aa r X i v : . [ h e p - t h ] N ov U-geometry : SL (5) J EONG -H YUCK P ARK AND Y OONJI S UH Department of Physics, Sogang University, Mapo-gu, Seoul 121-742, Korea [email protected] [email protected]
Abstract
Recently Berman and Perry constructed a four-dimensional M -theory effective action which manifests SL (5) U-duality. Here we propose an underlying differential geometry of it, under the name ‘ SL (5) U-geometry’ which generalizes the ordinary Riemannian geometry in an SL (5) compatible manner.We introduce a ‘semi-covariant’ derivative that can be converted into fully covariant derivatives afteranti-symmetrizing or contracting the SL (5) vector indices appropriately. We also derive fully covariantscalar and Ricci-like curvatures which constitute the effective action as well as the equation of motion. PACS : 04.60.Cf, 02.40.-k
Keywords : M -theory, U-duality, U-geometry. ontents SL (5) ⊂ SL (10)
19B Useful formulae 22
Duality is arguably the most characteristic feature of string/ M -theory [1–3]. While Riemannian geometrysingles out the spacetime metric, g µν , as its only fundamental geometric object, T-duality in string theoryor U-duality in M -theory put other form-fields at an equal footing along with the metric. As a conse-quence, Riemannian geometry appears incapable of manifesting the duality, especially in the formulationsof low energy effective actions. Novel differential geometry beyond Riemann is desirable which treats themetric and the form-fields equally as geometric objects, and makes the covariance apparent under not onlydiffeomorphism but also duality transformations.Despite of recent progress in various limits, eleven-dimensional M -theory remains still M ysterious,not to mention its full U-duality group which was conjectured to correspond to a certain Kac-Moody alge-bra, or an exceptional generalized geometry called E [4–7]. Yet, lower dimensional cases turn out to bemore tractable with smaller U-duality groups [1–3, 8–25]. Table 1 summarizes U-duality groups in variousspacetime dimensions. 1pacetime Dimension D = 1 D = 2 D = 3 D = 4 D = 5 6 ≤ D ≤ U-duality Group SO (1 , SL (2) SL (3) × SL (2) SL (5) SO (5 , E D Table 1: Finite dimensional U-duality groups in various spacetime dimensionsIn particular, Berman and Perry managed to construct M -theory effective actions which manifest a fewU-duality groups, firstly for D = 4 , SL (5) [18], secondly with Godazgar for D = 5 , SO (5 , [12], andthirdly with Godazgar and West for D = 6 , E as well as D = 7 , E [20]. Their constructed actions werewritten in terms of a single object called generalized metric which unifies a three-form and the Rieman-nian metric. Further, they are invariant under so-called generalized diffeomorphism which combines thethree-form gauge symmetry and the ordinary diffeomorphism. Yet, the invariance under the generalizeddiffeomorphism was not transparent and had to be checked separately by direct computations, since theactions were spelled using ‘ordinary’ derivatives acting on the generalized metric. The situation might becomparable to the case of writing the Riemannian scalar curvature in terms of a metric and its ordinaryderivatives explicitly, and asking for its diffeomorphism invariance.It is the purpose of the present paper to provide an underlying differential geometry especially for thecase of D = 4 , SL (5) U-duality by Berman and Perry [18], under the name, ‘
U-geometry ’. The approachwe follow is essentially based on our previous experiences with T-duality [26–32] where, in collaborationwith Jeon and Lee, we developed a stringy differential geometry (or
T-geometry ) for O ( D, D ) T-dualitymanifest string theory effective actions, called double field theory [34–37]. While Hitchin’s ‘generalizedgeometry’ formally combines tangent and cotangent spaces giving a geometric meaning to the B -field [38–44], double field theory (DFT) generalizes the generalized geometry one step further, as it doubles thespacetime dimensions, from D to D + D ( c.f. [45–48]) and consequently manifests the O ( D, D ) T-dualitygroup. Yet, DFT is not truly doubled since it is subject to the so called strong constraint or section condi-tion that all the fields must live on a D -dimensional null hyperplane.Specifically, through [26–32], we introduced an O ( D, D ) T-duality compatible semi-covariant deriva-tive [26, 27] . We extended it to fermions [28], to R-R sector [29], as well as to Yang-Mills [30]. Then weconstructed, to the full order in fermions, ten-dimensional supersymmetric double field theories (SDFT)for N = 1 [31] as well as for N = 2 [32]. Especially the N = 2 D = 10 SDFT unifies type IIA and IIBsupergravities in a manifestly covariant manner with respect to O (10 , T-duality and a ‘pair’ of localLorentz groups, besides the usual general covariance of supergravities or the generalized diffeomorphism.The distinction of IIA and IIB supergravities may arise only after a diagonal gauge fixing of the Lorentzgroups: They are identified as two different types of solutions rather than two different theories.For an extension of Hitchin’s generalized geometry to M -theory, we refer to the works by Coimbra,Strickland-Constable and Waldram [9, 10] which utilize the extended tangent space [8, 11], but did not For a complementary alternative approach we refer to [49–51] ( c.f. [52–57]) where a fully covariant yet non-physical deriva-tive was discussed. After projecting out the undecidable non-physical parts, the two approaches become equivalent. integralmeasure of the SL (5) U-geometry. In section 3, we discuss in detail the semi-covariant derivative as wellas its full covariantization. Section 4 contains the derivations of a fully covariant scalar curvature and afully covariant Ricci-like curvature , which constitute the effective action as well as the equation of motion.In section 5, U-geometry is reduced to Riemannian geometry. We conclude with some comments in sec-tion 6. We point out an intriguing connection to
