Geometric algebra and M-theory compactifications
aa r X i v : . [ h e p - t h ] J a n (c) 2018 Romanian Journal of Physics (for accepted papers only) GEOMETRIC ALGEBRA AND M–THEORY COMPACTIFICATIONS
CALIN IULIU LAZAROIU , ELENA-MIRELA BABALIC Department of Theoretical Physics,“Horia Hulubei” National Institute for Physics and Nuclear Engineering,Reactorului 30, RO-077125, POB-MG6, M˘agurele-Bucharest, Romania
Email : [email protected], [email protected]
Compiled October 18, 2018
We show how supersymmetry conditions for flux compactifications of super-gravity and string theory can be described in terms of a flat subalgebra of the K¨ahler-Atiyah algebra of the compactification space, a description which has wide-rangingapplications. As a motivating example, we consider the most general M-theory com-pactifications on eight-manifolds down to
AdS spaces which preserve N = 2 super-symmetry in 3 dimensions. We also give a brief sketch of the lift of such equationsto the cone over the compactification space and of the geometric algebra approach to‘constrained generalized Killing pinors’, which forms the technical and conceptual coreof our investigation. 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
Despite their physical relevance, the most general N = 2 warped product com-pactifications of 11-dimensional supergravity down to a three-dimensional Anti-de-Sitter space have not previously been studied in detail. In such compactifications, theinternal eight-manifold M carries a Riemannian metric g as well as a one-form f anda four-form F , the latter two of which encode the 4-form field strength of the eleven-dimensional theory. The internal Majorana pinor is a section of the real pin bundle S of M , which is a real vector bundle of rank . The condition that such a backgroundpreserves exactly N = 2 supersymmetry in 3 dimensions amounts to the requirementthat the real vector space of solutions to the following algebro-differential system(the so-called constrained generalized Killing (CGK) pinor equations ): Dξ = Qξ = 0 (1)has dimension two. Here, Q is an endomorphism of S given by: Q = 12 γ m ∂ m ∆ − F mpqr γ mpqr − f p γ p γ (9) − κγ (9) RJP v.1.1 r2012c Romanian Academy Publishing House ISSN: 1221-146X Calin Iuliu Lazaroiu, Elena-Mirela Babalic (c) 2018 RJP while D = ∇ S + A is a linear connection on S , where ∇ S is the connection inducedon S by the Levi-Civita connection of ( M, g ) and A = d x m ⊗ A m ∈ Ω ( M, End( S )) is an End( S ) -valued one form on M , with: A m = 14 f p γ mp γ (9) + 124 F mpqr γ pqr + κγ m γ (9) ∈ Γ( M, End( S )) . The quantity κ is a positive parameter related to the cosmological constant Λ of the AdS space through Λ = − κ . The requirement of having N = 2 supersymmetry inthree-dimensions does not impose any chirality condition on ξ . If one imposes sucha condition as an extraneous technical assumption (for example, if one adds the con-dition γ (9) ξ = + ξ to the system (1)), then one obtains a drastic simplification leadingto the well-known results of [1]. For a number of reasons (having to do, in particular,with efforts to generalize F-theory) we are interested in studying the problem with-out imposing any such chirality constraint. This leads to unexpected complications,which can be resolved upon re-formulating the problem by using geometric algebratechniques.
