JHEP08(2010)083 ons Springer that furnish August 2, 2010 August 6, 2010 August 17, 2010 : : : Received Accepted 10.1007/JHEP08(2010)083 Published -dimensional gauge theory doi: 5-branes using a null reduction. Published for SISSA by b 0) tensor multiplets. We briefly discuss possible , [email protected] , 0)-supersymmetric tensor multiplet. The on-shell conditi , 1007.2982 and Constantinos Papageorgakis 1 a, D-branes, M-Theory Using 3-algebras we obtain a nonabelian system of equations [email protected] On leave of absence from King’s College London. 1 Theory Division, CERN, 1211 Geneva 23, Switzerland Department of Mathematics, King’sThe College Strand, London, London WC2R 2LS, U.K. E-mail: b a Open Access Abstract: Neil Lambert Nonabelian (2,0) tensor multiplets and 3-algebras a representation of the (2 applications to D4-branes using a spacelike reduction and M Keywords: ArXiv ePrint: are quite restrictive soalong that with the six-dimensional system abelian can be (2 reduced to five JHEP08(2010)083 4 4 5 6 7 7 8 1 2 8 10 14 11 12 vebrane it does not seem al gauge theory language, the formulation of an M5- on between 3-algebras and ally local electric and mag- of a 3-algebra. Even though n of Lagrangian descriptions r about their possibly deeper eory of multiple M5-branes is mal symmetry due to the self- are difficult to reconcile with a ]. 7 – 1 – ]. These descriptions initially relied on the intro- 6 – 1 1 I µνλ A A µ A A µ B A ˜ A X C H In this note we begin the investigation of a potential relati For a review on M2 and M5-brane basics see [ 2.6 Summary 3.1 Lorentzian case 3.2 The Euclidean case and ‘D4 to D4’ 2.1 Closure on 2.2 Closure on 2.3 Closure on 2.4 Closure on 2.5 Closure on Ψ 1 There has been significantfor recent multiple progress M2-branes in in M-theory the [ formulatio 1 Introduction 4 Null reduction and BPS5 states Conclusions A Notation, conventions and useful relations 3 Relation to five-dimensional SYM Contents 1 Introduction 2 A nonabelian (2,0) tensor multiplet duality of the three-formgiven field-strength. by a In conformal addition fieldnetic the states theory and th in no six-dimensions couplingLagrangian with constant. description. mutu All of these features connections to M-theory in general. multiple M-theory fivebranes. Comparedbrane to theory M2-brane is systems difficultpossible at to best: write down Even a for six-dimensional the action case with of confor a single fi duction of a novel algebraicone structure, can going recast under the thethe former name presence of in 3-algebras terms is of intriguing a and completely one might convention wonde JHEP08(2010)083 (2.1) ] which Ψ. This 17 − – 15 0) tensor mul- Ψ = , nd a different null bra. We find that 012345 5-brane Lagrangians ian (2 e models [ tions, sporting lightlike ]. Although here we will ch transforms nontrivially ng dimension. Our ansatz 0) system. In addition this be eliminated at the price six-dimensional interacting 9 ]. For recent work that also , ǫ agating vector field that we 9 , ces to five-dimensional super- ions of a free six-dimensional from the expected nonabelian le M5-branes. , 8 ions for the abelian M5-brane ions if the structure constants µνλ 8 prove to be quite restrictive and ee-form field strength, we intro- heory with conformal symmetry t of equations of motion for the e associated to a 3-algebra. is selfdual. The supersymmetry e [ H ] µνλ νλ Γ B 0) tensor multiplets. In particular we 1 2 µ [ , ∂ 1 3! + = 3 ǫ ]. In this way we hope that new light can be I ]. – 2 – , X 18 µνλ ]. We conclude with some further remarks on the µ 14 Ψ H ∂ – 19 Ψ I I µν Γ 10 Γ Γ µ ¯ ¯ ǫ ǫ i i and the Fermions Ψ are antichiral: Γ ǫ = = 10 and i.e. they are totally antisymmetric and obey the related I = Ψ = Γ 2 µν δ ǫ δX ,..., δB ]: 20 = 6 012345 I = 8 3-algebras in three dimensions. 5, N ,..., is chiral: Γ ǫ = 0 0) supersymmetry. This is similar to the Lorentzian M2-bran µ , It is interesting to note that there have been proposals for M We proceed by studying the closure of the supersymmetry alge Here we will simply study the equations of motion of a nonabel These are the 0) tensor multiplet [ 0) supersymmetric set of equations of motion. On the other ha 2 , , involves ‘structure constants’ with four indices that can b that require the introductionof of sacrificing a manifest new six-dimensional scalar Lorentz field, invarianc which can tiplet. Starting with thewe propose set an of ansatz supersymmetry forversions transformat a of nonabelian the generalisation. scalars, Apart fermionsduce and a the gauge antisymmetric thr fieldunder as the well as nonabelian a gauge non-propagating symmetry vector and field has whi a negative scali (2 dyonic-instanton string BPS solutions [ possible connection of this theory to the