AdS . Summary • Notation : small Latin alphabet letters denote the SL (5) fundamental indices, as a, b = 1 , , , , . • Assuming the section condition, ∂ [ ab ∂ cd ] = 0 , we define a semi-covariant derivative (3.1) and (3.2),relevant for the SL (5) covariant generalized Lie derivative, ˆ L X (2.8), ( c.f. [9]), ∇ cd T a a ··· a p b b ··· b q := ∂ cd T a a ··· a p b b ··· b q + ( p − q + ω )Γ cdee T a a ··· a p b b ··· b q − P pi =1 T a ··· e ··· a p b b ··· b q Γ cdea i + P qj =1 Γ cdb j e T a a ··· a p b ··· e ··· b q , (1.1)where the connection is given in terms of an SL (5) generalized metric, M ab , by Γ abcd = (cid:2) B [ ab ] ce + ( B beac − B aebc + B acbe − B bcae ) (cid:3) M ed ,B abcd = A abcd + A e ( ab ) e M cd = B ab ( cd ) ,A abcd = M cd M ef ∂ ab M ef − ∂ ab M cd = A [ ab ]( cd ) = B [ ab ] cd . (1.2)This connection is uniquely determined by requiring the compatibilities with the generalized metric, ∇ ab M cd = 0 , and with the generalized Lie derivative, ˆ L X ( ∂ ab ) = ˆ L X ( ∇ ab ) , in addition to a certain‘kernel’ condition, J abcdefgh Γ efgh = 0 (3.7). Generically our semi-covariant derivative is not byitself fully covariant, i.e. δ X ∇ ab = ˆ L X ∇ ab , though there are some exceptions (3.34 – 3.37). • The characteristic feature of the semi-covariant derivative is that, by (anti-)symmetrizing or contract-ing the SL (5) vector indices properly, it can generate fully covariant derivatives (3.40 – 3.45): ∇ [ ab T c c ··· c q ] , ∇ ab T a , ∇ ab T [ ca ] + ∇ ac T [ ba ] , ∇ ab T ( ca ) − ∇ ac T ( ba ) , ∇ ab T [ abc c ··· c q ] ( divergences ) , ∇ ab ∇ [ ab T c c ··· c q ] ( Laplacians ) . (1.3)3 While the usual field strength, i.e. R abcdef = ∂ ab Γ cdef − ∂ cd Γ abef + Γ abeg Γ cdgf − Γ cdeg Γ abgf , turnsout to be non-covariant, the following are fully covariant . – SL (5) U-geometry
Ricci curvature (4.19), R ab := R ( acdb ) cd + R d ( acdb ) c + Γ cd ( ae Γ b ) ecd − Γ ( ac b ) d (Γ ecde + Γ edec )+ Γ c ( acd Γ b ) dee + Γ acdd Γ bcee . (1.4) – SL (5) U-geometry scalar curvature (4.14), R := M ab R ab = R abcabc + Γ abcd Γ cdab − (Γ cacb + Γ cbac )(Γ dbda + Γ dabd ) . (1.5) • The four-dimensional SL (5) U-duality manifest action is, with M = det( M ab ) , c.f. (4.22), Z Σ M − R . (1.6)Up to surface integral, this agrees with the action obtained by Berman and Perry [18], c.f. (A.12) and(A.13). • The equation of motion corresponds to the vanishing of an Einstein-like tensor (4.24), R ab + M ab R = 0 , (1.7)and hence actually, just like the pure Einstein-Hilbert action, R ab = 0 . • From a specific parameterization of the generalized metric in terms of a metric, a scalar and a vector(or its hodge dual three-form potential) in four dimensions, c.f. (5.1) and (5.2), M ab = g µν / √− g v µ v ν √− g ( − e φ + v ) , C λµν = √− g ǫ λµνρ v ρ , (1.8)it follows that, the U-geometry scalar curvature reduces, upon the section condition, to Riemannianquantities (5.8), R = e − φ h R g − ∂ µ φ∂ µ φ + 3 ✷ φ + e − φ (cid:0) ▽ µ v µ (cid:1) i , (1.9)and the action becomes, up to surface integral, as we will see in (5.9) and (5.10), Z Σ M − R = Z d x e − φ √− g (cid:0) R g + ∂ µ φ∂ µ φ − e − φ F κλµν F κλµν (cid:1) . (1.10)4 Section condition, Generalized Lie derivative and Integral measure
The only fundamental object in the SL (5) U-geometry we propose is a × non-degenerate symmetricmatrix, or generalized metric , M ab = M ( ab ) . (2.1)Like in the Riemannian geometry, this with its inverse may be used to freely raise or lower the positions ofthe five-dimensional SL (5) vector indices, a, b, c, · · · .The spacetime is formally ten-dimensional with the coordinates carrying a pair of anti-symmetric SL (5) vector indices, x ab = x [ ab ] . (2.2)We denote the derivative by ∂ ab = ∂ [ ab ] = ∂∂x ab , (2.3)such that ∂ ab x cd = δ ca δ db − δ da δ cb . (2.4)However, the theory is not truly ten-dimensional, as it is subject to a section condition : All the fields arerequired to live on a four-dimensional hyperplane, such that the SL (5) d’Alembertian operator must betrivial [19], ∂ [ ab ∂ cd ] = 0 , (2.5)when acting on arbitrary fields, Φ , Φ ′ , as well as their products, ∂ [ ab ∂ cd ] Φ = ∂ [ ab ∂ c ] d Φ = 0 , ∂ [ ab Φ ∂ cd ] Φ ′ = ∂ [ ab Φ ∂ c ] d Φ ′ − ∂ d [ a Φ ∂ bc ] Φ ′ = 0 . (2.6)For example, for the generalized metric we have ∂ [ ab (cid:0) M ef ∂ c ] d M ef (cid:1) = 0 , M ef ∂ [ ab M ef M gh ∂ c ] d M gh = 0 . (2.7)Generalizing the ordinary Lie derivative, the SL (5) covariant generalized Lie derivative is definedby [10, 19] ˆ L X T a a ··· a p b b ··· b q := X cd ∂ cd T a a ··· a p b b ··· b q + ( p − q + ω ) ∂ cd X cd T a a ··· a p b b ··· b q − P pi =1 T a ··· c ··· a p b b ··· b q ∂ cd X a i d + P qj =1 ∂ b j d X cd T a a ··· a p b ··· c ··· b q . (2.8) c.f. [10] where the flat SO (5) invariant metric was used to raise or lower the indices. T a a ··· a p b b ··· b q , have the total weight , p − q + ω : Each upper or lowerindex contributes to the total weight by + or − respectively, while ω denotes any possible extra weight of the tensor density.It follows from a well-known relation, δ ln(det K ) = Tr( K − δK ) which holds for an arbitrary squarematrix, K , that under the infinitesimal transformation generated by the SL (5) covariant generalized Liederivative (2.8) for ω = 0 , we have δ X det( K ab ) = X cd ∂ cd det( K ab ) + ∂ cd X cd det( K ab ) = ∂ cd (cid:2) X cd det( K ab ) (cid:3) ,δ X det( K ab ) = X cd ∂ cd det( K ab ) ,δ X det( K ab ) = X cd ∂ cd det( K ab ) − ∂ cd X cd det( K ab ) . (2.9)This shows that, det( K ab ) , det( K ab ) and det( K ab ) acquire the extra weights, ω = +1 , ω = 0 and ω = − respectively, while, of course, p = q = 0 . In particular, since det( M ab ) is a scalar density with the totalweight one as an SL (5) singlet, we naturally let it serve as the integral measure of the SL (5) U-geometry.