2. THE GEOMETRIC ALGEBRA APPROACH TO PINORS
The standard construction of the pin bundle S of a (pseudo)-Riemannian mani-fold ( M, g ) of signature ( p, q ) and dimension d = p + q can be described most brieflyby saying that S is a bundle of modules over the Clifford bundle Cl( T ∗ M ) of thecotangent bundle of M — where T ∗ M is, of course, endowed with the metric ˆ g in-duced by g ∗ . One problem with this approach (which manifests itself in many subtleaspects of spin geometry as constructed in [3]) is that the Clifford bundle is deter-mined by ( M, g ) only up to isomorphism and hence the association of Cl( T ∗ M ) to ( M, g ) is not functorial. The issue can be resolved by using a particular real-ization of the Clifford bundle (going back to Chevalley [4] and Riesz [5]) whichis known as the K¨ahler-Atiyah bundle of ( M, g ) . This removes the ambiguities ofthe standard approach to spin geometry, since the K¨ahler-Atiyah bundle of ( M, g ) is functorially determined by ( M, g ) . The Chevalley-Riesz realization identifies theunderlying vector bundle of Cl( T ∗ M ) with the exterior bundle ∧ T ∗ M of M , trans-porting the Clifford product of the former to a non-commutative but unital and as-sociative fiberwise multiplication on the latter which we denote by ⋄ and call the geometric product of ( M, g ) . The geometric product makes ∧ T ∗ M into the K¨ahler-Atiyah bundle ( ∧ T ∗ M, ⋄ ) , a bundle of associative algebras which is naturally (i.e.functorially) determined by ( M, g ) . The geometric product (which depends on g ) isnot homogeneous with respect to the natural Z -grading (given by rank) of the exte- ∗ This is equivalent [2] with giving S as the vector bundle associated with a Clifford c -structure of M via a representation of the Clifford c rior bundle. However, it admits an expansion into a finite sum of binary operations △ k ( k = 0 . . . d ) which are homogeneous of degree − k with respect to that grading: ⋄ = [ d ] X k =0 ( − k △ k + [ d − ] X k =0 ( − k +1 △ k +1 ◦ ( π ⊗ id ∧ T ∗ M ) , (2)where π is the parity automorphism, which is defined through: π def . = ⊕ dk =0 ( − k id ∧ k T ∗ M . The binary products △ k : ∧ T ∗ M × M ∧ T ∗ M → T ∗ M are known as generalizedproducts . Expansion (2) can be viewed as the semiclassical expansion of the geo-metric product when the latter is identified with the star product arising in a certain‘vertical’ geometric quantization procedure in which the role of the Planck constantis played by the inverse of the overall scale of the metric g . In particular, the classi-cal limit corresponds to g → ∞ and ⋄ reduces to the wedge product ∧ = △ in thatlimit. The other generalized products △ k ( k > ) depend on g , being determined onsections of ∧ T ∗ M by the recursion formula: ω △ k +1 η = 1 k + 1 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 [6].The pin bundle S can now be viewed as a bundle of modules over the K¨ahler-Atiyah bundle of ( M, g ) , where the module structure is defined by a morphism ofbundle of algebras which we denote by γ : ( ∧ T ∗ M, ⋄ ) → (End( S ) , ◦ ) . Since we areinterested in pinors of spin / , we assume that γ is fiberwise irreducible. Notations and conventions.
We let ( e m ) m =1 ... denote a local frame of T M , de-fined on some open subset U ⊂ M and e m be the dual local coframe (local frameof T ∗ M ), which satisfies e m ( e n ) = δ mn and ˆ g ( e m , e n ) = g mn , where ( g mn ) is theinverse of the matrix ( g mn ) . The space of smooth inhomogeneous globally-defineddifferential forms on M is denoted by Ω( M ) def . = Γ( M, ∧ T ∗ M ) . The fixed rank com-ponents of the graded module Ω( M ) are denoted by Ω k ( M ) = Γ( M, ∧ k T ∗ M ) , with k = 0 , . . . , dim M . A general inhomogeneous form ω ∈ Ω( M ) expands as: ω = d X k =0 ω ( k ) = U d X k =0 k ! ω ( k ) a ...a k e a ...a k with ω ( k ) ∈ Ω k ( M ) , (3)where e a ...a k def . = e a ∧ . . . ∧ e a k and the symbol = U means that equality holds only af-ter restriction of ω to U . A bundle of real pinors over M is an R -vector bundle S over M which is (compatibly) a bundle of modules over the Clifford bundle Cl( T ∗ M ) . Similarly, a bundle of real spinors is a bundle of modules over the even Clifford bun-dle Cl ev ( T ∗ M ) . Of course, a bundle of real pinors is automatically a bundle of realspinors. Hence any pinor is naturally a spinor but the converse need not hold. Inthis paper, we focus on the case of pinors and in particular on the case when S is abundle of simple modules over Cl( T ∗ M ) . We let γ m def . = γ ( e m ) ∈ Γ( U, End( S )) and γ m def . = η mn γ n ∈ Γ( U, End( S )) be the contravariant and covariant ‘gamma matrices’associated with the local orthonormal coframe e m of M and let γ m ...m k denote thecomplete antisymmetrization of the composition γ m ◦ . . . ◦ γ m k . Twisted (anti-)selfdual forms.
