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Lorentzian Lie (3-) and toroidal compactification of M/ theory

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Please note that terms and conditions apply. JHEP03(2009)045 + 1)- p March 6, 2009 er of pairs of : February 5, 2009 -brane with an January 28, 2009 p : : , Published escribes supersymmetric Received Accepted 10.1088/1126-6708/2009/03/045 s are given and corresponding th the Kaluza-Klein modes by bras associated with the loop For example, D . Simple systems with nontrivial onal Taiwan University, doi: kyo, b can be identified with D( g Published by IOP Publishing for SISSA [email protected] , and Shotaro Shiba b . g Yutaka Matsuo a D-, M-Theory We construct a class of Lie 3- with an arbitrary numb [email protected] SISSA 2009 Department of Physics, CenterLeung for Center Theoretical Sciences for and Cosmology and Particle Astrophysics, Nati Taipei 10617, Taiwan, R.O.C. Department of Physics, Faculty ofHongo Science, 7-3-1, University Bunkyo-ku, of Tokyo 113-0033, To E-mail: Japan [email protected] b a c Abstract: Pei-Ming Ho, (infinite dimensional) affine symmetry ˆ Lorentzian Lie (3-)algebra andcompactification toroidal of M/ generators with Lorentzian signature metric.BLG models Some are example studied. Wemassive show vector that multiplets such after a the system ghostinteraction in fields general are are d Higgsed realizedalgebras. by The infinite massive dimensional fieldsthe are Lie toroidal then 3-alge compactification naturally triggered identified by wi the ghost fields. with gauge symmetry

Keywords: JHEP03(2009)045 7 7 9 1 3 5 22 26 29 10 13 15 16 18 30 14 22 ]. For the BLG 6 nerators , 5 ries ] and its direct sums. c is given up, we also 7 [ brackets give realizations 4 ples, a Lie 3-algebra with a A = 0 ntal identity (a generalization ge symmetry is based on a Lie ume theory in M-theory. This ijkl a is n undertaken [ F 6= 0 ic is also imposed to avoid ghosts, ij ab J ] constructed a three-dimensional su- 4 -field background on a 3- where C ]. The BLG model with this algebra realizes – 1 – 9 , 8 ] and Gustavsson [ 3 – 1 6= 0 ijk a f + 1) via Kac-Moody algebra via 3-algebra p p to D( p If we relax the condition on dimensionality, Nambu-Poisson Similarly, when the requirement of a positive definite metri 5.3 M2 to M5 revisited 4.3 Inclusion of 5.1 D 5.2 M2 to D 3.4 Solutions 3.5 Summary of the 3-algebra solutions 4.1 Component expansion 4.2 Elimination of ghosts 3.1 Fundamental identities 3.2 Lorentzian extension3.3 of Nambu bracket Constraints from the fundamental identities for 1 Introduction Recently, Bagger, Lambert [ 6 Conclusion and discussion 5 Application to toroidal compactification of M/string theo 4 BLG model for Lorentzian 3-algebra with 3 Analysis of Lie 3-algebra with two or more negative-norm ge Contents 1 Introduction 2 Lorentzian BLG model the world-volume theory of M5-branes in the percomformal field theory as aBLG model multiple-M2-brane is world-vol characteristic of3-algebra, the and novel thus feature various that studies on the thismodel gau algebra to have work, bee the Lie 3-algebraof needs Jabobi to identity). satisfy If the the fundame the positivity only of non-trivial the example invariant metr of finite dimensional 3-algebr of infinite dimensional Lie 3-algebra [ Nambu-Poisson bracket can be defined. found physically meaningful models. Among the various exam JHEP03(2009)045 ] 1 16 ore – ] but 14 11 Lie 3-algebra. ch the removal of , we derive the BLG ith Lorentzian gauge 4 etely decoupled. It was igniture. It was known dimensional symmetries ian 3-algebras for which analyzed in the context entzian extension [ Lorentzian y. As we noted, such D- he Higgs mechanism which he Lorentzian BLG models nstants directly. Although pected that the appearance onsidered extensively by de er of Lorentzian pairs. We the context of BLG models. ivial interacting models. we find a class of interesting ection n pairs was made in [ with additional compactified duces the multiple- D2-brane = 8 SUSY in the BLG model, it gives us insight about how cludes an interesting example string modes in a compactified n that adding Lorentzian pairs o as tities. We also study the BLG ] to construct M2-brane model which N associated with the compactifica- ture of the generalized Lorentzian ture constants. We find it fruitful 10 ions to string/M theory. We note ]) we have to treat such an infinite d [ 12 , we first review the Lorentzian BLG 2 , we give a detailed study of the constraint 3 – 2 – limit. It was suggested that we need 3-algebra instead of ] in string/M theory and we hope that our method N 13 ) in large 2 / 3 N ( O ] is possible. In section ] when the number of Lorentzian pairs is two. Here we present m 19 , 11 [ 18 ]. We describe the typical structure of the 3-algebra for whi 16 et. al – 14 In this paper, we study some generalizations of such Lorentz This paper is organized as follows. In section Our observation of the relation between the D-brane system w The Lie 3-algebra with zero-norm generators was also studie 1 the ghost field [ Lie algebra to have such scaling. realized that the inclusion oftion the Lorentzian of generators a is spatialworld-volume dimension, theory and in this type Lorentzian IIA model string repro theory. ghost fields can stillMedeiros be decoupled. Such algebras have been c The corresponding BLG model has ghosts, but they can be compl negative-norm generator was constructed and was referred t of Lorentzian Lie algebras.through It gauge enables theories usbrane to with system analyze Lorentzian is some Lie by of itself algebra t an symmetr interesting objectproduces to the study. correct entropy In s gives a simple direct interpretation to such systems. model [ from the fundamental identity. Such study for two Lorentzia dimensional gauge symmetry on D-branes.of infinite It dimensional was symmetry generally should bespace. ex related to However, closed the explicitimplement analysis Kaluza-Klein was mass not waswere made also not because studied t in known. various contexts Similar [ infinite we generalize their resultuse by a considering strategy an towe do arbitrary analyze not numb the claim constraint3-algebras that through for we such could the analysis, classify with structurethat all potential the co possible applicat many algebras, 3-algebras which we found can be realized by Lor straightforward and explicit analysis in termsto of consider the struc generalizations withto more circumvent the Lorentzian strict pairs, constraintsmodel as from associated fundamental with iden suchwhich 3-algebras. contains Our the construction massivedimensions. in Kaluza-Klein This towers seems associated to becorresponds consistent to with additional compactifications. our expectatio A3-algebra typical fea is indeed that we have a massive spectrum with and we need an infinite dimensional realization to havesymmetry nontr and higher dimensionalIn branes is fact, not most restrictedof of to a the Yang-Mills examples systemthat considered whose in here gauge some brane can symmetry configurations be has (see Lorentzian directly for s example [ JHEP03(2009)045 (2.9) (2.7) (2.8) (2.6) (2.1) (2.2) (2.3) (2.4) (2.5) (2.10) ]. The denote 16 – G , 14 EF ABF A , f ) A,B,C,... ∧ F , we construct the 5 CS L CD CDE A + f s which are the simplest tion ∧ r fields where each gauge han one Lorentzian pairs. BLG model [ B = pot . G L )Φ AB x vative is A + ( = 0 ebra. Finally we comment that CDF int D f 2 specify the longitudinal directions µCD L CE T F , EFGB h , A 1 D + i f Lorentzian , E A ] G Ψ K ABE L f ABC = 0 ABD CDB f , X + CDA µ + f f J f X G + 3 1 − ]= L – 3 – , X ( C A I + x ED , Φ CFE X 3 i : , T h µ [ f d , B ∂ CD , E ] F Ψ] i AB , I K Z = h ] can be also regarded as the typical example of the , T dA J 2 9 A X ABC A ABD ∧ T = , , X µ , T f i , X )) f [ J i 8 10 the transverse directions. The indices I = x Ψ B AB , D + µ X I , X [ A Φ( G I , T D xL µ X µ IJ ,..., A 3 µ X D [ d Γ Γ T ( h D , , h FDE h = 3 ABCD ¯ ¯ Z 1 Ψ Ψ f 1 2 12 h h f 2 F i i 2 4 2 1 − T − ======ABC I, J, K f Ψ S X int CS pot Ψ. L L L , L L I is the M2-brane tension. The indices X 2 T Invariant metric Tri-linearity Skew symmetry Fundamental identity In order to define the BLG model action, the Lie 3-bracket • • • • for Φ = for a Lie 3-algebra must satisfy the following constraints: components of Lie 3-algebra generators. The covariant deri We demonstrate that suchfield system absorbs typically two has degrees massive of vecto freedom from scalar fields. In sec model associated with the simplest Lie 3-algebra with more t where BLG model (or supernontrivial Yang-Mills examples theory) of based generalizedthe on Lorentzian loop description Lie algebra of (3-)alg M5-brane [ compactification through the Lorentzian 3-algebra. 