We propose an SL (5) compatible semi-covariant derivative , in analogy to the one introduced for O ( D, D ) T-duality [26, 27], ∇ cd T a a ··· a p b b ··· b q := ∂ cd T a a ··· a p b b ··· b q + ( p − q + ω )Γ cdee T a a ··· a p b b ··· b q − P pi =1 T a ··· e ··· a p b b ··· b q Γ cdea i + P qj =1 Γ cdb j e T a a ··· a p b ··· e ··· b q , (3.1)with the connection specifically given by Γ abcd = (cid:2) B [ ab ] ce + ( B beac − B aebc + B acbe − B bcae ) (cid:3) M ed ,B abcd = A abcd + A e ( ab ) e M cd = B ab ( cd ) ,A abcd = M cd M ef ∂ ab M ef − ∂ ab M cd = A [ ab ]( cd ) = B [ ab ] cd . (3.2) A similar expression to (3.1) yet with a different connection first appeared in [9, 10] for the case of p = 2 , q = 0 having thetrivial total weight, p − q + ω = 0 .
6s shown below, this connection is the unique solution to the following five conditions we require, Γ abcd + Γ abdc = 2 A abcd , (3.3) Γ abcd + Γ bacd = 0 , (3.4) Γ abcd + Γ bcad + Γ cabd = 0 , (3.5) Γ cabc + Γ cbac = 0 , (3.6) J abcdefgh Γ efgh = 0 , (3.7)where for the last constraint (3.7) we set J abcdefgh := δ [ e [ a δ f ] b ] δ [ g [ c δ h ] d ] + δ [ e [ c δ f ] d ] δ [ g [ a δ h ] b ] + δ h [ a M b ][ c M g [ e δ f ] d ] + δ h [ c M d ][ a M g [ e δ f ] b ] . (3.8)The first condition (3.3) is equivalent to the generalized metric compatibility, ∇ ab M cd = 0 ⇐⇒ Γ ab ( cd ) = A abcd . (3.9)The second condition (3.4) is natural, from ∂ ( ab ) = ∇ ( ab ) = 0 . The next two relations, (3.5) and (3.6), arethe necessary and sufficient conditions which enable us to replace freely the ordinary derivatives, ∂ cd , bythe semi-covariant derivatives, ∇ cd , in the definition of the generalized Lie derivative (2.8), such that ˆ L X T a a ··· a p b b ··· b q = X cd ∇ cd T a a ··· a p b b ··· b q + ( p − q + ω ) ∇ cd X cd T a a ··· a p b b ··· b q − P pi =1 T a ··· c ··· a p b b ··· b q ∇ cd X a i d + P qj =1 ∇ b j d X cd T a a ··· a p b ··· c ··· b q . (3.10)Eq.(3.7) is the last condition that fixes our connection uniquely as spelled in (3.2). We may view the threeconstraints, (3.5), (3.6) and (3.7), as the torsionless conditions of the SL (5) U-geometry.It is worthwhile to note that, the connection satisfies Γ abcd = A abcd + Γ [ ab ][ cd ] , Γ abee = 2Γ ebae = − eabe = A abee = 2 M ef ∂ ab M ef , (3.11)and, from (2.7) due to the section condition, we have ∂ [ ab Γ c ] dee = 0 , Γ abee Γ cdf f + Γ bcee Γ adf f + Γ caee Γ bdf f = 0 . (3.12)7urther, J abcdefgh (3.8) satisfies J abcdefgh = J [ ab ][ cd ][ ef ] gh = J cdabefgh ,J eaebklmn = J ebeaklmn = M nl (cid:0) δ ma δ kb + δ mb δ ka − M ab M km (cid:1) − M nk (cid:0) δ ma δ lb + δ mb δ la − M ab M lm (cid:1) , (3.13)and J abcdefgh J efghklmn = J abcdklmn + (cid:0) M ad J ebecklmn − M bd J eaecklmn + M bc J eaedklmn − M ac J ebedklmn (cid:1) , (3.14)which are all consistent with the conditions (3.6) and (3.7). For example, the closeness (3.14) gives J abcdefgh J efghklmn Γ klmn = 0 .The uniqueness of the connection can be proven as follows.First of all, it is straightforward to check that the connection (3.2) satisfies the five conditions (3.3), (3.4),(3.5), (3.6), (3.7). We suppose that a generic connection may contain an extra piece, say ∆ abcd , which weaim to show trivial. The first four conditions, (3.3), (3.4), (3.5), (3.6) imply ∆ abcd = ∆ [ ab ][ cd ] , (3.15) ∆ [ abc ] d = 0 , (3.16) ∆ e ( ab ) e = 0 . (3.17)Contacting a and d indices in (3.16), we further obtain ∆ e [ ab ] e = 0 . Thus, with (3.17), we have ∆ eabe = 0 , ∆ eaeb = 0 . (3.18)The last condition (3.7) now implies ∆ [ ab ][ cd ] + ∆ [ cd ][ ab ] = 0 . (3.19)Finally, utilizing (3.15), (3.16) and (3.19) fully, we note ∆ abcd = − ∆ cdab = ∆ dacb + ∆ acdb = − ∆ bcad − ∆ acbd = ∆ abcd + 2∆ cabd . (3.20)Therefore, as we aimed, ∆ cabd = 0 . (3.21)Namely, the connection given in (3.2) is the unique connection satisfying the five conditions (3.3), (3.4),(3.5), (3.6) and (3.7). This completes our proof of the uniqueness.8 .2 Full covariantization Under the infinitesimal transformation of the generalized metric, given in terms of the generalized Liederivative, δ X M ab = ˆ L X M ab = ∇ ac X bc + ∇ bc X ac − M ab ∇ cd X cd , (3.22)we have δ X A abcd = ˆ L X A abcd − ( ∂ ab ∂ ce X fe ) M fd − ( ∂ ab ∂ de X fe ) M fc , (3.23)and consequently, δ X Γ abcd = ˆ L X Γ abcd − ∂ ab ∂ ce X de + H abcd . (3.24)Here we set the shorthand