Assuming that M is oriented, we let ν denote thevolume form of ( M, g ) . We concentrate on the case when ν ⋄ ν = +1 (this happens, inparticular, when M is an eight- or nine-manifold endowed with a Riemannian metric— the two cases which will be relevant for our application). With this assumption,we have the bundle decomposition: ∧ T ∗ M = ( ∧ T ∗ M ) + ⊕ ( ∧ T ∗ M ) − , where the spaces of sections of the bundles ( ∧ T ∗ M ) ± of twisted (anti-)selfdualforms are the following C ∞ ( M, R ) -submodules of Ω( M ) : Ω ± ( M ) def . = Γ( M, ( ∧ T ∗ M ) ± ) = { ω ∈ Ω( M ) | ω ⋄ ν = ± ω } . When g is a Riemannian metric, the bundle morphism γ : ∧ T ∗ M → End( S ) is in-jective iff. d , . When d ≡ , , we have γ ( ν ) = ǫ γ id S , where ǫ γ ∈ {− , } is asign factor known [6] as the signature of γ . In those cases, we have γ | ( ∧ T ∗ M ) − ǫγ = 0 while the restriction γ | ( ∧ T ∗ M ) + ǫγ is injective. To uniformly treat all cases, we set: ( ∧ T ∗ M ) γ def . = (cid:26) ∧ T ∗ M , if d , ∧ T ∗ M ) + ǫ γ , if d ≡ , and Ω γ ( M ) def . = Γ( M, ( T ∗ M ) γ ) , which is a subalgebra of the K¨ahler-Atiyah algebra. Dequantization.
In our applications — when M is a Riemannian eight-manifold(the compactification space of M -theory down to 3 dimensions) or a nine-manifold(the metric cone over an eight-dimensional compactification space) — the fiberwiserepresentation given by γ is equivalent with an irreducible representation of the realClifford algebra Cl(8 , or Cl(9 , in a 16-dimensional R -vector space, which hap-pens to be surjective. Due to this fact, the map γ − . = ( γ | ( ∧ T ∗ M ) γ ) − : End( S ) → ( ∧ T ∗ M ) γ can be used to identify the bundle of endomorphisms of S with the bun-dle of algebras (( ∧ T ∗ M ) γ , ⋄ ) . In particular, every globally-defined endomorphism T ∈ Γ( M, End( S )) admits a dequantization ˇ T def . = γ − ( T ) ∈ Ω γ ( M ) , which is a (generally inhomogeneous) differential form defined on M . Furthermore, the de-quantization of a composition T ◦ T equals the geometric product ˇ T ⋄ ˇ T of thedequantizations of T , T ∈ Γ( M, End( S )) . The Fierz isomorphism.
When the Schur algebra [6] of
Cl( p, q ) is isomorphicwith R (i.e. when γ is surjective), one can define an isomorphism of bundles ofalgebras ˇ E : ( S ⊗ S, • ) ∼ → (( ∧ T ∗ M ) γ , ⋄ ) called the Fierz isomorphism , where ( S ⊗ S, • ) is a bundle of algebras known as the bipinor bundle . On sections, this inducesan isomorphism of C ∞ ( M, R ) -algebras ˇ E : (Γ( M, S ⊗ S ) , • ) ∼ → (Ω γ ( M ) , ⋄ ) whichidentifies the bipinor algebra (Γ( M, S ⊗ S ) , • ) with the subalgebra (Ω γ ( M ) , ⋄ ) ofthe K¨ahler-Atiyah algebra. Both ˇ E and the multiplication • of the bipinor algebradepend on the choice of an admissible [7, 8] pairing B on S . In our application(when M is an eight- or nine-dimensional Riemannian manifold), B is a certainadmissible bilinear pairing on S which is positive-definite and symmetric. We define ˇ E ξ,ξ ′ def . = ˇ E ( ξ ⊗ ξ ′ ) ∈ Ω γ ( M ) , where ξ, ξ ′ ∈ Γ( M, S ) . Constrained generalized Killing forms.