2 Lorentzian BLG model In this section, we review the basic features of the original BLG action for multiple M2-branes is of M2-branes; JHEP03(2009)045 . ij h (2.13) (2.12) (2.15) (2.17) (2.11) theory that the , v, l 2 ij µν 2 ˆ f F ]) kl 2 J be a given Lie is the center of h 1 λ ˆ they act only as . X 4 G − specific way that v , ] (2.14) I ν − ˆ ˆ X A ]= such that [ lgebra has a negative- , ˆ , k ˆ Ψ] I u Ψ , and Killing form µ } , . Let ′ µ i k ˆ  I X . The decoupling of the A , T ij u ( [ A I j u X f µ 1 2 [ Ψ X − ]. This is motivated by the generator I Γ µ , T i u − µ s ∂ otherwise = 0 16 u, v, T ˆ µ ¯ 2 T ¯ ˆ Ψ A { ΨΓ ] [ requirement of invariant metric, Γ ν i 2 J v 2 ∂ , = iλ ˆ ]. The Lagrangian becomes, after ¯ X = 0 (2.16) Ψ − A , i + 2 17 I u I ˆ = 0 Ψ+ ν T 2 ˆ Ψ ] ˆ µ X − u A [ J ˆ , µ ) D 2 Ψ k ˆ I ) ∂ v µ X µ T , ∂ K u X , k I , µ ¯ ˆ := µ ΨΓ ij ij ˆ X Γ X I ∂ 10 i ( 2 f h [ µν , – 4 – 1 4 )( 2 λδ ˆ 4 I = , structure constants + u F λ , gh i ]= i 2 X j L T ) j := µ , I u = 0 ˆ I Ψ] + ∂ + ˆ , T , T Ψ+ , ˆ X I λ i Φ] u i µ J ′ µ ,  T + ( X ˆ µ ˆ ′ λ = D h A 0), BLG model has a ghost field. The mode expansion of X 2 µ ˆ I [ µ u, T A I A u ∂ [ I [ u ˆ − X X µν I X ¯ ˆ ′ µ − ΨΓ ˆ α> F ˆ i 2 A X ˆ IJ Φ I u µ µ µνλ + ˆ ∂ X ¯ ˆ D ǫ (for ΨΓ 2 ( µ , , 2 1 i ) 2 ∂ 2 1 I + αv +  is not produced in any 3-commutators. This property ensures ˆ − 9. This can be regarded as D2-branes theory in type IIA string X = 1 Φ := , ′ µ u − −  ] = 0 µ i ˆ A B ˆ D u D ( = = ,..., u, v 2 1 , T h L Ψ. As we see, fortunately, the ghost fields decouple, that is, gh A − , = 3 L I = X v, T [ L I, J The simplest Lorentzian Lie 3-algebra was defined as follows The origin of the decoupling of the ghost fields comes from the where Higgs mechanism in BLGintegration model over first considered in [ for Φ = algebra. We denote its generators as Langrange multipliers. Their equations of motions are and we can set without breaking any or gauge symmetry [ where system is invariant under the translation of the scalar field the Langrangian becomes (up to total derivatives) This 3-algebra satisfies the fundamentalso identities we and the can use itnorm as generator the gauge symmetry of BLG model. Since this a Lorentzian generators appear in the 3-algebra. Namely, the which is the compactification of M-theory on a circle. Now we define a Lie 3-algebra whose generators are the 3-algebra and JHEP03(2009)045 ) 3.1 (3.2) (3.1) (3.4) (2.20) (2.18) (2.19) ). We give the . For the a nerators ,χ u I µ , ,...,M C b ] 10 dimensional v and a b 16 ) i η . = 1 1 e µ − t the assignment of ∂ ) tive-norm generators t µ a, b eeping the metric ( Γ R ( i a ]. cture constant imply that = = (( , v rentzian 3-algebra invented 20 a st fields as we have reviewed. [ owing simple form a is given by [ u ˜ v for indices of the 4-tensor, for . pect to all indices. a , , ab u b δ I , µ v η, δχ a C b = I for u S ED = i b h X a v + µ E , v b Ψ ∂ a u ] by gauging this global symmetry. Namely = 0 (3.3) u a b + and h πR . ABC 19 R i χ f , e – 5 – u are in the center of the 3-algebra. This condition + , BCD , δ = 2 ¯ Ψ a i 18 a I := ij v e v λ for − δ Λ a i f µ i Q = = ∂ through i = = j ABCD new a ), and Lorentzian pairs as f I µ , e ,χ ˜ u i L µ e = 0. The equations of motion by variation of h ) invariant: C , v a and so on. 3.3 v ,...,N i b a , δC u = Ψ P a I I u v = 1 + j . Therefore the index in the 4-tensor is limited to e X i j i = Λ ( e e i j f I i v ) correctly. e O := δX = B,C,D triggers the compactification of 11 dimensional M-theory to 2.16 i ˜ I e u ijab ] by adding pairs of generators with Lorentzian metric. Posi f X 16 – We also assume that the generators We note that there is some freedom in the choice of basis when k Another important feature of the Lorentzian BLG model is tha 14 the invariance of the metricthe and condition that the this skew 4-tensor symmetry is of anti-symmetric the with stru res example assume that the invariant metric for them is given by the foll type IIA theory. The compactification radius of M-direction In terms of the four-tensor defined by ghost fields can be made more rigorous [ VEV to is necessary to apply the HiggsIn mechanism terms to get of rid the of 4-tensor the gho this condition is written as For various aspects of the Lorentzian model, see for example simplicity of the notation, we write and the form of 4-tensor ( 3 Analysis of LieIn 3-algebra the with following, two or wein [ more consider negative-norm some ge generalizations of the Lo by adding extra gauge fields are denoted as for arbitrary we have an extra gauge symmetry: It enable us to put assignment ( JHEP03(2009)045 , i ˜ e , it { i (3.9) (3.8) (3.7) (3.6) (3.5) e (3.10) ) mix. l, since ]. . = 2, one 3.6 describes 21 nontrivial . It comes 3.1 P M P . . abcd L abcd ijkl L F alysis: l c = P P . s can indeed be iden- = 0. For k t b anges of basis give rise = 0 and so on. asis. In this sense, the hich will be erased after n indices automatically )), some components of P P abcd a j a ). So it can take arbitrary − abcd f cohomology, namely only P , v abcd , f L a = , 3.3 or its direct sums [ + l L u b t n below in section l i abc c ) = P = P k S a K ijkl k 1 b , ǫ i P abc − i = P abc ij ab K R . J K ijkl ikl i a j c e f = abci F P + ( + ij + ab S + , f l c J 1 with ij ij ab P − ), there is no constraint on ac ijk a k J J a a f j b k P b = , v – 6 – P 3.24 P a ikl b u , − f − O, R )–( abij . So if we assume positive definite metric for t is proportional to ij bc − ijkl ijk J l b b ijkl , f F j a 3.11 f = 1 for simplicity and keep only the matrix RP i P ijkl a k F P a k ), the structure constant in terms of the new basis ) is identical to the fundamental identity of a 3-algebra ) vanish identically due to the anti-symmetry of indices. ijk − a a P R F P f P P + , t = + ), various components of the structure constants ( 3.6 = + ikl = c 3.11 P i ( abc f ijkl 1 2 ij O jkl 3.4 ab a f J K + F = 1, we need to put − aijk , Q ijkl = = = = describe the usual rotations of the basis. The matrix = M F , f = 1 ij ab R S jkl i ˜ 4. This term, however, is not physically relevant in BLG mode abc a ijkl J ˜ ijkl f O ˜ (i.e. smaller number of Lorentzian pairs ( ˜ nonvanishing but we have to keep K t F ≥ F O and ij ab M = M J O are given as } ijkl a f v , ˜ a ˜ u We will find thattified many with solutions of well-known 3-algebra theclassification fundamental after of identitie the such Lorentzian redefinition 3-algebrasolutions of has which a b can character not o reduceto to physically known new examples system. after all ch In the fundamental identity ( with the structure constant For example, if we put For example, for value for they appear only in theHiggs interaction terms mechanism. of the ghost fields w A constraint for may put from the fact thatvanishes the due to contraction the with restriction respect of the to structure Lorentzia constant ( By a change of basis ( (which implies For lower the structure constants ( automatically implies that We introduce some notation for the 4-tensor, There are a few comments which can be made without detailed an • • • • the mixing of the Lorentzian generators where We rewrite the fundamental identity in terms of this notatio The matrices JHEP03(2009)045 ). = 4 3.13 (3.20) (3.22) (3.21) (3.19) (3.24) (3.11) (3.23) (3.25) (3.16) (3.12) (3.18) (3.14) (3.15) (3.13) (3.17) N ) implies ), ( ume . ). It implies 3.11 3.12 ijkl 3.8 , , , , . , , , , , , , , , F . We do not claim = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 e fundamental iden- ) = 0 lj in ab 3.5 ab l m i k l abc abc ijn cde abe nkl abd a a J J mj ijnp jkln bc f jkim ijkm f K K K K K il J cd F F F F J ki cd ijl ijn ): jil jkln d b c i , it is impossible to construct abf J f f lm f ikm klmn m ab a − abc ijkm F 3 inm ry compactification. s we will see in the later sec- a K f klmn J 3.6 f + F f − + K F − lj c cd F − ) − ∂ + J )+ − − ] for example). The realization of − 2 k bcd l li abd nkl bd il − b ab f )+ ) i 22 bde b J f mk K J bc jkn K b kln a ∂ ijml mi J jl ab f ac K ki f ae 1 ijm + jknm c ijl ijn c klnp a F J J f J f f f iln F a a as a compact three dimensional manifold i ijm F