notations, H abcd := I abcd + I cdab − I cdba − I abdc ,I abcd := ∂ ab ∂ ce X de − M ac ∂ f b ∂ fe X de + M bc ∂ f a ∂ fe X de = I [ ab ] cd . (3.25)Before we proceed further, it is worthwhile to analyze the properties of H abcd . Firstly, it satisfiesprecisely the same symmetric properties as the standard Riemann curvature, H abcd = H [ ab ][ cd ] = H cdab , (3.26) H abcd + H bcad + H cabd = 0 . (3.27)Secondly, from ∂ ea ∂ eb X ab = 0 , ∂ c ( a ∂ b ) d X cd = 0 . (3.28)it follows that H acbc = 0 . (3.29)Besides, H abcd can be expressed in terms of J abcdefgh given in (3.8) as H abcd = 4 J abcdefgh ∂ ef ∂ gk X hk , (3.30)and hence, with (3.14) and (3.29), it further satisfies H abcd = J abcdefgh H efgh , J abcbefgh H efgh = H abcb = 0 . (3.31)Now for an arbitrary covariant tensor density, satisfying δ X T a a ··· a p b b ··· b q = ˆ L X T a a ··· a p b b ··· b q , (3.32)9traightforward computation may show δ X ( ∇ ab T c c ··· c p d d ··· d q ) = ˆ L X (cid:0) ∇ ab T c c ··· c p d d ··· d q (cid:1) − P pi =1 T c ··· e ··· c p d d ··· d q H abec i + P qj =1 H abd j e T c c ··· c p d ··· e ··· d q . (3.33)Hence, the semi-covariant derivative of a generic covariant tensor density is not necessarily covariant.Yet, for consistency, the metric compatibility of the semi-covariant derivative (3.9) is exceptional,according to (3.26), ∇ ab M cd = 0 , δ X ( ∇ ab M cd ) = ˆ L X ( ∇ ab M cd ) = 0 . (3.34)Other exceptional cases include a scalar density with an arbitrary extra weight, ∇ ab φ = ∂ ab φ + ω Γ abcc φ , δ X ( ∇ ab φ ) = ˆ L X ( ∇ ab φ ) , (3.35)the Kronecker delta symbol, ∇ ab δ cd = 0 , δ X ( ∇ ab δ cd ) = ˆ L X ( ∇ ab δ cd ) = 0 , (3.36)and, with (3.24), (3.30) and (3.31), the ‘kernel’ condition of the connection, J abcdefgh Γ efgh = 0 , δ X ( J abcdefgh Γ efgh ) = ˆ L X ( J abcdefgh Γ efgh ) = 0 . (3.37)In particular, from (3.34) and (3.35), the SL (5) U-geometry integral measure, M − = det( M ab ) having ω = 1 , is covariantly constant, ∇ ab M − = 0 , (3.38)which is also a covariant statement as δ X ( ∇ ab M − ) = ˆ L X ( ∇ ab M − ) = 0 . (3.39)The crucial characteristic property of our semi-covariant derivative is that, by (anti-)symmetrizing orcontracting the SL (5) vector indices appropriately it may generate fully covariant derivatives: From (3.27)10nd (3.29), the following quantities are fully covariant , ∇ [ ab T c c ··· c q ] , (3.40) ∇ ab T a , (3.41) ∇ ab T [ ca ] + ∇ ac T [ ba ] , (3.42) ∇ ab T ( ca ) − ∇ ac T ( ba ) , (3.43) ∇ ab T [ abc c ··· c q ] : ‘divergences’ , (3.44) ∇ ab ∇ [ ab T c c ··· c q ] : ‘Laplacians’ , (3.45)satisfying δ X ( ∇ [ ab T c c ··· c q ] ) = ˆ L X ( ∇ [ ab T c c ··· c q ] ) , δ X ( ∇ ab T a ) = ˆ L X ( ∇ ab T a ) , etc. Note that thenontrivial values of q in (3.40), (3.44) and (3.45) are restricted to q = 0 , , , only, since the anti-symmetrization of more than five SL (5) vector indices is trivial.Of course, from the metric compatibility, ∇ ab M cd = 0 (3.9), the SL (5) indices above may be freelyraised or lowered without breaking the full covariance: For example, ∇ [ ab T c c ··· c q ] is also equally fullycovariant along with (3.40).Further, in particular, for the case of q = 0 , the divergence (3.44) reads explicitly, ∇ ab T ab = ∂ ab T ab + ( ω − abcc T ab , (3.46)and hence, ∇ ab T ab = ∂ ab T ab for ω = 1 , (3.47)which will be relevant to ‘total derivatives’ or ‘surface integral’ in the effective action.Successive applications of the above procedure to a scalar as well as to a vector —or directly from(B.2)— lead to the following second-order covariant derivatives, ∇ [ ab ∇ cd ] φ = 0 , ∇ [ ab ∇ cd T e ] = 0 , ∇ [ ab ∇ c ] d T d = 0 , (3.48)which turn out to be all trivial , i.e. identically vanishing, due to (3.12), (3.4), (3.5), (3.6) and the sectioncondition (2.6). Similarly, for arbitrary scalar and vector, we have an identity, ∇ [ ab φ ∇ cd T e ] = 0 . (3.49)11 Curvatures