Using properties of the Fierz isomor-phism, the algebraic constraint Qξ = 0 and the generalized Killing pinor equations Dξ = 0 translate [6] into the following conditions on the inhomogeneous differentialforms ˇ E ξ,ξ ′ , which hold for any global sections ξ, ξ ′ ∈ Γ( M, S ) satisfying (1): ˇ D ad ˇ E ξ,ξ ′ = ˇ Q ⋄ ˇ E ξ,ξ ′ = 0 . (4)Here, ˇ Q def . = γ − ( Q ) ∈ Ω( M ) is the ‘dequantization’ of the globally-defined endo-morphism Q ∈ Γ( M, End( S )) and ˇ D ad = e m ⊗ ˇ D ad m is the ‘adjoint dequantization’of D (see [6]). The operators ˇ D ad m are even derivations of the K¨ahler-Atiyah algebrawhich are defined through: ˇ D ad m def . = ∇ m + [ ˇ A m , ] − , ⋄ , where ˇ A m def . = γ − ( A m ) and ∇ is the connection induced on ∧ T ∗ M by the Levi-Civita connection of ( M, g ) . The Fierz identities between the form-valued pinorbilinears ˇ E ξ,ξ ′ take the concise form [6]: ˇ E ξ ,ξ ⋄ ˇ E ξ ,ξ = B ( ξ , ξ ) ˇ E ξ ,ξ , ∀ ξ , ξ , ξ , ξ ∈ Γ( M, S ) , defining a certain subalgebra of the K¨ahler-Atiyah algebra of ( M, g ) .Equations (4) generalize the usual theory of Killing forms in a number of dif-ferent directions and can be taken as a starting point for a mathematical theory whichis of interest in its own right. When expanding the geometric product into general-ized products as in (2), these seemingly innocuous equations become a highly non-trivial system whose analysis would be extremely difficult without recourse to the synthetic formulation given above in terms of K¨ahler-Atiyah algebras. In particular,the geometric algebra formulation given here allows one to easily extract structuralproperties of such equations and to study them using techniques familiar from thetheory of non-commutative algebras and modules over such – thereby providing aninteresting connection between spin geometry and noncommutative algebraic geom-etry. We stress that equations (4) apply in much more general situations than thoseconsidered in this brief summary.
3. THE CGK EQUATIONS FOR METRIC CONES
As explained in [9], it is convenient to lift ξ to the metric cone over M , whichcan be viewed as the warped product ( ˆ M , g cone ) ≈ ((0 , ∞ ) , d r ) × r ( M, g ) (of warpfactor equal to r ): d s = d r + r d s . The one-form θ def . = d r = ∂ r y g cone has unit norm with respect to the cone metric. The pin bundle ˆ S of the cone can beidentified with the pullback of S through the natural projection Π : ˆ M → M . Wedefine the lift ˆ D of D to be the connection on ˆ S obtained from D by pullback to thecone. Then ˆ D can be expressed as: ˆ D = ∇ ˆ S, cone + A cone , (5)where ∇ ˆ S, cone is the connection induced on ˆ S by the Levi-Civita connection of g cone . Since the metric cone ( ˆ M , g cone ) over ( M, g ) has signature (9 , and since − ≡ , the Clifford algebra Cl(9 , corresponds to the normal non-simple casediscussed in [6]. In particular, its Schur algebra equals the base field R and the cor-responding pin representation γ cone : ( ∧ T ∗ ˆ M , ⋄ cone ) → End( ˆ S ) is surjective. Wehave two inequivalent choices for γ cone , which are distinguished by the signature ǫ ∈ {− , } . The morphism γ cone : ( ∧ T ∗ ˆ M , ⋄ cone ) → (End( ˆ S ) , ◦ ) is completely de-termined by the morphism γ : ( ∧ T ∗ M, ⋄ ) → (End( S ) , ◦ ) once the signature ǫ hasbeen chosen. In the following, we shall work with the choice ǫ = +1 . Setting ǫ = +1 and rescaling the metric on M as g → (2 κ ) g , we find: ∇ ˆ S, cone m = ∇ Sm + κγ m , A cone9 = 0 , A cone m = 14 f p γ mp + 124 F mpqr γ pqr . The generalized Killing pinor equations D m ξ = 0 ( m = 1 . . . ) for pinors ξ ∈ Γ( M, S ) defined on M amount to the flatness conditions: ˆ D a ˆ ξ = 0 , ∀ a = 1 . . . , for pinors ˆ ξ ∈ Γ( ˆ
M , ˆ S ) defined on ˆ M . Indeed, the last of the cone flatness equations ˆ D ˆ ξ = 0 is equivalent with the requirement that the section ˆ ξ of ˆ S is the pullbackof some section ξ of S through the natural projection Π from ˆ M to M , while theremaining equations amount to the generalized Killing conditions D m ξ = 0 on M .Furthermore, the algebraic constraint for ξ is equivalent with the following equationfor ˆ ξ : ˆ Q ˆ ξ = 0 , where ˆ Q ∈ Γ( ˆ
M ,
End( ˆ S )) is the pullback of Q ∈ Γ( M, End( S )) . We refer the readerto [9] for much more detail about the geometric algebra realization of the cone for-malism of [10] and for the applications of this realization to the theory of constrainedgeneralized Killing pinors and forms.