acf a lij c − f ∂ km + ab f ijmn + − km jkm f ab K iln c a J N li bc + f J F f abc k J ijmn acd + nk l abd ab l + ǫ acd + + + + J jl ) + ( F ad jnl K b K K – 7 – i J bcf f ki 6= 0. The situation is totally different if the 3- be mi + ) in the notation ( ca X a,b,c + ijl ijln imkl b imk . This expression actually solves ( J K mjk c + b J knm a ikn f jnlm a a f l F f F = f lij 2.9 d − f li ijkl cd i F P ade − f } − jm im jkm J ab ac jm knmp b ab F 3 6= 0 first. As we already mentioned, eq. ( + K ijln f J J k cde jl ikn J ab a l . Then by the skew-symmetry of indices they can be F ] which is related to the description of M5-brane (for the abc nl l ab bcd ], ,f F f J 9 a − K J + + − 2 + nkl ( K b K ijkl , 23 ki ijln + + f ab 8 − ,f F mi ijl i def bc J a 1 F ijkn mjk mjk f c c J ijn f a mjkl − K is exactly the same as the right hand side of ( for some F f f nlm { + f a nklm ( F f i a abc ikm im im l a ab ab F ijk a im f ab J J P K ( f ( ( nlmp J ijkn ijn a F F f ijkl ǫ 6= 0 for some and its direct sum. So without losing generality, one may ass for the terms which include nontrivial contraction with ijkn = F ijk can be set to zero by a redefinition of basis. a ijkl f ijkl ijk ǫ a ǫ f ijk a ∝ f = ). Solutions which we found are summarized in section ijkl ijkl 2.9 F F Therefore, at least when the 3-algebra is finite dimensional In the following, we give a somewhat technical analysis of th Suppose 3.2 Lorentzian extension ofLet us Nambu examine bracket the case with algebra is infinite dimensional [ Lorentzian algebra with nontrivial However, this form of various aspects of M5-brane in BLG context, see also [ that our analysis exhauststions, all they the play possible an important solutions. physical role But in3.1 a string/M theo Fundamental identities We rewrite the fundamental identity ( that that such the three-algebra was given as follows. We take tity ( and where Nambu-Poisson bracket [ written as JHEP03(2009)045 . ) 3 ate T well- 3.24 , and . For (3.27) (3.26) (3.31) (3.32) (3.33) (3.29) (3.28) (3.30) = a p H )–( not N + . The fact a 3.11 y N ∼ a , y ) l ~ ned on : the Hilbert space + or example, the inner H s which are ra with the additional . If we write the flat . ~m 3 ) , and one can choose a , ) invariant. The simplest ) ~p T + N ~m + ~n does not matter whether or ( xplicitly we define the extra n l ~ . 3.25 t, + δ y all the fundamental identi- ( c ij erefore even if we include extra + we consider the case ~n δ ( m , b δ ~m 3 c i n l . The fundamental identity for the Z n + )= a y abc . y ,χ ∈ ~n ( ǫ abc ) ( j ǫ 2 o ~n δ ) as the basis of = ) ) χ k ( and these generators remains in c y ) l a δ ( πi b πi y i ,χ u , ~n H ( j χ i abc m a ǫ a y ,χ a = (2 n = (2 , i yχ = – 8 – ) 3 i i χ l ~ i πin abc d ~m ) satisfies the fundamental identities ( ~m n 2 ǫ ~n h e ). 3 ,χ N ) ,χ + ,χ Z = o 3.33 o πi ~n o ~m ( 3.11 ~n c ) := := δ 3), where the periodicity is imposed as )–( y ijkl i ,χ , ( ,χ , u j ]. If we take ~n b b F 2 ~n = = (2 , χ i 10 ,χ 3.30 ,χ , u , u l~p i ~ a a a ~m , χ = 1 u is covered by the local coordinate patches where the coordin u u 8 h ~n~m [ ,χ n n n a F h h h ~n ( 3 N is then given by χ is spanned by the periodic functions on S a h y := := := H l H ~ as ~n and abc ~n~m ab ~n~m 3 a ). J 3 K f T but the Nambu bracket among T ) is not well-defined if the function is not globally well-defi 3 3.24 T 3.26 )–( . The basis of 3.11 Z We have to be careful in the treatment of the new generators. F We are going to show that it is possible to extend this 3-algeb The idea to extend the 3-algebra is to introduce the function ∈ , such generators are given by the functions 3 a basis mutually orthonormal with respect to the inner produc satisfies the fundamental identity ( It is known that the structure constant which consists of functions which are globaly well-defined o that the structure constants ( product ( Nambu-bracket comes from the definitionnot of the derivative functions and in it thegenerators bracket the is well-defined analog globally. of Th structure fundamental constants identity as holds. More e is well defined. Namely generators with the Lorentzian signature. For simplicity, examples are It is not difficultties ( to demonstrate explicitly that they satisf T p coordinates on with the invariant metric and the structure constant: The Hilbert space transformation between the two patches keeps the 3-bracket defined on JHEP03(2009)045 (3.40) (3.39) (3.35) (3.38) (3.37) (3.36) M5 model totally anti- ] although it . that the BLG 16 , 5.3 9 5.3 . So far, the only , ABCD 1 8 = 0 (3.34) S F ] where a subalgebra et itself. The naive i b × 16 = 0 2 d in [ , v ant ebra. a S l identity may be violated v ijkl in section iture. It may be useful to h 7 in [ ons of Nambu-Poisson type Lorentzian extensions which the Nambu-Poisson bracket. F ely this turns out not to be ies the following redefinition , § and the Nambu-Poisson type ). Namely we introduce extra , a = and c e original Nambu bracket after v n , 4 i l ~ v = 0. The fundamental identi- b 3 3.1 b A v S ~n ~n~m abc , , a , u f a ~n~m as ( ijkl ab K u J F on the right hand side is necessary to − h = 0 = 0 − . ~p + a ~n n = ~m l v χ ~ ) χ such as ) ~p i χ χ mkl kl b l ~ b l ~ ~n f ~m f N j abc~n ~n~m ( ~n ,χ ~n~m ab a abstractly a a ijm F K f J . We have to come back to our original defini- ( ni a v – 9 – a f f h u = = = = was originally defined through ill-defined function = i i i i l ~ a c i ~n ~m ) is new, we will see later in section u ~n ,χ , u ,χ b ,χ b ~m ,χ ~n a 3.38 , u , u u ,χ a ,χ a h ~n u a )–( u χ h u h h h , ab 3.35 δ 3) and define , = 2 i , b ). , v = 1 a ) are now simplified to be the following: u 3.29 a ) appeared and the relation with the Lorentzian BLG model and h ( a 3.24 v 3.38 )–( )–( 3.11 We also need to be careful in the definition of the three-brack It is straightforward to obtain similar Lorentzian extensi While the 3-algebra ( 3.35 , we have to neglect this fact to define the metric and the 3-alg a symmetric in all four indices.the This introduction condition of is the broken extra in generators th tion of 3-algebra where this symmetry is manifest. This impl of the 3-algebra: ties ( Nambu bracket needs to be modified to make the structure const In the following, we restrict ourselves to the case implies that we can define the inner product 3-algebras. The examples we considercan here be would obtained exhaust from the them. 3.3 Constraints from the fundamental identities for of ( was discussed. We will give more comments on this issue later model based on it turns outwas to not be noticed the explicitly. same as A the glimpse M5 of models this define fact appeared i Lie 3-algebras defined on different generators This 3-algebra may be regardedThe as additional the factors “central which extension” are of proportional to by the redefinition oftrue. the algebra. So we Inrepeat have this our a example, emphasis consistent that, fortunat 3-algebra although with Lorentzian sign make the metric invariant. One might worry if the fundamenta y nontrivial Lie 3-algebra with positive definite metric are while keeping ( JHEP03(2009)045 ) → ) , we , We 3.39 α a n (3.50) (3.49) (3.48) (3.47) (3.46) (3.45) (3.44) (3.43) (3.42) (3.41) (3.51) (3.53) (3.52) ab δ g . ) = cbe α a γ B ) and ( ad ⊕···⊕ 1 jki A ( g − , = → ). It is also obvious g kj ) bde ab ) alone to derive ( J B ijk ik 3.40 cd ac . ). J 3.41 , A ) , − i abf kj − cd ij ab 3.43 es displayed above and find a eses are fully antisymmetrized. J kj K J cd b and let ( dce J ik a i i , cde cde n ik B . ab , , J , , , , I α . It is possible to have some of the K K J I ), we get ( ab ) and ( empty. In this case, for M A ∈ = = 0 = 0 = = = 3 = 0 = 0 = 0 = 0 is a real number. l , ) ) ) ) − 3.41 f il ) 6=1 ) ) ) α a kl 3.42 ij bc a k kij de bcd d mil k ∪···∪ k a jk ijk j a γ bcd cde c I α i de J J f bce f f 1 K f f i, j, k c c K j i K ( ( I K B a α ( ( a ijk b ( c l l bc ab ( i k ijk γ ab ( ab a a f ij ad ijk jkm a ( b = J J f i ab J f K f K while – 10 – A = α I 3 2 + ijk g K and all α + + 0 otherwise f 3 g 6= 1. ijk ) a 1 ljk c ( f g a f dbe mjl b b ). ( f = B ) and replacing the indices as ( blocks = il a ac J g n ijk kim α a 3.39 A 3.41 f f ) is to use a direct sum of Lie algebras does not have to equal + + n = 0 for all cde 3.39 mkl b B f ijk a ab f ) implies ( A ijm ( a ) can be easily derived from ( f 1 6 3.41 and 3.44 := 1 ijk g e ) f is defined by cd is the structure constant for ) and ( = ) as follows. Taking ( B ijk b α