The commutator of the SL (5) compatible semi-covariant derivatives (3.1) leads to the following expres-sion, [ ∇ ab , ∇ cd ] T e ··· e p f ··· f q = ( p − q ) R abcdkk T e ··· e p f ··· f q − P i T e ··· g ··· e p f ··· f q R abcdge i + P j R abcdf j g T e ··· e p f ··· g ··· f q + (cid:16) ab [ cg δ hd ] − cd [ ag δ hb ] − Γ abkk δ gc δ hd + Γ cdkk δ ga δ hb (cid:17) ∇ gh T e ··· e p f ··· f q , (4.1)where R abodef denotes the standard curvature, or the field strength of the connection, R abcdef := ∂ ab Γ cdef − ∂ cd Γ abef + Γ abeg Γ cdgf − Γ cdeg Γ abgf = ∇ ab Γ cdef + Γ abgg Γ cdef + Γ cdeg Γ abgf − Γ abcg Γ gdef − Γ abdg Γ cgef − [( a, b ) ↔ ( c, d )] . (4.2)Similarly, straightforward computation shows that the Jacobi identity reads (cid:16) [ ∇ ab , [ ∇ cd , ∇ ef ]] + [ ∇ cd , [ ∇ ef , ∇ ab ]] + [ ∇ ef , [ ∇ ab , ∇ cd ]] (cid:17) T g ··· g p h ··· h q = − P i T g ··· m ··· g p h ··· h q ( Q abcdefmg i + Q cdefabmg i + Q efabcdmg i )+ P j (cid:0) Q abcdefh j m + Q cdefabh j m + Q efabcdh j m (cid:1) T g ··· g p h ··· m ··· h q + ( p − q ) ( Q abcdefmm + Q cdefabmm + Q efabcdmm ) T g ··· g p h ··· h q , (4.3)where we set Q abcdefgh := ∇ ab R cdefgh + Γ abmm R cdefgh + 2Γ ab [ cm R d ] mefgh − ab [ em R f ] mcdgh = ∂ ab R cdefgh − R cdefgm Γ abmh + Γ abgm R cdefmh = −Q abefcdgh . (4.4)Hence, the Jacobi identity implies Q abcdefgh + Q cdefabgh + Q efabcdgh = 0 . (4.5) In (4.1), for simplicity, we assume a trivial extra weight, i.e. ω = 0 . R abcdef + R cdabef = 0 , R [ abcd ] ef = 0 . (4.6)On the other hand, from [ ∇ ab , ∇ cd ] M ef = 0 and (3.11) separately, nontrivial identities are R abcdef + R abcdfe = R abcdgg M ef , R abcdgg = 0 , (4.7)and hence, combining these two, we note R abcdef = R [ ab ][ cd ][ ef ] = − R [ cd ][ ab ][ ef ] . (4.8)This implies that the last line in (4.3) is actually trivial as Q abcdefgg = 0 , and furthermore that there ex-ists essentially only one scalar quantity one can construct by contracting the indices of R abcdef , which is R abcabc .Now we proceed to examine any covariant properties of the curvature, R abcdef , as well as the scalar, R abcabc . Since ∇ ab is semi-covariant rather than ab initio fully covariant , we expect it is also in a waysemi-covariant, which is also the case with T-geometry for double field theory [27]. In fact, we shall seeshortly that R abcabc and hence R abcdef are not fully covariant, but they provide building blocks to constructfully covariant quantities which we shall call fully covariant curvatures.Under the transformation of the generalized metric set by the generalized diffeomorphism, the connec-tion varies as (3.24), δ X Γ abcd = ˆ L X Γ abcd − ∂ ab ∂ ce X de + H abcd , (4.9)while the section condition (2.6) implies ∂ ab ∂ cd X cd = 2 ∂ ac ∂ bd X cd ,∂ ab ∂ ch X gh Γ gd ( ef ) + ∂ ab ∂ dh X gh Γ cg ( ef ) − ∂ ab ∂ gh X gh Γ cd ( ef ) = ∂ ab ∂ cd X gh Γ gh ( ef ) . (4.10)Using the formulae above, it is straightforward to compute the variation of the curvature, δ X R abcdef − ˆ L X R abcdef = (cid:0) ∇ ab H cdef + Γ abgg H cdef − Γ abcg H gdef − Γ abdg H cgef (cid:1) + ∂ ab ∂ ch X gh Γ gd [ ef ] + ∂ ab ∂ dh X gh Γ cg [ ef ] − ∂ ab ∂ gh X gh Γ cd [ ef ] − [( a, b ) ↔ ( c, d )] . (4.11)13s expected, R abcdef itself is not fully covariant. Yet, for consistency, the trivial quantity, R abcd ( ef ) = 0 ,is fully covariant, since H ab ( cd ) = 0 from (3.26).In order to identify nontrivial fully covariant curvatures, from (4.9), we replace ∂ ab ∂ ce X de in (4.11) by ∂ ab ∂ ce X de = − ( δ X − ˆ L X )Γ abcd + H abcd , (4.12)and using (3.11), (3.26), (3.27), (3.29), (B.6) and (B.7), we may organize the anomalous part in the varia-tion of the scalar, R abcabc , as ( δ X − ˆ L X ) R abcabc = − ( δ X − ˆ L X ) (cid:16) Γ abcd Γ cdab − Γ cacb Γ dbda + Γ abcc Γ dadb + Γ abcc Γ abdd (cid:17) . (4.13)Therefore, the following quantity is a genuine fully covariant scalar curvature of SL (5) U-geometry , ( c.f. [10]), R := R abcabc + Γ abcd Γ cdab − (Γ cacb + Γ cbac )(Γ dbda + Γ dabd )= R abcabc + Γ abcd Γ cdab − Γ cacb Γ dbda + Γ abcc Γ dadb + Γ abcc Γ abdd , (4.14)satisfying with ω = 0 , δ X R = ˆ L X R = X ab ∂ ab R . (4.15)Further, under arbitrary variation of the generalized metric, δM ab , the connection transforms as δA abcd = − ∇ ab δM cd + M cd M ef ∇ ab δM ef + Γ ab ( ce δM d ) e ,δ Γ abcd = δ (Γ abce M ed ) = δB [ ab ] cd + ( δB bdac − δB adbc + δB acbd − δB bcad ) , (4.16)which induces δR abcdef = ∇ ab δ Γ cdef + Γ abgg δ Γ cdef − Γ abcg δ Γ gdef − Γ abdg δ Γ cgef − [( a, b ) ↔ ( c, d )] . (4.17)Now, from (4.17) alone —without referring to the details of (4.16)— we may be able to derive the trans-formation of the fully covariant scalar curvature as follows δ R = 2 δM ab R ab + ∇ ab (cid:16) M bc M de δ Γ adec − δ Γ abcc (cid:17) , (4.18) This is analogue to the variation of the