4. APPLICATION TO N = 2 COMPACTIFICATIONS OF M-THEORY DOWN TO THREEDIMENSIONS
In this example, one obtains useful simplifications of the problem by usingthe geometric algebra reformulation (see [9] and the previous Section) of the coneformalism, which is particularly relevant when seeking a geometric interpretation interms of reductions of structure group. Using this variant of the cone formalism aswell as a software implementation of our approach using
Ricci [11] and
Cadabra [12], one can extract and analyze the cone reformulation of (1). Since the detailedtheory of the K¨ahler-Atiyah algebra of cones is somewhat involved and since theequations obtained in this manner for the application at hand are quite complex,we cannot reproduce them here given the space limitations. Instead, we refer theinterested reader to [9] and [13].
5. CONCLUSIONS
We summarized an approach to the theory of constrained generalized Killing(s)pinors which is inspired by geometric algebra, a formulation of spin geometrywhich resolves the lack of naturality affecting certain traditional constructions. Us-ing this approach, we showed how generalized Killing pinor equations translate suc-cinctly into conditions for differential forms constructed as bilinears in such pinors.We also touched upon the applications of this approach to the study of N = 2 com-pactifications of M -theory down to three dimensions, which are discussed in moredetail in [9] as well as in [13]. Acknowledgments. “M theory on Eight-Manifolds” , Nucl. Phys.
B 477 (1996) 155[hep-th/9605053].2. A. Trautman, “Connections and the Dirac operator on spinor bundles” , J. Geom. Phys. (2008)2, 238–252.3. H. B. Lawson and M. L. Michelson, “Spin Geometry” , Priceton University Press (1989).4. C. Chevalley, “The Algebraic Theory of spinors and Clifford Algebras” , Collected works, vol. 2— ed. P. Cartier and C. Chevalley, Springer (1996).5. M. Riesz, “Clifford Numbers and spinors: with Riesz’s Private Lectures to E. Folke Bolinder anda Historical Review by Pertti Lounesto” , Kluwer (1993).6. C. I. Lazaroiu, E. M. Babalic, I. A. Coman, “Geometric algebra techniques in flux compactifica-tions (I)” , arXiv:1212.6766 [hep-th].7. D. V. Alekseevsky, V. Cort´es, “Classification of N -(super)-extended Poincar´e algebras and bilin-ear invariants of the spinor representation of Spin ( p, q ) ” , Commun. Math. Phys. (1997) 3,477 – 510 [arXiv:math/9511215 [math.RT]].8. D. V. Alekseevsky, V. Cort´es, C. Devchand and A. V. Proyen, “Polyvector Super-Poincar´e Alge-bras” , Commun. Math. Phys. (2005) 2, 385-422 [hep-th/0311107].9. C. I. Lazaroiu, E. M. Babalic, “Geometric algebra techniques in flux compactifications (II)” ,arXiv:1212.6918 [hep-th].10. C. Bar, “Real Killing spinors and Holonomy” , Commun. Math. Phys. (1993) 509–521.11. J. M. Lee, D. Lear, J. Roth, J. Coskey and L. Nave, “Ricci — A Mathemat-ica package for doing tensor calculations in differential geometry” ∼ lee/Ricci/ .12. K. Peeters, “Introducing Cadabra: A Symbolic computer algebra system for field theory problems” [hep-th/0701238].13. E. M. Babalic, “Revisiting eight-manifold flux compactifications of M-theory using geometric al-gebra techniques”“Revisiting eight-manifold flux compactifications of M-theory using geometric al-gebra techniques”