empty. An example has α ( ijk ijk f g 1 a 3.40 a f f 3.40 A I The constraints above are not all independent. We can use ( First, a solution for ( Similarly, ( Note that the number ) and subtracting the derived equation from ( ba In the above, we used theFor notation instance, that all indices in parenth ( sets have 3.4 Solutions In this subsection, weclass try of to solutions. solve the fundamental identiti and ( that ( Here divide the values of indices into where JHEP03(2009)045 , ) b 6= 3.43 c (3.57) (3.59) (3.55) (3.56) (3.54) (3.58) . ijk and a tric system f ), eq. ( l for this 3- a = 0, there is ) as follows, . In this case, = 1 after the . A derivation 6= a is the structure b I 3.57 ijk c g a f 3.50 ∈ M ijk a f . )–( and ients to be nonzero. k a . e i, j, k g . ) solves the constraints 6= 0 for ant vanish, one obtains k 3.39 b a e ijk , and α I I f generators such that the n f ijk ebra for c I ijk ∈ α α a f ∈ . f eed nonvanishing γ , we have )] α = 0 for j i, j k I k e i, j X X ( ( , , ijk 1 a a ∪···∪ D a I f or 1 ij , δ ]= i I J j ∈ e a , = , e ]= I a i are identical. Assuming ( j I v ] + [ ∈ , e α a j , e . However, since we put i, j, k a ]). We will study the BLG model for this i . In general, if γ J b u , e [ I , e i, j ) 11 a i a, b, c a ′ are different. If two of the indices are the same, e X ∈ – 11 – ( u , ijk i [ a = a D 0 otherwise f ). v 1 or ′ , ( α a a, b, c v disjoint sets a a γ ′ I 3.41 = ]) = [ v n j a ∈ − X )–( ijk i a , e i f − e if is a derivation for both Lie algebras ]= . ([ 3.39 k a ij ab ) with ab ]= g D . , e J k J 6= 0 only if j α ), we can solve all the constraints ( such that I 3.4 , e ). ), ij , e ab j i g J e . Hence we consider the case 3.57 [ , e c i to 3.42 I 3.42 e [ g , arbitrary anti-symmetric matrix . It demonstrates the essential feature that the supersymme ∈ a 4 i I ) is simplified as ) is trivial if two of the indices ) already solves ( ) a set of solutions to the fundamental identity. The BLG mode 3.52 3.43 ) is now trivial if all indices =0 if 3.57 ij 3.52 ab J 3.44 According to ( Starting with ( Eq. ( If all the other components of the 3-algebra structure const In order to obtain somethingThe simplest new, class we of have solutions to can allow be other found coeffic when For simplicity, let us assume that there is a suitable basis o is a map from where the indices are divided into it is equivalent to ( constant for a Lie algebra D acquires mass proportional to eigenvalues of imposes no constraint on while ( case in section from ( algebra is not new, however. For each range of index, say we always have Therefore it is reduced to the standard Lorentzian Lie 3-alg restriction of indices to for this range By a suitable rotation ( (this case is a special case of solutions in [ then no interaction. In order to have the interacting system, we n Eq. ( solution ( JHEP03(2009)045 ) 6= can b 3.45 ).) In (3.63) (3.67) (3.61) (3.65) (3.60) (3.66) (3.64) (3.62) 6= i abc ) to be d . K c 3.61 I = 3.45 ∪ a and b corresponds to I . However, the ab are different. If a ∪ g J , ijk a a ). If e f I abc instead corresponds . on k ac C ∈ ab Λ 3.58 i abc ac i . Since all derivations of D j ab C g ]+ a, b, c, d, e ab + . is an outer automorphism. c 2 D ] satisfies the Jacobi identity. and Λ I , , ). . ijk ab c a cd cb ∪ f J ms is quite nontrivial. We will come I ) = 0 (3.68) ab can be solved from ( D b D k cb = 0 , I ∈ abc Λ , are not summed over in ( i i ab abc k ij ∪ by ac C j ca ( ] + [ D J ij a b K ), I ab f a, b ad . ca ), i j ) can be easily satisfied if adb D for + Λ k ba 3.9 ∈ D D ) vanishes unless e , K i Λ . ) is trivial due to ( ij ) and the constraint ab i ijk b ba b ), ( 3.61 J 3.47 e + − )+ v f 3.62 D ( i ba k outer derivation ij 3.45 i ab ab k ba 3.8 3.62 is a central element in adb ij Λ a J )= Λ Λ – 12 – i C f i abc ] is an inner automorphism. The solution of ( ] + [ ( e i j bc b abc i b acd ( k ab bc K X ) is trivial if all indices ac ac C X ab K D D − , J + Λ , = − D = Λ + a i 3.47 ab ]. It will be more interesting if corresponds to an inner automorphism, so ac e u J ijk · ij )+ a ij ab abc ad D ab f , J are different, ( J C = = k J acd k ac e i = [ =0 unless ′ a i ′ abc ) C Λ i ), eq. ( e u ( . (Note that the indices e k ba K j ab i a abc ab are all different, ( Λ I i abc C D 3.66 − ∈ = Λ K a, b, c, d, e k k ab i := abc ), this implies that ), one can define a derivations K a, b, c, d ) and ( abc 3.58 3.42 K are not summed over in this equation.) The term Λ ) = [(Λ · 3.58 ( given by an inner automorphism ( generating an inner automorphism is not interesting becaus ab = 0 unless ab D ab a, b, c J k ab J ) says that the Lie bracket [ , it is The simplest case is when If all indices Due to ( For When all indices a We have to keep in mind that the existence of such automorphis 3.45 2 = , ( be both set to zero after a change of basis ( the Lie bracket of the two automorphisms generated by Λ case of (Indices back to this issue below. One can then check that this follows from ( where Λ e As a result of ( Therefore, in the following we will focus on the case when is in general given by this case Together with ( where the antisymmetric tensor to an infinitesimal outer automorphism (an c a Lie algebra is always a Lie aglebra, the Lie bracket [ JHEP03(2009)045 (3.75) (3.76) (3.74) (3.77) (3.78) (3.70) (3.79) (3.69) (3.71) (3.72) (3.73) is an infinite er derivation g , abc C first example is when ]+ of matrices composed of ab he generators is an outer . , mal outer automorphisms. ) D c , I a, the general solution of the . cb 5 ∪ , 3.68 ) = 0 e is found when D ’s. The same discussion applies d b , , I v , b a ab c abc , k δ v ∪ D C e ] + [ ab ( abcd a = i abc ca D I ijk ) and ( L a ad i f D b , K ∈ D , a + i , v + v , i − ba 3.66 a e b j )+ ! u D ijk e v a matrices, respectively. An arbitrary scaling h i abc f ij ij ab ab C adb n , J K J − , a ] + [ C , – 13 – 0 A B I × ( bc ij or Ψ do not participate in interactions in the BLG ac

n ∈ g ] = ] = ] = ] = D i I c j , k D ), which are repeated here for the convenience of satisfying ( = , e X ac , e , e , u b i i g b j and D j )+ i, j, k 3.62 ). , e , u , e , u n , e a i a a i acd = [ e u u ) for a derivation [ e =0 unless u [ × [ i i C [ h 3.50 is an outer automorphism. In both of these examples, the e ( e ijk ( a m i abc ab 0 otherwise f i abc B ab , ) and ( C D K ( D m ) and ( × 3.60 = = := j m e 3.49 ), ( ijk abc a is inert to the outer derivation. Hence the appearance of out ij in the expansion of ab f K i i J are ), ( e e 3.57 are central elements in 3.48 abc ), ( A,B,C C Before closing this subsection, let us comment on infinitesi 3.46 of the off-diagonal block coefficients of where as a result of the Jacobi identity of the Lie bracket of model, unless dimensional Lie algebra. This example is studied in section The nontrivial part of the metric is given by the reader, 3.5 Summary ofTo the summarize 3-algebra the solutions result offundamental our identity construction for of our a ansatz new 3-algebr For finite dimensionalthe Lie Lie algebra, algebra we isautomorphism. have Abelian, The two 2nd and examples. example anyupper is triangular nontrivial when The blocks linear the Lie map algebra of is t that to ( in these cases is irrelevant to physics. A nontrivial exampl where is given by ( JHEP03(2009)045 , ), is n g ij field (4.1) ). In g + 3.27 B encodes · · · . 4.38 g abc + 1 K ): This is the g 5.1 = g 6= 0. We will first ): By this general- ij ab with constant 5.2 tzian symmetry is dis- result by studying the in the form ( J examples of Lorentzian ): This is the simplest ra is given by the affine p r of ghost fields. It also ted by the ghost fields. ntzian Lie algebra. (section T 4.1 en lluminating, hence we will tion ]. For this simplest choice, norm fields. After including re constants to zero, erivations), and 6= 0 the 3-algebra can be directly 11 ly positive definite. he direct analysis of interacting ij ab ): This is the simplest nontrivial ry, in the following. ], the 3-algebra constructed above J . 