Riemannian scalar curvature, δR = δg µν R µν + ∇ µ (cid:0) g νρ δ Γ µνρ − g µν δ Γ ρρν (cid:1) . Ricci curvature of SL (5) U-geometry , ( c.f. [10]), R ab := R ( acdb ) cd + R d ( acdb ) c + Γ cd ( ae Γ b ) ecd − Γ ( ac b ) d (Γ ecde +Γ edec )+ Γ c ( acd Γ b ) dee + Γ acdd Γ bcee , (4.19)satisfying R ab = R ba , M ab R ab = R , (4.20)and δ X R ab = ˆ L X R ab . (4.21)Naturally, the four-dimensional SL (5) U-duality manifest effective action reads Z Σ M − R , (4.22)where Σ denotes the four-dimensional hyperplane where the theory lives to satisfy the section condi-tion (2.6). As shown through (A.12) and (A.13) in Appendix A, up to surface integral, this action agreeswith the action obtained by Berman and Perry [18].From (4.18), the action transforms under arbitrary variation of the generalized metric, δ (cid:18)Z Σ M − R (cid:19) = Z Σ M − δM ab (2 R ab + M ab R ) . (4.23)Hence, the equation of motion corresponds to the vanishing of the following Einstein-like tensor, R ab + M ab R = 0 , (4.24)and hence, it follows R ab = 0 . (4.25)This also (indirectly) verifies the covariance of the Ricci-like curvature (4.21), since any symmetry of theaction —in this case the generalized diffeomorphism— is also a symmetry of the equation of motion. Further, from the invariance of the action under the generalized diffeomorphism (3.22), a conservationrelation follows ∇ c [ a R b ] c + ∇ ab R = 0 , (4.26)which may be also directly verified using e.g. (4.5). Note the plus sign in (4.24) in comparison to the Riemannian Einstein tensor, R µν − g µν R . As discussed in section 5, upon the section condition the U-geometry action (4.22) reduces to a familiar Riemannian ac-tion (5.9) of which the equations motion, c.f. (4.25), are surely fully covariant. See also e.g. [58] for general analysis and proof. Parametrization and Reduction to Riemann
We parametrize the generalized metric, i.e. a generic non-degenerate × symmetric matrix, by M ab = g µν / √− g v µ v ν √− g ( − e φ + v ) , (5.1)where φ , v µ and g µν denote a scalar, a vector and a Riemannian metric in Minkowskian four-dimensions,such that v µ = g µν v ν , v = g µν v µ v ν and g = det( g µν ) . The vector can be dualized to a three-form, C λµν = √− g ǫ λµνρ v ρ , (5.2)which may couple to a membrane.The existence of the scalar might appear odd especially if the spacetime dimension were eleven ratherthan four. However, without the scalar, the (off-shell) degrees of freedom would not match in the abovedecomposition of the generalized metric,
15 = 1 + 4 + 10 = 4 + 10 . (5.3)Moreover, with a parametrization of an sl (5) Lie algebra element, i.e. a generic × traceless matrix, H ab = a µν b µ c ν − a λλ , (5.4)the infinitesimal sl (5) U-duality transformation, δM ab = H ac M cb + H bc M ac , amounts to δφ = − (cid:0) a λλ + √− g b λ v λ (cid:1) ,δv µ = a µλ v λ − a λλ v µ + √− g (cid:0) − e φ + v (cid:1) b µ + √− g c µ ,δg µν = a µν + a νµ − a λλ g µν + √− g (cid:0) b µ v ν + b ν v µ − b λ v λ g µν (cid:1) . (5.5)Clearly this confirms that the scalar is inevitable for the closeness of the U-duality transformations: Setting a λλ ≡ and b λ ≡ for δφ ≡ would break the SL (5) U-duality group to its subgroup, R ⋊ SL (4) . In our convention, ǫ = 1 . In (5.5), the four-dimensional Greek letter indices are raised or lowered by the Riemannian metric from the default positionsin (5.4), for example a µν = a µλ g λν . ( X µν , X µ ) = ( ǫ µνρσ Λ ρσ , ξ µ ) and upon the choice of the ‘section’ by ( ∂ µν , ∂ µ ) ≡ (0 , ∂ µ ) ,each component field transforms as ( c.f. [19]) δφ = ξ λ ∂ λ φ = L ξ φ ,δv µ = ξ λ ∂ λ v µ + ∂ µ ξ λ v λ − √− g ǫ µρστ ∂ ρ Λ στ = L ξ v µ − √− g ǫ µρστ ∂ ρ Λ στ ,δg µν = ξ λ ∂ λ g µν + ∂ µ ξ λ g λν + ∂ ν ξ λ g µλ = L ξ g µν . (5.6)In particular, as expected, the covariant divergence of the vector is a scalar, δ ( ▽ µ v µ ) = ξ λ ∂ λ ( ▽ µ v µ ) .The inverse of the generalized metric and their determinants are M ab = √− g ( g µν − e − φ v µ v ν ) e − φ v µ e − φ v ν − e − φ / √− g , det( M ab ) = e φ / √− g , det( M ab ) = e − φ √− g , (5.7)which are consistent with (2.9), and in particular assures us that M − = det( M ab ) corresponds to the SL (5) invariant measure of the U-geometry.The fully covariant scalar curvature (4.14) now reduces to Riemannian quantities, R = e − φ h R g − ∂ µ φ∂ µ φ + 3 ✷ φ + e − φ (cid:0) ▽ µ v µ (cid:1) i , (5.8)and hence the action (4.22) becomes, up to surface integral, Z Σ M − R = Z d x e − φ √− g (cid:0) R g + ∂ µ φ∂ µ φ − e − φ F κλµν F κλµν (cid:1) , (5.9)where F κλµν is the field strength of the three-form potential, F κλµν = 4 ∂ [ κ C λµν ] . (5.10) With the Bianchi identity of the Riemann curvature, ▽ µ (cid:18) √− g ǫ µρστ ∂ ρ Λ στ (cid:19) = 12 √− g ǫ µρστ (cid:2) ▽ µ , ▽ ρ (cid:3) Λ στ = 12 √− g ǫ µρστ (cid:16) − R λσµρ Λ λτ − R λτµρ Λ σλ (cid:17) = 0 . Comments
