 k + 1)! π ( ω ) ∧ k +1 η , (4)where π is the grading automorphism defined through: π ( ω ) def . = d X k =0 ( − k ω ( k ) , ∀ ω = d X k =0 ω ( k ) ∈ Ω( M ) , where ω ( k ) ∈ Ω k ( M ) . (5)The Clifford bundle is thus identified with the bundle of algebras ( ∧ T ∗ M, ⋄ ) , whichis known as the K¨ahler-Atiyah bundle of ( M, g ) . The binary C ∞ ( M, R ) -bilinearoperations ∧ k which appear in the expansion above are the (action on sections ofthe) contracted wedge products , defined iteratively through: ω ∧ η def . = ω ∧ η , ω ∧ k +1 η def . = g ab ( e a y ω ) ∧ k ( e b y η ) = g ab ( ι e a ω ) ∧ k ( ι e b η ) , where ι denotes the so-called interior product (see [5]). We will mostly use, instead,the so-called generalized products △ k , which are defined by rescaling the contractedwedge products: △ k def . = 1 k ! ∧ k . (6)The K¨ahler-Atiyah bundle also admits an involutive anti-automorphism τ (known as the main anti-automorphism or as reversion ), which is given by: τ ( ω ) def . = ( − k ( k − ω , ∀ ω ∈ Ω k ( M ) . (7)
3. LIFTING THE CGK EQUATIONS TO THE METRIC CONE OVER THECOMPACTIFICATION SPACE
The CGK pinor equations can be lifted to the metric cone ( ˆ
M , g cone ) over M as explained in [6] and outlined in [4], to which we refer the reader for the notationsused below. As in loc. cit., we work with an admissible [9, 10] bilinear pairing B on the pin bundle S of M which is symmetric and with respect to which all γ m areself-adjoint. We let ˆ B denote the pull-back of B to the pin bundle ˆ S of the cone. Wework with that pinor representation γ cone on the cone which has signature +1 . The basic form-valued bilinears on the cone.
Recall that s denotes the dimensionof the space of solutions to the CGK pinor equations. Choosing a basis ( ˆ ξ i ) i =1 ...s of such solutions on the cone, we set (see [5, 6]): ˇ E cone ij def . = ˇ E coneˆ ξ i , ˆ ξ j = 12[ d +12 ] ˇ E cone ij ∈ Ω + , cone ( ˆ M ) , and: ˇ E cone ij = d X k =0 ˇ E ( k ) , cone ij = U d X k =0 k ! ˇ E ( k ) , cone a ...a k ( ˆ ξ i , ˆ ξ j ) e a ...a k cone , + , ˇ E ( k ) , cone a ...a k ( ˆ ξ i , ˆ ξ j ) def . = ˆ B ( γ a k ...a ˆ ξ i , ˆ ξ j ) = ˆ B ( ˆ ξ i , γ a ...a k ˆ ξ j ) . The symbol Ω + , cone ( ˆ M ) denotes the space of twisted self-dual forms on the cone ,which is a subalgebra of the K¨ahler-Atiyah algebra of ( ˆ M , g cone ) (see [5, 6] and [4]). CGK equations on the cone.
As explained in [6], it is computationally conve-nient to replace the algebra (Ω + , cone ( ˆ M ) , ⋄ cone ) of twisted selfdual forms of the cone(which is the effective domain of definition of γ cone ) with a certain isomorphic model (Ω < ( ˆ M ) , ♦ cone+ ) whose precise definition is given in loc. cit. When s = 2 ( N = 2 supersymmetry in three dimensions), the CGK pinor equations on ˆ M admit two lin-early independent solutions ξ and ξ . We have Ω < ( ˆ M ) = ⊕ k =0 Ω k ( ˆ M ) , so we areinterested in pinor bilinears ˇ E ( k )ˆ ξ , ˆ ξ with k = 0 . . . for two independent solutions ˆ ξ , ˆ ξ ∈ Γ( ˆ
M , ˆ S ) of the CGK pinor equations lifted to the cone (which are equivalentwith the original CGK pinor equations on M ): ˆ D a ˆ ξ = ˆ Q ˆ ξ = 0 , (8)where the definition of ˆ D a = ∇ ˆ S, cone a + A cone a , where a = 1 . . . , and ˆ Q can be foundin [6] and [4].To extract the translation of these equations into constraints on differentialforms, we implemented certain procedures within the package Ricci [11] for ten-sor computations in