4.3 24 ): We give a brief explanation how ] can be related to the analysis in , 16 = 8 supersymmetric vector multiplets. = 0 5.3 16 , , . Although we have assumed that N 9 i abc g 14 , = 8 supersymmetric massive K 8 = N ’s and choices of central elements in ), others= 0 (section – 14 – ab ijk a J f ] can be related to the Lorentzian 3-algebra ( 6= 0 (section = 16 ’s are generators of a Lie algebra i , ,...,n e 9 ijkl ijkl , F 8 = 1 F 6= 0, others= 0 (section ) where the gauge symmetry is defined by Lorentzian Lie i, j ijk 1 ( f 2.17 , ij ij J J ab ǫ ab ǫ = ) and how the analysis in [ = ij ab ij ab J J 3.33 )–( is the Killing form of the Lie algebra = 2, = 2, ij g 3.31 M M ( construction of M5-brane [ special case of above exampleLie where algebra. the It Lorentzian Lie illuminates algeb how Kaluza-Klein mass is genera ization we describe the compactification on general torus this paper. algebra. This is possible since the 3-algebra can be written example which contains the interaction. We will present our finite dimensional example where someplayed. character Namely of the BLG the model Loren defines the Yang-Mills system ( such case, one can skipilluminate the the structure discussion of of the eliminating Yang-Mills one system pai with Lore flux on it. Based on this analysis, we will analyze the BLG model for some Compared with the 3-algebra discovered in [ ’s correspond to infinitesimal outer automorphisms (outer d 1. 2. 4. Lie 3-algebra associated with general loop algebras (sec 3. Lie 3-algebra associated with affine Kac-Moody Lie algebra 5. Lorentzian 3-algebra with ab 3-algebras: both the commutation relations among J 4 BLG model for Lorentzian 3-algebra with In this section, we describe generic features of BLG model wh after the Higgs mechanismother is structure used constants, we to have eliminate anmodel the interacting from theory. negative- BLG T model itselfconsider is its somehow equivalent complicated version, and the less super i Yang-Mills theo where We note that this is the simplest example considered in [ we see that BLG model gives rise to a free contains more information. While start with the “minimal” choice, namely we set other structu positive definite in thegeneralized derivation to above, a it generic Killing is form obvious which that is not necessari JHEP03(2009)045 (4.4) (4.5) (4.9) (4.8) (4.3) (4.7) (4.6) (4.2) (4.11) (4.10) te in the ). For sim- a , v a , u i )) e λkb , ns are I = ( j B , . α . ) , A X jk I j a , C J ij v a shing part of the structure =: X ν ˜ I a v Λ J A, B a µ ij C µbi X Λ C B . + B µ C ) + ij = 0 as the equation of motion. +Ψ ˜ + ij A a ] J a I λjb b λjb u µ u + I a B B X + C a ν ν ν ) becomes , X I . [ b ∂ ˆ , J C ∂ µja ( D , ij X µ A ab + ia 2.7 J +Ψ B ǫ ˜ ′ µ i A β i µia µia µia µ e ab ij , A e 2 1 C ǫ I i i ( ′ µ B B J B C B 1 2 2 1 ij ij A X ab ab = ˜ Λ ǫ =: ǫ J J – 15 – ij 2 2 1 , 1 ), with the indices , a = − =: =: =: b + + , I B B u iu i I a I I A b − i − a I a 2.6 a Ψ=Ψ Ψ X Λ u u B µij λ λ X X X a X A A f ˜ µu Λ µ µ µ C C A )–( ) can be rewritten in terms of the component gauge B B . µu µ ∂ ∂ ∂ ν ν A ij ˜ ˜ ∂ Λ Λ ∂ ∂ A J − 2.1 2.6 ′ µ ′ µ = = = γ, = = = i A A = a a ij 2 1 ) ( ( v u X I A A ,...,n I a ) ) I B I Ψ X =: µνλ µνλ µ A µiu δ δX ǫ ǫ µ X X ˜ = 1 b A µ A µ u D δ a ( = D =: D u i, j ( ( Λ appears only in the Chern-Simons term. It does not participa CS ′ µ L A 2 and , = 1 a, b The Chern-Simons action ( In the original BLG model, the gauge symmetry transformatio As usual, we expand the relevant parts of the fields as dynamics but only imposes the flatness condition The gauge field and similar expressions for Ψ. 4.1 Component expansion The BLG action is defined by ( fields as We introduce the components of the gauge parameters as In terms of the modes, the covariant derivative ( where plicity, we first studyconstant the of special the 3-algebra case is when the only nonvani JHEP03(2009)045 (4.16) (4.13) (4.12) (4.15) (4.14) (4.17) associated with µ ,C ] or the introduction ǫ, ′ µ ǫ, 16 A ns are ǫ, becomes IJK IJK = 8 SUSY system with supersymmetry, since the Γ Γ A IJK K γ, b N Γ X K . µjb j J BCD j 2 B X f , . X , , J ij b I K i D j a ) become J X µ X Φ X Φ = 1 = 0 I a ˜ − J ij A ij C jb a a X J µ β , X Ψ γJ ij C ab A I ab µ B J ǫ ǫ jb ∂ − 1 2 X ab µ ij β a ǫ 1 6 J CDB gives Γ ij = 0; . , + 2 1 Φ f J a − ǫ − ij – 16 – jb Ψ ab , D I , + J ǫ ǫ b , ] + Ψ β b j b I Γ I ǫ Ψ Φ ij ǫ, µ ib I Γ I Ψ Ψ Ψ C I J X i = 0 γ , ab α, β , µ Γ I I I Γ i a [ , Γ I a µ µ µ X a , µ ab Γ I a A µ v I X X X ∂ ǫ αǫ ∂ ∂ λ I Γ ) A I I I Γ X Γ Ψ I i I 2 Γ Γ Γ a I µ = X ]: ======0 X , we solve them by the assignment [ µ µ µ X ∂ µ µ Γ Γ 2 I i X a a a µ ′ µ µ ¯ ¯ ǫ ǫ Γ Γ Γ 19 µ D i i D ¯ ¯ ¯ ǫ ǫ ǫ Φ µib X Φ Φ , ∂ i i i D δ δ δ δC δA = = = . So, in terms of the components, the nontrivial parts of the 18 δB = ( = = = = = A I A A i a a ′ µ µ B Ψ Ψ µib Ψ Ψ µ A δ δX δ CDB δC δ δ δA ˜ are Abelian. A f δB δ CD α, γ Ψ. The gauge transformations for the gauge fields , I = Λ X A B ˜ Λ In the original BLG model, the supersymmetry transformatio By the definition of the BLG model, we obtain an As already reviewed in section the parameters where Φ = Then the gauge symmetry transformation in terms of the modes of extra gauge symmetry [ where supersymmetry transformation (namely, for Ψ and It is clear thattransformation this of choice these does fields not is break closed. gauge symmetry nor ghost fields. 4.2 Elimination ofVariation ghosts of the Lagrangian by fields JHEP03(2009)045 , I a is a X (4.28) (4.18) (4.26) (4.19) (4.27) (4.23) (4.24) (4.22) (4.20) (4.21) (4.25) . The IJ is fixed P P ia β = , it does not 2 , P µia µ kb into two parts: the remains the same B B } , i e ab , i . CS { µ kb ne the relevant parts Q Ψ L 2) B 2) and µ µia . The matrix jk , ociated with ) ˆ ab . B D ↔ ab 2 J ian µ j δ Q grangian. J ation for Γ hese VEV’s: = 1 ( j X i jk ) , Ψ ¯ bJ Ψ a 2 = (1 µja ) =  ij i π 2 J 2 b B J ( ab λ ~ k λjb + ǫ , . Γ a µja 2 ij i B = 0 ( X remain massless. Actually the latter ) ) , ~π ~π ν ¯ B I b Ψ ) 1 1 2 a = 1 a ˆ 2 D λ − λ ∆ ~ λ ~ 2 λ ~ J + ) i ) 2 , X P µia k µja i 2 2 ) 2 1 ) λ B ~ − B 2 (Γ J become massive after putting VEV to · := ( ij ij J λ ~ j + , 1 I X J , , J · is invertible. We focus our attention on the latter part. ia 2 λ µ ~ j X , µia J 1 2 a , namely ( j 1 X ( µia ab J – 17 – J b ˆ ) a π Φ D ǫ λ ~ IJ I a λ B − ( λ ~ ( δB − ij 3 νλ I a P λ , , β + λ J 1 I i IJ λ F − + 2 µ I , ia i ( λ C ~ P 2 X ab 1 2 . µja β i ν | C | 2 ǫ X ) =1 ) X µ 2 ∂ µia µi 2 ′ B a µ I b ′ µ + ∆ λ ˆ ~ IJ λ ~ B D | B ˆ | λ ~ i D X ij − A 2 Γ ( , ( ij | ) µ ′ Φ 2 = = 1  2 J a ˆ µ 1 i 1 IJ becomes D λ ~ J ∆ 2∆ λ ~ ∂ | δ ( X ( ( µνλ ′ ′ µνλ µia µia X 1 1 2 2 i ǫ ǫ bb B := := := := := X 2 1 ǫ is the mass term for the gauge potential + − − δB ′ i k 2 1 2 1 − IJ aa X ~π Φ Γ = ∆ ǫ − µ P L = ˆ is not invertible, one can first decompose the linear space L D to give the action for = ′ B := ij 2) is the dual basis of L X J , is trivial and the part on which X ab L L ij Q J = 1 and a ( CS a L ~π The Lagrangian is simplified considerably after inserting t To see the mass term for gauge fields more explicitly, we combi After this gauge transformation, the Chern-Simons Lagrang If the matrix 3 part on which where potential implies that six components of projector with codimension two which satisfies from The second term in can be removed by redefinition of while the two components in the plane spanned by The Since this redefinition takes the form of the gauge transform change the form of Chern-Simons term. The gauge symmetry ass while the kinetic term for by this manipulation and will not survive in the gauge fixed La JHEP03(2009)045 (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) (4.36) (4.37) (4.38) in the γ . massive scalars µ ∂ n compactification of . = , 6 2 µ  2 s, 2 T µi j ǫ, µ B k th B Γ 2 ection. ). A different point is that J a-Klein modes. This feature α, δC µi Γ µ , B which appears in the covariant . ∂ 1 ) transformations: IJ 2 . 2.11 . This algebra is similar to that v µ P ) . R ij ∆ = I D j 2 , C = 8) after the Higgs mechanism, 6= 0 R J ij , ′ µ X ∆ , ) AB ij C 2 2 N ij f := vector fields J J T J ( C , ij SL(2 1 2 CD 2 , J ij h u AB , γ, δA +∆ ∈ J + f j f − ǫ 2 = 0 ab J Ψ µja 2 . The mass spectrum of this supersymmetric k J Γ i i massive B ) – 18 – ] = 0 ] = ] = µ ijk and 2 , X ij C i B B Γ n , g νλ J . In the second term, we used partial integration. b IJ F Ψ ijk IJ 1 λ ~ − , T ( , T , T ja , we integrate over them, and J P , f f P ) 1 I A A B ab Γ I ν i = g Γ µi = eigenvalues of X B := IJ µ X , T , T , T 6= 0 2 B λ = 1 1 µ Γ 4 1 . A µia v ¯ ¯ ǫP ǫ ˆ u ′ a m [ D [ ijk i D i T ijk − 1 ( [ λ ~ f f  )-extension of Lie 3-algebra ( = = = = 0 , − 1 } i fermion fields Ψ I i 11 ′ µ µ , δB ′ 1 , v ja j , i 6= 0 Ψ ) Q 1 n 2 δ Φ δC δA λ u δX ij , v ijk a B → 2 f ν 6= 0, one may include interacting non-Abelian gauge symmetry u γJ ˆ B D { and 8 − L ijk a I i = ) f = = ( ′ i X ja Φ ) δ ], that is, a ( νλ =: ( F 16 A,B,... J i – The gauge symmetry is now reduced to Abelian transformation In the end, we find that we have The original supersymmetry remains the same ( X 14 IJ where ( They are, however, mostly trivial since the gauge field Since there are no derivatives of derivative is required to be flat by the equation of motion. P We note that this mass formula is invariant under SL(2 system is given by This property is natural if we want to associate the system wi M-theory, so that the mass spectrumbecomes corresponds more to explicit the in Kaluz the example considered in the next s 4.3 Inclusion of By turning on action. For simplicity, we set In this case, we can rewrite it as where of [ JHEP03(2009)045 . } 2 , 1 ] is a (4.39) (4.40) (4.42) (4.41) (4.43) (4.44) (4.48) (4.49) (4.47) (4.46) , v 16 2 , – 1 14 , u i , instead of . T } { i , u, v i µν e , F { µν = 1; F i i ] h ie algebra and Lorentzian J 2 1 . gives an infinitesimal outer ie 3-algebra j v, 1 λ , , u, v e 4 h J , X ij = I ′ J ra. So, let us consider the Yang- = 0 = 0 X i− [ otherwise = 0 lj , ) and spinor fields Ψ based on this ] ]= Ψ] J ljm , v i J other commutators = 0 ; (4.45) , ],) BLG model with Lorentzian Lie i f I e other inner products = 0 kil i 16 , , X u, e f kil X α [ [ I ,...n k f A I + α v, − X Γ L k [ I + v li , h u = 1 ij v, + J ¯ X 2 1 Ψ I 4 f ij h = lim λ ( jkl + 1 J ′ int f – 19 – = 2 I f + L u iλ + I jkl u i ij , X + j + I has Lorentzian generators, while that of [ f J + X e lk } , k v, u Ψ X i , + i = 1 2 + J µ ij ij L ij Ψ α i , (according to [ i ′ f J δ , v µ ijl 2 e + 2 lkm , D + I i f D , v I = = f ′ i is an inner automorphism µ , u X ]= e i u pot i , X j Γ ijl h j k ij µ uij , L ′ T f = = , e e J { , e ¯ i Ψ D i i k + ′ , h h e I e [ e = ij i h ij 2 2 1 X , f . One may then redefine the basis X f δ } + L − k A ijk = α f T ]= . The Jacobi identity is written as = i =: { j j ′ ′ L , e , e i i ′ ′ e e [ h ,...,N . Then the metric (or Killing form) and structure constant is takes the adjoint representation } 2 I = 1 , v i X 2 In this subsection, we denote generators of this algebra as As we explained in section , u i T This is the simplest “Lorentzian extension” of Lie algebra, where This extension is trivial if which are consistent with the fundamental identity for the L for some parameter such that the algebrapairs: becomes the direct sum of the original L { this Lie 3-algebra where Mills theory coupled with scalar fields 3-algebra results in super Yang-Mills theory with Lie algeb standard (positive-definite) Lie algebra. In the following, we will focus on the nontrivial case where extended algebra: automorphism. JHEP03(2009)045 (4.50) (4.51) (4.61) (4.60) (4.58) (4.62) (4.59) (4.63) (4.56) (4.52) (4.57) (4.53) (4.54) (4.55) , namely i conditions e ]. In analogy, 25 . I u ) I j X X µj µ µi plays the role of a gauge A C ji A 1 j j ǫ J ij X ji orientifolds [ J , + , J 1 responding to the gauge sym- − I j , 2 X µu − u I v X F . Φ , µ ) X 2 1 µk i j i µ C ǫ e 1 λ e ∂ 1 2 ji i ji µi ( γA I J J u X , . A ki − j I u . X J − − 2 e + − µ , ) 1 , X I ] k k ′ ij I u + ∂ u I I u i j δ Φ J µj X µj D 2 D ˆ µ + D γ X X µk 2 A µ λ µj , µ B 2 λ ki A µi ji γA µ C ) ( ˆ A )= j – 20 – D = J I i i 1 J 0 ji ˆ A ǫ ( D =: e i − λ I u : J ij jk ( X + + is used to imply the adjoint action of µ =2 jk n J u f J ′ X i k µv + µj ∂ X f I ) ) as the derivative of a certain noncommutative space i := [ := D Φ j I − + e A J , 2 1 j ) to be in the direction of = u + I I I ǫ ǫ u i v ji X µu i i ˆ u − µ ), µ D , A µ J ǫ = 0. So we take it as constant as before, F X X X ˆ (or ˆ jk µ µ X , ˆ D D µ µ µ = I D u + f ∂ u ∂ ∂ ∂ C becomes 4.54 I X i X D 2 I = = 0 = =: ( = = = L X =: ( =: ∂ µ i X i u v u µi ˆ ) ) Φ ) D µu Φ I I ( should thus be defined as I δ δ A δA 2 1 i . The situation here is reminiscent of the result of quotient X X gives X e u µ µ µ I v = X D D D ( X ( ( X L ]. The gauge transformation is written as Ψ. , ,x is the derivation defined by I i e u X D The kinetic term for ) = [ x ( i in the context of Matrix Models in dealing with and in the direction of where After imposing this VEV, for Φ = The variation of We are thus led to interpret since we have taken the VEV of e and similar expressions formetry Ψ. generated by The covariant derivative cor where On the right hand side of ( JHEP03(2009)045 u 4 . ˆ sm. D 2 . By λ to be u 1 (4.72) (4.66) (4.64) (4.67) (4.69) (4.68) (4.65) (4.70) (4.71) λ . This is ν 1 C X d thus and Ψ in u J , X tion. it is not present ) jµ lobal symmetry of ) with ∆= A ν , to absorb C 4.31 .

j , 2 v − ) µ µ ′ µν − J C ”-direction on the D-brane jν F u = X A ) becomes 1 u j µ n that for the compactification ˆ µνu X C D ( ( F . ′ 4.61 ij J j =2 + n ′ J , X . 2 J X λ ′ ) + 2 1 C J µj i . ν µνi A νk . X -direction in the transverse directions. ∂ − F ′ µ u ij µi A 2 ( νj ] ) = 0 A ′ A  2 µj ] A , one may shift J ν ij 2 1 J j A ) µi ( i µ 1 λ µνλ C 2 i – 21 – ǫ , X 4 A µ jk ′ =2 [ J n jk ′ I ij ( f ∂ − X = f J 2 2 J µ X 2 ), we have slightly different Lagrangian, 2 2 − [ λ = C ∂ = ′ ), where we gauged the translation of − λ 2 i , can be redefined away if it corresponds to an inner automorphi A 2 1 ki µi =2 − µ ′ µ J λ n 4.18 L J µν A ,J B X C 2.18 ′ ν ν ν I − with exactly the same mass as eq. ( ∂ , F ∂ = 0, the second term in ( ∂ 2 1 ′ 4 − 1 J i λ µν − − gives a free equation of motion for F X ν νi X ν h = µ A C B 2 1 B is the projection of the µ µ µ 1 λ ∂ ∂ ∂ pot 4 I u using the gauge symmetry, the last term above is simply L becomes topological and pure gauge. Hence we should set X = = = 1 − i µ X that of a covariant derivative on a noncommutative space, an C , analogous to ( µνi µνv µνu µ is indeed the covariant derivative in the compactified direc ij , F F F J ′ µ C u A ˆ D is an auxiliary field. From the viewpoint of the SYM, although ′ µ is an inner automorphism, i.e. A J If we fix the gauge by The potential term is The kinetic term for the gauge field becomes If 4 mimics a covariant derivative. We willon see a in circle, the next sectio consistent with our comment above that where potential and It gives the mass term for This is the mass term for vector bosons. from the beginning,translation one of can add this term as a way to gauge the g variation of where a constant. It can be interpreted as the projection of the “ If we gauge away Variation of gauge field If we start from the BLG action ( worldvolume, while JHEP03(2009)045 . ij J (5.3) (5.2) (5.1) . We 2 (4.76) (4.75) (4.79) (4.78) (4.74) (4.77) (4.73) λ h their 4.3 1 λ s. we consider ries 5.2 , i , . ′ i imensional gauge group I j , Ψ] , Ψ n ′ the mass matrix X u , + I , in section j ij ˆ c D ) m an extension of a Lie algebra, r a simple case where Lie 3- X Ψ 1 e M2-branes in flat spacetime 2 ory studied in section [ T ′ c ij Γ J I i ( J = 0, they become ab ′ Γ ¯ ) Ψ I i , 1 u i if , 2 ¯ Ψ X Γ , , h i ′ µj 2 2 + i + ( g 1 i λ A i n 2 2 Ψ 2 1 ] ¯ itself is a Lorentzian Lie algebra. The iλ Ψ + µ ′ µi k λ 2 , ˆ m A D G Ψ + λ δ A 2 + , µ 1 ij 2 , ′ L ) ] ab 2 Γ λ I j ′ a 2 ) m , ′ J + X J I ¯ i − , Ψ [ ( – 22 – ′ h mT mvg 2 2 I X int , X Ψ i 2 2 ′ µ λ Γ µ L I = , ˆ ˆ D i = D − ] = ] = 0 X ( + i = 0, [ ¯ µ Ψ b a a 1 2 Ψ n m m 2 1 h 1 Ψ 2 µν 4 . L 1 λ − ¯ L , T ΨΓ F X 2 J i v, T 2 1 iλ 2 a u, T 2 m [ + =2 =2 1 λ λ ′ ′ T n n 1 4 h =2 X ,J =2 ,J n n X X is the affine Lie algebra ˆ ′ ′ ′ ′ λ X ] = [ X I L − I I I ) where Lie algebra G ======that the SYM theory with the gauge symmetry generated by the v, u [ via Kac-Moody algebra 2.11 A Ψ L X int int L L 5.1 L L L -dimensional (noncommutative) torus with background field d + 1) = 0, the second term becomes the mass term for the fermions wit p 1 ( i X to D p To summarize, in the gauge On the fermionic parts, after setting the VEV to Ψ simplest example is when which is of the form of a massive super Yang-Mills theory with consider the Kac-Moody algebra as anand example show of in the Lorentzi section on a base spacethe of BLG higher model dimensions. with thecompactified Finally, full to on 3-algebraic a be structure complete to describ Kac-Moody algebra is equivalent to a SYM theory with a finite d In the gauge masses given by the matrix 5 Application to toroidalIn compactification this of section we M/string first theo consider an example of the general the and algebra is defined as ( Before we go to the general discussion, let us briefly conside 5.1 D JHEP03(2009)045 ]. (5.5) (5.7) (5.8) (5.9) (5.4) 28 (5.10) (5.11) . The ,...,p i is Hermitian. ] has