Like in double field theories (bosonic DFT [27], N = 1 SDFT [31] and N = 2 SDFT [32]), according to(4.20) and (4.25), the U-geometry Lagrangian vanishes on-shell strictly, M − R = 0 . However, this doesnot necessarily mean that the Riemannian action (5.9) is trivial, as the difference is given by a nontrivialsurface integral. Hence, in contrast to Riemannian geometry, U-geometry as well as T-geometry appear toclearly distinguish the bulk Lagrangians from the York-Gibbons-Hawking type boundary terms [59, 60],by removing their ambiguity, c.f. [61].In fact, the parametrization of the generalized metric (5.1) we have considered above possesses thespacetime signature, ‘ ’, e.g. as seen from M ab = E a ¯ a E b ¯ b ¯ η ¯ a ¯ b , E a ¯ a = e µi / √ e √ e v ν e νi √ e e φ/ , ¯ η = diag ( − + + + − ) . (6.1)Alternatively, if we had assumed the Minkowskian signature with ¯ η = diag ( − + + + +) , such that φ hadbeen replaced by φ + iπ or e φ → − e φ , the kinetic term of the four-form field strength in the resultingaction (5.9) would have carried the opposite wrong sign to break the unitarity. Therefore, we conclude thatthe spacetime signature of the generalized metric ought to be , and the relevant internal local Lorentzgroup should be O (2 , . This seems to point to the four-dimensional anti-de Sitter space, AdS .It is desirable to verify (4.21) and (4.26) directly in a covariant manner, for which one might need moreidentities for the curvature in addition to (4.8) and (4.26).Supersymmetrization, reduction to double field theory ( c.f. [21]) and extensions to other U-dualitygroups ( c.f. [9, 10]), especially E [4, 5, 7], are of interest for future works. It is intriguing to note that,the SL (5) U-duality group naturally gets embedded into SL (10) (see [18] and also our Appenix A), whichmay well hint at higher dimensional larger U-duality groups. Acknowledgements
We wish to thank David Berman for the kind explanation of his works during
CQUeST EU-FP Workshop , Seoul, 2012 . JHP also benefits from discussions with Bernard Julia during an Isaac newton Institute 2012Program,
Mathematics and Applications of Branes in String and M-theory , and also with Pei-Wen Kao.The work was supported by the National Research Foundation of Korea and the Ministry of Education,Science and Technology with the Grant No. 2012R1A2A2A02046739, No. 2012R1A6A3A03040350,No. 2010-0002980 and No. 2005-0049409 (CQUeST). We thank Chris Blair and Emanuel Malek forpointing out numerical errors in (5.8), (5.9) from the previous arXiv version.18 ppendices A & BA SL (5) ⊂ SL (10) As a shorthand notation [18], we let the capital letters,
A, B, C, · · · represent pairwise skew-symmetric SL (5) indices, such that for the derivative, ∂ A ≡ ∂ a a , (A.1)and for tensors carrying pairwise skew-symmetry indices, T A A ··· A m B B ··· B n ≡ T [ a a ][ a a ] ··· [ a m a m ][ b b ][ b b ] ··· [ b n a n ] . (A.2)Being ten-dimensional, the capital letters are essentially for SL (10) , as the sl (5) infinitesimal transforma-tion, w ab with w aa = 0 , acts now as an sl (10) element: w AB = w a [ b δ a b ] + δ a [ b w a b ] , w AA = 0 . (A.3)We may further set a generalized metric for the SL (10) indices, M AB = M [ a a ][ b b ] := ( M a b M a b − M a b M a b ) . (A.4)It follows that, the inverse is given by M AB = M [ a a ][ b b ] = ( M a b M a b − M a b M a b ) , (A.5)satisfying M AB M BC = δ AC = δ [ c [ a δ c ] a ] = ( δ c a δ c a − δ c a δ c a ) , (A.6)and the determinant reads det( M AB ) = ( ) [det( M ab )] . (A.7)Henceforth, we use M AB and M AB to raise and lower the sl (10) capital letter indices.For (3.2), A abcd = M cd M ef ∂ ab M ef − ∂ ab M cd , (A.8)we further set A ABC := 2 A a a [ b [ c δ c ] b ] = δ CB ( M DE ∂ A M DE ) − ( ∂ A M BD ) M CD , (A.9)such that A ABC = A ACB = M BC ( M DE ∂ A M DE ) − ∂ A M BC , (A.10)19nd A ABB = 2 M DE ∂ A M DE = 4 A a a bb = 4Γ a a bb = 8 M bc ∂ a a