Mathematica R (cid:13) . The dequantizations: ˇ A cone a = γ − ( A cone a ) ∈ Ω < ( ˆ M ) , ˇ Q cone = γ − ( ˆ Q ) ∈ Ω < ( ˆ M ) , of A cone and ˆ Q are given by ˇ A cone9 = 0 and: ˇ A cone m = 14 ι cone e cone m F cone + 14 ( e cone m ) ∧ f cone ∧ θ ∈ Ω < ( ˆ M ) , ∀ m = 1 . . . , ˇ Q cone = 12 r d∆ − f cone ∧ θ − F cone − κθ ∈ Ω < ( ˆ M ) , while the ˆ B -transpose of ˆ Q dequantizes to the cone reversion of ˇ Q cone : ˆ τ ( ˇ Q cone ) = 12 r d∆ + 16 f cone ∧ θ − F cone − κθ . The forms f cone and F cone above are the cone lifts (see [6]) of f and F respectively,while ∆ stands for the pullback Π ∗ (∆) = ∆ ◦ Π of the warp factor through the naturalprojection Π : ˆ M → M , even though the notation does not show this explicitly. Theone-form θ is defined through: θ def . = d r ∈ Ω ( ˆ M ) , where r is the radial coordinate along the metric cone ˆ M .A basis for the space spanned by the forms k ! ˆ B ( ˆ ξ , ˆ γ a ...a k ˆ ξ ) e a ...a k cone ∈ Ω < ( ˆ M ) (of rank k ≤ ) which can be constructed on the cone from ˆ ξ and ˆ ξ is given by(where we have raised all indices using the cone metric to avoid notational clutter) : V a = ˆ B ( ˆ ξ , ˆ γ a ˆ ξ ) , V a = ˆ B ( ˆ ξ , ˆ γ a ˆ ξ ) , V a = ˆ B ( ˆ ξ , ˆ γ a ˆ ξ ) ,K ab = ˆ B ( ˆ ξ , ˆ γ ab ˆ ξ ) , Ψ abc = ˆ B ( ˆ ξ , ˆ γ abc ˆ ξ ) , Φ abce = ˆ B ( ˆ ξ , ˆ γ abce ˆ ξ ) , Φ abce = ˆ B ( ˆ ξ , ˆ γ abce ˆ ξ ) , Φ abce = ˆ B ( ˆ ξ , ˆ γ abce ˆ ξ ) . To arrive at these bilinears we used the identity: B ( ˆ ξ i , ˆ γ a ...a k ˆ ξ j ) = ( − k ( k − B ( ˆ ξ j , ˆ γ a ...a k ˆ ξ i ) , which follows from the fact that γ ta = γ a and implies that certain of the forms ˇ E ( k ) , coneˆ ξ i , ˆ ξ j vanish identically.Here and below, we have taken ˆ ξ and ˆ ξ to form a ˆ B -orthonormal basis of the R -vector space K ( ˆ D, ˆ Q ) of solutions to the CGK equations on the cone: ˆ B ( ˆ ξ i , ˆ ξ j ) = δ ij , ∀ i, j = 1 , , and we noticed that the pairing ˆ B = Π ∗ ( B ) on ˆ S = Π ∗ ( S ) has the same symmetryand type properties as the admissible pairing B on S — namely, both B and ˆ B are symmetric and nondegenerate (and they can be taken to be positive-definite) andmake the eight- (respectively nine-) dimensional gamma ‘matrices’ γ m and ˆ γ a intoself-adjoint operators. From now on — in order to avoid notational clutter — weshall suppress the superscripts and subscripts “ cone ”. In particular, we shall denotethe cone lifts F cone and f cone simply by F and f . With these notations and con-ventions, the truncated model ( ˇ K <, cone ( ˆ D, ˆ Q ) , ♦ cone+ ) of the flat Fierz algebra on thecone admits the basis: ˇ E < = 132 ( V + K + Ψ + Φ ) , ˇ E < = 132 ( V − K − Ψ + Φ ) , ˇ E < = 132 (1 + V + Φ ) , ˇ E < = 132 (1 + V + Φ ) and can be generated by two elements (see Subsection 5.10 of [5]), which we chooseto be: ˇ E < = 132 ( V + K + Ψ + Φ ) , ˇ E < = ˆ τ ( ˇ E < ) = 132 ( V − K − Ψ + Φ ) . The overall coefficient comes from the prefactor [ d +12 ] when d = 9 . As explainedin [5], the Fierz relations for the inhomogeneous forms ˇ E ij (which in this case aretwisted selfdual ˇ E ij = ˇ E Using the procedures which we have implemented and thepackage Ricci [11] for tensor computations in Mathematica R (cid:13) , we find that the first equation (that with the minus sign) in (15) amounts to the following system whenseparated on ranks: ι f ∧ θ K = 0 ,rι d∆ K + 13 ι f ∧ θ Ψ − ι Ψ F − κ ι θ K = 0 , ι f ∧ θ Φ − F △ Φ + r (d∆) ∧ V + 2 κ V ∧ θ = 0 ,rι d∆ Φ − V ∧ f ∧ θ + 16 ι V F − ∗ ( F △ Φ ) + 13 ∗ ( f ∧ θ ∧ Φ ) − κ ι θ Φ = 0 ,r (d∆) ∧ Ψ − f ∧ θ ∧ K − K △ F − ∗ ( f ∧ θ ∧ Ψ) + 16 ∗ (Ψ △ F ) + 2 κ Ψ ∧ θ = 0 , while the second equation (that with the plus sign) in (15) amounts to: − ι F Φ + rι d∆ V − κ ι θ V = 0 , ι V ( f ∧ θ ) − ∗ ( F ∧ Φ ) = 0 ,rι d∆ Ψ + 13 ( f ∧ θ ) △ K + 16 ι K F + 16 ∗ ( F ∧ Ψ) − κ ι θ Ψ = 0 , 13 ( f ∧ θ ) △ Ψ + 16 Ψ △ F + 16 ∗ ( K ∧ F ) + r (d∆) ∧ K − κ K ∧ θ = 0 , 13 ( f ∧ θ ) △ Φ + 16 F △ Φ − ∗ ( F ∧ V ) + ∗ ( r (d∆) ∧ Φ ) − κ ∗ (Φ ∧ θ ) = 0 . Differential constraints. Using the same Mathematica R (cid:13) package, we find thatthe differential constraints (16), when separated on ranks, amount to: d V = 12 Φ △ F + ι f ∧ θ Φ , d K = ( f ∧ θ ) △ Ψ + Ψ △ F , dΨ = 32 F △ K − F △ ∗ Ψ + 2 ∗ ( f ∧ θ ∧ Ψ) − f ∧ θ ∧ K , dΦ = − F ∧ V + 12 e m ∧ ∗ (( ι e m F ) △ Φ ) − e m ∧ ∗ ((( e m ) ∧ f ∧ θ ) △ Φ ) . According to our notational conventions, e m in the equations above stands for thecone lifts e m cone etc. Furthermore, ∗ def . = ∗ cone is the ordinary Hodge operator of ( M, g cone ) and ι stands for ι cone . The generalized products △ p def . = △ cone p are con-structed with the cone metric on ˆ M . Fierz relations. Let us consider the Fierz identities (14) for the basis elements ˇ E 4. CONCLUSIONS The geometric algebra approach developed in [5, 6] provides a synthetic andcomputationally efficient method for translating generalized Killing (s)pinor equa-tions into conditions on differential forms constructed as (s)pinor bilinears. Thisapproach is highly amenable to implementation in various symbolic computationsystems specialized in tensor algebra — and we touched upon two such implementa-tions which we have carried out using Ricci [11]. It affords a more unified and sys-tematic description of flux compactifications and generally of supergravity and stringcompactifications. We illustrated our techniques with the most general flux compact-ifications of M-theory preserving N = 2 supersymmetry in three dimensions, a class of compactifications which had not been studied in full generality before — showingon the one hand how to obtain a complete description of the differential and algebraicconstraints on pinor bilinears and on the other hand how to write all Fierz identitiesbetween the form bilinears. A detailed analysis of the resulting equations, geome-try and physics is the subject of ongoing work. The methods introduced in [5, 6]have much wider applicability, leading to promising new directions in the study ofsupergravity and string theory backgrounds and actions. Acknowledgments. This work was supported by the CNCS projects PN-II-RU-TE (contractnumber 77/2010) and PN-II-ID-PCE (contract numbers 50/2011 and 121/2011). The authors thank theorganizers of the 8-th QFTHS Conference for hospitality and interest in their work. C.I.L thanks theCenter for Geometry and Physics, Institute for Basic Science and Pohang University of Science andTechnology (POSTECH), Pohang, Korea for providing excellent conditions at various stages duringthe preparation of this work, through the research visitor program affiliated with Grant No. CA1205-1.The Center for Geometry and Physics is supported by the Government of Korea through the ResearchCenter Program of IBS (Institute for Basic Science). 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