a positive- J µ = 0 a n ) A ] (5.6) , X T ν I a,n I ( µ,ν spective in which we X , A [ X can be identified with , µ 5 ] µ ; here try. What we are going A J g 2 [ i nC -brane system and provides rator. On the other hand, , X − p I − strength are given as n. µ in this subsection) as ) . X [ A m dy symmetry was obtained in [ h u, v, ν − v . ) is the spinor field. Both are in µ 2 I v ∂ = 1 v 4 x λ C c,n , i X I ( − I u + + ν X + +Ψ X ) i v u, v A ) I u h u µ µ I u u b,m ∂ X B a,n ( ( X µ µ , is the Killing form of a compact Lie algebra µ , + n := A + +Ψ a + , D n ab Ψ] a a I n n g T m , inA µν m ) T – 23 – T δ I X . X ) ) where we use the super Yang-Mills system on D2 I u g + a µ ab in the ). While X a,n ) [ g ( X and a,n bc a,n D I I 0 ( ) , F ( µ h 4.3 f a,n = L Z A X 2 1 ( a,n have a negative-norm generator. Φ] ¯ ) i + ΨΓ ( − , ∈ b I n µ = = , µ 2 I I iλ an u i− X I µ , T A u, v Ψ=Ψ . The world volume index is given as µ [ is the center of Kac-Moody algebra and usually taken as a X X a i µν inA A m g X ˆ µ µ n,m D v T ∂ + ∂ − Ψ+ h ) are the scalar field and Ψ( , F ), µ g Φ = = µν =: ( D by using the Higgs mechanism for one Lorentzian pair. µ -brane system whose gauge symmetry is ˆ µ F ) ∂ u h g p ) 2 an dim( I ¯ ( ,...,D ΨΓ 1 λ ) i X 4 2 I Φ := µ − + = 1 µ X gives the level (or ,..., D µ ( D I u = D Ψ. The convention here differs from that in section ( = 1 ) ( L , x I ( I X X a, b, c + 1)-brane system with Lie algebra We start from the action If we start from the BLG model directly, we have a different per In fact, the following analysis can be carried out for any D We follow the method in section We note that a different type of Lie 3-algebra based on Kac-Moo p 5 . This algebra has an invariant metric will treat more general argument given in the next subsectio D( for Φ = covariant derivative and the field strength are defined (only definite metric, the generators We note that thequantized generator c-number. Herethe we generator identify it as a nontrivial gene g where with gauge symmetry ˆ a general mechanism of theto gauge show theory is with that affine gauge the symme D where We consider the following component expansion, the adjoint representation of Various components of the covariant derivative and the field JHEP03(2009)045 ) et . We gives 4.71 (5.20) (5.19) (5.21) (5.18) (5.22) (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) 2 µν D . We also R , , ) m iny/R − − e c,n . ) ( , , ν x ) # ( A m ) gauge fields as fields in ) = 0 e paragraph after ( . − 2 µya µ a,n b, ˜ ( ( components, the equations b,m F C ( , ν with the radius 2 Ψ µ ν the additional action is 1 ) µ A λ , ] as reviewed in section ∂ 1 ) = 0 µ ) A ) disappear from the action, and Ξ u, v µ m S C m + 19 − X v − C ν m a,m , − 2 ν ν ( X ∂ Ψ ) . b, µ ∂ = I , ( C I a c 18 ) )= − u µ A ˜ µ bc X − X ν ) Ψ f ab ν , x,y µb x,y C ( I v ( ˜ C + µ a,m , ∂ A a ( µ ∂ ) ya a img µ ( ∂ ˜ , δB ˜ Ψ , X bc ( A µ I A u a,n = 0 µ µν f ( m 1 λ Ξ , X ab µ ∂ u , ν D , X − ∂ A Ψ 2 – 24 – + µ → = I ν a , 1 λ ) µ img − ∂ µ ˜ u 4 can be rewritten as X iny/R ,B , ν B iny/R C µ Ψ − I µ − − ν m ν − Ξ ∂ µ e x,y ) X ID C ∂ ∂ ( ) e X ( µ ∂ δ = 1 ) ′ x ∂ µ D + a a,n − − ( x − =1 λ ( ) ( ˜ I I X v ν ν D ν = X ) = 0. The equation of motion by the variation of " =Γ X A C B a,n Ψ. From the kinetic part for are free, ( is essentially flat, we can ignore it for simplicity (namely s is introduced to parametrize a,n µ µ µ µ µν µ µ I I ( u µ µ ∂ ∂ ∂ ∂ . µ y B D πR A X dy X additional C µ δD 2 C µ S = = = = C ∂ m m µ X X ) Z v v u ) ) and ) ∂ = const. =: 1 2 I a,n u ( µν µν I u ) X ) = ) = − F F µ X ( ( , Ψ µν D u F x,y x,y ( ( ( ( X I a µa ˜ is a new field. It gives rise to a new gauge symmetry, X ˜ A µν D We identify the infinite components of the scalar, spinor and Since the gauge field = 0). After this, the ghost fields + 2 dimensions, µ C where an extra coordinate the flatness condition of the system is unitary. p rename by which we can gauge fix to derive the last one. For general world volume dimensions, need to introduce the extra gauge symmetry as commented in th The kinetic term of the scalar field where and similar expressions for We fix their values as of motion for For the first two relations, we need to use the method [ JHEP03(2009)045 (5.27) (5.25) (5.28) (5.24) (5.30) (5.23) (5.26) (5.29) In the 6 to give the should satisfy . i ] ) except that I ˜ I Ψ J y X ˜ 5.5 X D , y I ˜ X ¯ ˜ [ ΨΓ , , ] rm for . J i tic term for Ψ. 2 πR dy 2 ) ˜ 2 ) X I I , ˜ . I X ˜ Z Lagrangian ( X which is consistent with the y ˜ + 1 and the gauge symmetry yc y X i 2 ′ [ ˜ D p A h D ( , , ). On the other hand, Γ µb ˜ → Ψ nλλ ˜ . , int πR + ( A dy ) πR ) ˜ dy ). It can be fixed by applying the ) Ψ]+ 2 p a y 2 2 L 5.26 , ) 2 µy bc I D I ˜ + ) is Z f y F ˜ Z 2.20 ˜ X X [ 1 1 . + I µ =1 −···− ′ pot − − =1 1 and + 2 5.24 +Γ I X X D L D D µa I,J ( here. We see from the kinetic term of Ψ in the ¯ ˜ µ ΨΓ − 2 ˜ µν 2 1 h A /λλ 2 + + 2 dimensional world volume. The Kaluza- ˜ I D 4 1 F y and obtain ( λ − D ( µ Ψ p ∂ − πR – 25 – dy =1 y I 2 = 1 X L D Γ = − → i πR dy ¯ ˜ = diag. (+ + R and Γ i Ψ(Γ − 2 Z ] ya D πR } 1 dy J µ X ˜ = ν 2 A − Z πR =1 dy L Γ µ D I X 2 D , , X 2 ∂ Z µ I + 1 λ 2 Γ iλ 1 2 4 Z ), when combined with the kinetic term for gauge fields, A { X := [ i 2 L − − , ]. ] J 5.22 µya 29 = = = = ˜ Ψ] = F , A Ψ L , X X I satisfies I L L L X µ X [ [ + 2 dimensional world volume. I h p =1 ¯ ΨΓ D X I,J ) that Γ =1 D as usual. So we choose Γ 2 I X 4 λ ). 5.5 IJ : 2 iλ δ g = 4.31 → } J g Finally, we can rewrite the interaction term, Similarly, we can rewrite the commutator term, The second term in ( Γ We should notice the definition of Γ , 6 I = ˆ Γ G kinetic energy on Here again the second term can be combined with the kinetic te where Lagrangian ( result ( we change the dimension parameter end, the Lagrangian thus obtained is the same as the original properly reproduces the kinetic term for T-duality transformation [ This relation seems strange if we compare with ( { Klein mass from the compactification radius ( Here the second term can be produced properly if we identify Here, this time, the second term can be combined with the kine JHEP03(2009)045 . i ). ix e 3.79 = (5.36) (5.37) (5.35) (5.34) (5.39) (5.32) (5.33) (5.31) (5.38) of the i θ )–( Z 3.70 . In this case d θ T ]. It is straightforward 27 , -dimensional torus. More , are noncommutative alge- i d he trivial bundle, ] tive torus i l turn on noncommutativity the BLG model for multiple J ummarized in ( 26 previous subsection, namely , structure constants Z ˜ i X ~m , T I ˜ , X [ ~n . . . , ) , + ] . ~n d ˜ ~m Ψ] J j~n ~n . + m , i d ˜ )( + + ~m , I l X 0 ~ ~ Z Z ~m ~x , δ 0 ~ ˜ + j · i~m l X 0 I ~ ~ δ ( [ δ Z δ ˜ ijk I · · · i~m ~m l~m ij ~ X a ′ ij e [ f g 1 has the property ∝ i h iθ m ¯ ˜ m 1 0 ΨΓ ) e T = = g = 2 was studied in [ with generators Z – 26 – ) = is the coordinate on a ) k~n πR i ) gauge group, and dy = = d 0 2 ) πR )( dy T j~n ~x k~n j g N 2 i ~m )( j~n ) of )( Z j ~m Z = T i )( )( i~m Z l j ~m ~ Z 1 ( i i ~m 1 i~m a =1 ( )( g − 5.33 ( l 0 ) and ~ T − i f =1 X D ( J has I X I,J D N f 0 2 4 2 g iλ λ ). The case of = = , and it depends on the rank of the gauge group and its twisting d θ int T pot 5.35 L L is in general not the same as the noncommutative parameter via 3-algebra ′ θ p -dimensional vector of integers. d denotes a generator of the U( is the generator for U( ) reduces to ( i is a i T T ~m 5.36 The simplest example of Since the structure constant ( We start by defining a Lie algebra maps a section of the twisted bundle to another section. For t has derivations i 0 generally, one can consider a twisted bundle on a noncommuta and ( where The parameter noncommutative torus braic elements satisfying Z where Here the compactification of D2-branes onM2-branes torus, corresponding but to we an start example from of the Lie 3-algebra s Here we consider essentially the same physical system as the 5.2 M2 to D and metric g to generalize it to arbitrary dimensions. The formulation here will be moreand general a than gauge above field as we background. wil JHEP03(2009)045 ] ). ’s )– 24 6=0 , a 5.50 (5.42) (5.46) (5.43) (5.45) (5.44) (5.40) (5.41) (5.50) (5.47) (5.49) (5.48) I ) from 5.43 16 , )–( to obtain 5.46 and 14 d 5.47 0 )–( g , which encode = . In the following abc 5.43 0 K g g lgebra ( a and , ab ~n , J tion for generic a discovered in [ ~n + ). It follows that the first 3 k + rs ~m k ~m T , T b 5.39 . k ijk ~m~n v f , , which is a subalgebra of the loop ab c ij is the identity of ~m~n , g v 0 + , f C , ). However, it is still a good example , c 0) v ~ b a ~m 0 ~ v ~n + abc v (0 . (Note that we have changed the range v abc δ corresponds to a nontrivial gauge field 0 + i 0 a ~n , T ab 3.79 C abc L δ = 1 the resulting SYM theory is the low ~m i v 0 + 0 otherwise ~m ab ab given by ( 0 ~n ~ + ~m ), this 3-algebra is not a good representative 0 ~ d 0 l L + ~ K − δ ~ )–( C + C 0 δ δ i a 0 0 0 i ~m 0 ~m ~ ~m 0 δ ~ 0 ~m + − ~ ij ,...,N , v δ J T δ ijk g T l~m 0 0 ~ 2 – 27 – , 0 i i 0 3.70 ~ ~m = a a ij f , ab u δ T g T ) C − m m ), so that a a ab , = 0 = i~m = 1 N ( C m m abc ] that the BLG model with the Lie 3-algebra ( ab ] = ] = ] = ] = K i~m i~m abc 0 b j k 16 ~n i ~n ~m ] = ] = ] = ] = 0 K 6= 0, and K b , u j i, j, k ~n , T i i , T ~m ~m , T a i ~m a j ~m , u , T , T , T a , u a, b i ~m a a , u 0 , T , T and u 0 [ i v 0 u l u T ~ [ [ [ [ u u T + 2)-branes. [ [ [ d = 0 if ). In the sense that one can construct the 3-algebra ( ,...,d ). 