M bc . (A.11)Now, we are ready to compare our action (4.22) with the action by Berman and Perry which waswritten in terms of the SL (10) notation. Up to total derivatives, our scalar curvature (4.14) agrees with theLagrangian by Berman and Perry [18] as R = ∇ ab (Γ cabc − Γ cacb ) − R Berman − Perry , (A.12)where R Berman − Perry = M ST ∂ S M P Q ∂ T M P Q − M ST ∂ S M P Q ∂ P M T Q + M MN M ST ∂ M M NT ( M P Q ∂ S M P Q ) + M ST ( M MN ∂ S M MN )( M P Q ∂ T M P Q )= − A ABC A ABC + 2 A ABC A BAC − A AC C A DDA + A AC C A ADD = − A abcd A abcd + 4 A abcd A acbd + A abcc A abdd + 6 A abcc A dabd − A cabc A dbad . (A.13)Note also R = − ∂ ab (2 A cabc + A abcc ) + A abcd A abcd − A abcd A acbd − A abcc A abdd − A cabc A abdd + 2 A cabc A dbad . (A.14)The remaining of this Appendix is devoted to the construction of another semi-covariant derivativewhich is for the group SL (10) and is different from the one in (3.1) for SL (5) . The alternative semi-covariant derivative is defined by employing A ABC (A.9) as the connection, D A T B ··· B m C ··· C n := ∂ A T B ··· B m C ··· C n + ( m − n ) A ADD T B ··· B m C ··· C n − P i T B ··· D ··· B m C ··· C n A ADB i + P j A AC j E T B ··· B m C ··· E ··· C n . (A.15)In contrast to (A.9), for the connection of Γ abcd defined in (3.2), an analogue expression, Γ ABC :=Γ a a [ b [ c δ c ] b ] , cannot be written entirely in a SL (10) covariant manner, i.e. in terms of ∂ A and M AB carrying the SL (10) indices only. In fact, generically, D A T B ··· B m C ··· C n = ∇ A T B ··· B m C ··· C n . (A.16) One might try to look for other connection alternative to the one we constructed in (3.2), by e.g. modifying the index-eighttensor, J abcdefgh (3.8), —used in the condition (3.7)— to a more symmetric index-eight ‘projection’, P abcdefgh P efghklmn = P abcdklmn , as in the case of T-geometry [27]. However, such modification would better not ruin the nice properties of H abcd as(3.24), (3.26), (3.27), (3.29).
20n any case, the alternative semi-covariant derivative is compatible with the SL (10) generalized metric, D A M BC = 0 , D A M BC = 0 , (A.17)and furthermore, the new connection, A ABC (A.9), is the unique connection which satisfies the abovecompatibility condition and the symmetric property, A ABC = A ACB , c.f. (A.10).The commutator of the above semi-covariant derivatives (A.15) has the expression, [ D A , D B ] T C ··· C m D ··· D n = ( m − n ) R ABEE T C ··· C m D ··· D n − P i T C ··· E ··· C m D ··· D n R ABEC i + P j R ABD j E T C ··· C m D ··· E ··· D n + (cid:0) A ABE − A BAE − A AF F δ BE + A BF F δ AE (cid:1) D E T C ··· C m D ··· D n , (A.18)where R ABC D denotes the standard field strength of the connection, R ABC D := ∂ A A BC D − ∂ B A AC D + A AC E A BED − A BC E A AED = D A A BC D + A AEE A BC D − A ABE A EC D + A BC E A AED − ( A ↔ B ) . (A.19)Arbitrary variations of the metric, δM AB , induces δA ABC = δ ( A AB D M DC ) = M BC M DE D A δM DE − D A δM BC + A A ( BD δM C ) D ,δR ABCD = D A δA BCD − A BC E D A δM ED + A BC E A AEF δM F D + A EE (cid:0) M CD M F G D B δM F G − D B δM CD + A B ( C F δM D ) F (cid:1) − A ABE (cid:0) M CD M F G D E δM F G − D E δM CD + A E ( C F δM D ) F (cid:1) − ( A ↔ B ) , (A.20)which may be useful to address a higher dimensional U-geometry in future.21 Useful formulae
The generalized Lie derivative and the semi-covariant derivative of Kronecker delta symbol are all trivial, ˆ L X δ ab = 0 , ∇ cd δ ab = 0 . (B.1)For a generic covariant tensor density satisfying (3.32), using (3.33), we have δ X ( ∇ ab ∇ cd T e e ··· e p f f ··· f q )= ˆ L X (cid:0) ∇ ab ∇ cd T e e ··· e p f f ··· f q (cid:1) − P pi =1 (cid:0) T e ··· g ··· e p f ··· f q ∇ ab H cdge i + ∇ ab T e ··· g ··· e p f ··· f q H cdge i + ∇ cd T e ··· g ··· e p f ··· f q H abge i (cid:1) + P qj =1 (cid:0) ∇ ab H cdf j g T e ··· e p f ··· g ··· f q + H abf j g ∇ cd T e ··· e p f ··· g ··· f q + H cdf j g ∇ ab T e ··· e p f ··· g ··· f q (cid:1) + H abcg ∇ gd T e ··· e p f ··· f q + H abdg ∇ cg T e ··· e p f ··· f q . (B.2)From J almnefgh H blmn Γ efgh = 0 (3.7), we have H almn (Γ blmn − Γ mnlb + Γ mnbl ) − ( H ambn + H bman )Γ lmln = 0 . (B.3)Contracting free indices, a, b , and from (3.26), we note H abcd Γ abcd = 0 . (B.4)This further implies with H [ abc ] d Γ abcd = 0 , H abcd Γ acbd = 0 . 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