2 ab , is the identity of U( J 1 0 from the convention used above.) , 5.50 ,...,d T ) vanish, hence , we showed explicitly that for )–( = 0 = 1 5.1 a, b, c 3.62 5.47 a, b a, b, c ) is exactly equivalent to the SYM theory defined with the Lie a This 3-algebra is actually precisely the Lorentzian algebr The 3-algebra is defined by the 3-brackets Now we consider the 3-algebra with the underlying Lie algebr It follows from the result of [ 5.46 the SYM theory for D( energy theory for D3-branes. Now we briefly sketch the deriva of indices It will be shown below that the constants where background. the information about derivatives of the Lie algebra algebra ( of the new class of 3-algebras defined in ( assuming that constructed from the (multiple) loop algebra defined by In section because it demonstrates the roles played by the new paramete terms in ( a Lie algebra by adjoining two elements ( ( we choose where ( empty. We take JHEP03(2009)045 (5.59) (5.60) (5.57) (5.56) (5.53) (5.52) (5.55) (5.54) (5.51) (5.61) . d θ ) (5.58) T ) Z e twisting of the as ( a ′ µa u ˆ ,...,d, A 1 ∧ , , j ~n a , round field strength with . Finally, after integrating i T v = 0 , , Ψ , ∧ d d d 5.1 =0 d A , , a X m m i ~m d d m ˆ d a, b D T Z Z m ( m Z ) A Z )+ Z Γ j~n · · · · · · , on the noncommutative torus, such · · · Z )( 1 1 ( ¯ · · · · · · 1 Ψ a is chosen to be zero for simplicity. If , m m h 1 i~m 1 m µa 1 D ( i 2 Z Z ˆ m m , µ Z A i i µab = 0 a Z Z A ab + T T i i ∧ ) ) A i C m T a , T ij ) ) µab a i~m i~m u X , ( ( v AB a i~m i~m I 1 2 I ] = ] = ( a ( ′ µa µa d v =0 b d – 28 – a a , F X Y A A Ψ X + m D d b , + Z AB ~m ~m ~m ~m a , A v ) are sections of a twisted bundle on b )+ X X X X F u D Z h )+Φ ∧ Z [ · · · ( ( = 0 a ∧ Z 1 I =0 v ′ µa a 9 a ˆ m 1 . ) := ) := ) := ) := ˆ ˆ X X Ψ( A u Ψ A,B Z Z Z Z ′ µab µI , Z + ( ( ( + 4 1 a F A µab I ˆ a a Ψ( ′ µa µa ˆ D A − u u X [ ˆ ˆ =0 I A A d a , a can be ignored. Here ) and ) and the notation X = =0 a,b d Ψ X Z a ( X v L 2 1 a,b d d 5.36 =0 =0 µa 2 1 a a + X X ˆ A = = ) I Z µ Ψ = = constant ( , the BL Lagrangian turns into that of a SYM theory i A X ˜ . I a A d θ X is the constant background field strength that determines th T ), Ψ ab Z ( C I i ’s are nonzero, it corresponds to turning on a constant backg X To proceed, we first define covariant derivatives As we have done it many times already, we fix the coefficients of The rest of the derivation is essentially the same as section Expanding the fields in the BLG model, we have µab where nonvanishing components of that and and the coefficients of A bundle on where we have used ( out the field JHEP03(2009)045 . e )– ith µab A (5.67) (5.63) (5.66) (5.64) (5.68) (5.69) (5.62) (5.65) (5.71) (5.70) 3.31 ), ( are scalar and a d 0 µ 3.27 − A . , 8) h we can reproduce LG model is to use 3) , is exactly the descrip- 2 ,..., , . ]). Various kinetic terms 8 a = 4 = 1 v ) . I a x ) ( ( I a x,y X . ( , , ), we have to redefine the above ) I a + IJ Z , X a , v ( ) C a I u . ) Z u ˆ 8) are generated from the extra term in ( X , , one may put x , ab 0 ] ]+ = const ] ) =: ( 8 µ I ν J I y a C − I a ˆ ( R ˆ ˆ ˆ A J b D D i a D X ,..., , , , a χ X D − I µ µ ) u I I a a , we can assign nonzero values to ˆ ˆ ˆ x µ a – 29 – D )+ D D , X = 4 ( X X ∂ λ y µI I i i ( ( i C X := [ := [ := [ are ghost fields. As we have seen reapeatedly, one := := := χ 2 )+ ) ( ) = 0 ) ) y I a y i µ x x ’s are covariant derivatives and the rest 7 → defined through the structure constants ( µI ( IJ ( µν IJ v ( ( ˆ I ˆ i i ] remain the same and we have the same conclusion. i D D ) X F F I F I a 3 i 9 ˆ C χ χ a x D T ) ) T y , ( ) X X x x I a ∂ 8 x , there is an interesting Lorentzian 3-algebra associated w ( ( ( a I i i X I ] while it was not explicitly understood. We would like to giv i )= 3.2 X X X x of the 16 ( , I d 9 ˜ )= )= X , ) and 8 x ( x,y x,y I a ( ( I a X ˜ ˜ X X are constant numbers. This is exactly the assignment by whic a λ ). We claim that the BLG model associated with this 3-algebra The key observation to define 6-dimensional fields on M5 from B . All the other analysis in [ a 3.33 ˜ fields. To turn on the background field Roughly speaking, only and expansion as where 5.3 M2 toAs M5 we revisited discussed in section If we add three pairs of Lorentzian generators ( on M5 world volume such as ( the “mode expansion” such as the Nambu-Poisson bracket on tion of M5-brane in [ a brief sketch on this point. ( Here, the fields X the M5-brane action from BLG model (for example, eq. (30) in [ where may put By change of basis in the transverse direction JHEP03(2009)045 , we , 3 T ore general -branes ] on (2009) 66 2 9 , M 8 el associated with (2008) 105 B 811 l discussions. Y. M. (2007) 045020 02 rogram “the Physical he infinite dimensional we studied, we naturally gebras which are relevant negative-norm generators e to consider the D-brane D 75 entzian Lie 3-algebras and nsional loop algebras. JHEP e toroidal compactification. mass term generated by the 3-algebra. he National Center for The- oint Research Program pro- gain reduced to the inclusion e kinetic term as mentioned. e even when we consider the , ]. aboration is made possible. Nucl. Phys. Lorentzian Lie (3-)algebras. p in Taiwan. The work of P.-M. ted by KAKENHI (#20540253) , -branes 2 SPIRES Phys. Rev. , -branes M ] [ 2 ’s 2 M M – 30 – arXiv:0711.0955 [ ]. ]. Comments on multiple Gauge symmetry and supersymmetry of multiple Modeling multiple ]. SPIRES SPIRES . On the other hand, if we use the M5 action in [ SPIRES 3 ] [ ] [ (2008) 065008 ] [ T Algebraic structures on parallel 6= 0. Another interesting possibility is the description of m D 77 ijkl F arXiv:0712.3738 arXiv:0709.1260 hep-th/0611108 [ [ [ Phys. Rev. We note that if we do not include these extra terms, the BLG mod We do not believe that our examples exhaust all possible 3-al The authors thank Yosuke Imamura and Ta-sheng Tai for helpfu [3] J. Bagger and N. Lambert, [4] A. Gustavsson, [2] J. Bagger and N. Lambert, [1] J. Bagger and N. Lambert, this 3-algebra would contain infinitecompactification number on of massless mod can produce the Kaluza-Klein massTherefore, correctly the since generation of we Kaluza-Klein have mass th of on pairs M5 can of be Lorentzian a norm generators in the Nambu-Poisson 6 Conclusion and discussion In this paper,studied we the considered BLG models some basedobtain generalizations on the the of string/M symmetry. the theory In compactifiction theHiggs Lor on examples fields the can torus. be The identifiedThe with dimension the of Kaluza-Klein the mass torusof in can the th be 3-algebra. identified with Wesystem also the where number argued of its that gauge one symmetry may is use described our by techniqu infiniteto dime M/string theories.case with For example, we did not fully examine t background, such as orbifolds, through different choices of Acknowledgments We appreciate partial financialvided support by from Interchange Japan-Taiwan Association J (Japan) by which this coll would like to thank the hospitalityH. of is the supported string in theory grou partoretical by Sciences, the Taiwan, R.O.C. National Y. Sciencefrom M. Council, is and MEXT, partially t Japan. suppor S.Sciences Frontier”, S. MEXT, is Japan. partially supported by Global COEReferences P JHEP03(2009)045 ]. ]. , ] SPIRES SPIRES -branes, 2 ] [ ]; ] [ , M , -algebras in 3 , ]. ]. SPIRES , ation [ (2008) 037 (1999) 1835 (2008) 003 superconformal gauge ]. 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