dynamics of2 multip A nonabelian (2,0)We tensor start by multiplet giving the covariant supersymmetry transformat provided a different angleshed on on D2-branes M5-branes by [ reformulating D4-branespaper in can terms be of viewed as a(2 a (2 no-go theorem for obtaining areduction genuine leads to a novel system with 4 space and 1 null direc various fields, as well as a numberupon of expanding constraints. the theory The around latter aYang-Mills particular vacuum along it redu with six-dimensional,will abelian essentially arrive (2 at aand reformulation (2 of the D4-brane t touch upon some of these issues see [ it closes on-shell upare to those translations and of‘fundamental gauge real identity’. transformat 3-algebras, The on-shell conditions yield a se only study the equationsintroduce of is a motion, nonabelian the analogue of additional the non-prop auxiliary field in [ where generator JHEP08(2010)083 , A T (2.6) (2.5) (2.4) (2.2) (2.3) ives ǫ A CDB 0) transformation ǫ , 5 f Γ J D ) we need to introduce IJ X lian fields and interac- . I C ]Γ nal (2 2.3 = 0. or space with a basis J X ] ǫ fficient to write the algebra λ B A C,D ,X C νλρ I µνλ lls, which are given by Ψ in ( IJ δ H X CDB H [ Γ , µ g [ λ i 2 ∂ D . I Γ B µνλ ] for − 2 1 Ψ Γ X J ǫ I C 1 2 A so that we can consider the ansatz: − = 0 X B µ 1 αβ ,X 3! ǫ I µ ˜ κ I B F A C 5 , X + C X µνλ A µ − Γ : ǫ Ψ ∂ I H ] I A µν αβ µ and Ψ have opposite chirality. Thus a natural λ µνλκ – 3 – X ∂ X Γ ∂ B is antisymmetric in ǫ µ Γ µνλ µ 1 2 I ∂ µν ∂ A Γ [ Ψ I Γ 1 2 I + A Γ ¯ ǫ Γ Ψ = = i ¯ ǫ Γ ǫ µ µ 1 i 3! ¯ ǫ I I CDB A ∂ i + , are ‘structure’ constants that we will determine in due µ f CDB X + X A Γ = 3 = α h A Ψ µ ǫ 5 Ψ D I I A D ] D Ψ = Γ Γ Ψ I λ Ψ δ X α I µνλ Γ CDB λ C D µ δX Γ Γ α h A ¯ ¯ ǫ ǫ C D δH i i µν [ Ψ and not mention I µλ I Γ Γ = = ¯ ǫ , Γ Γ and µ , and promote the derivatives to suitable covariant derivat i ¯ ¯ ǫ ǫ I α µνλ i i Ψ = Γ A δ H etc. = 0 = = 3 = = Γ δA δX , µ I A A A A CDB B 4. g µ A Ψ T ˜ δC δ , δX I A A is a gauge field. µνλ A A δ X A ,..., δH B µ = ˜ CDB A I = 0 f matrix to account for the fact that X In order to obtain a term analogous to the [ Upon reduction on a circle one expects that the six-dimensio We wish to try and generalise this algebra to allow for nonabe α µ . viz Here algebra closes on-shell with the equations of motion for course. Note that we can assume where rules reduce to those of the five-dimensional super-Yang-Mi a Γ tions. To this end we assume all fields take values in some vect guess is to propose the existence of a new field We note that, from thepurely point in of terms view of of supersymmetry, it is su in which case one must include the equation of motion JHEP08(2010)083 1. − (2.9) (2.7) (2.8) to be (2.10) (2.11) (2.12) (2.13) µ ] = C C [ , 2 5 . ] A X . [ 1. Therefore, if we [Ψ] = ] , − − X ect to their scaling di- [ BCD = 0. On the other hand : f . hat are all related to the µ A J C ] = 1 ] = 3 C X ] µνλA rise to the Fermion equation: , . C 2 = 0 H λ D [ . [ H I B , δ , C A µ B 1 ) s X = 0 . δ 1 , C A ǫ A A A 2 1 J B B CDB Γ ˜ ˜ = 0 but also comes from supersym- ] = 2 Λ g Λ λ τσρ D ] + X CDB Γ + µ + B , H 2 Ψ f X µ I C ǫ A A I ρ C (¯ A Γ C i X C ν λ we expect the expectation value of B µ has scaling dimension λ B ] = 1 Γ ˜ Λ D D µνλτσρ ˜ = 2 R I µ C ν A ǫ – 4 – µ [Ψ] = [ [ v A v C X 1 3! B = ) by investigating the closure of the supersymmetry = ˜ Λ λ B = µ ,C A I C A 2.6 C ] X + 2 ] = 0 µνλA 2 , A , δ µ ) A 1 H , , , δ Ψ 1 δ 1 C [ ǫ λ 2 1 δ ν [ µ D − Γ D λ 2 ] + 1 Γ ǫ (¯ X ] = i ǫ 2 [ I A µ A − C X ] = [ . = H [ R µ v ]. However the canonical choice is [ X As with the abelian case we also impose selfduality on Note that consistency of the above set of equations with resp what we expect is where Here the situation is rather simple since one clearly has [ 2.2 Closure on These agree on-shell if algebra on the scalars. A straightforward calculation give In particular we see that the new field metrising the first constraint. The second constraint comes from setting so one could stillchoice make of this work [ with some other assignment t Demanding that this is preserved under supersymmetry gives mensions gives compactify the theory on a circle of radius proportional to 2.1 Closure on We now proceed to test the ansatz ( JHEP08(2010)083 – 1 (2.14) (2.16) (2.17) (2.18) (2.15) (2.19) (2.20) A CDB h . A nimise the size of our deduce that µνλD C µ B ˜ H A hould amount to, and the λ C CDB h A ed that C ν C ν D ˜ A v We obtain l. In addition, acting with su- : . + − . µν EFG A ˜ A A F f , C . ν B . J F ˜ ≡ A = 0 A X BDC ] CDB CDB = 0 = 0 A ν I E f A h h µ C A A ˜ X I I D D A ,D DBC λ G µ X X f ). BDC − C λ µ µνλ D f D BDC BDC . ν C = D D H D f f – 5 – A C 2.13 ν B A λ λ λ C C C I A ) D D B ˜ A 1 C C C ˜ Λ ǫ Ψ X ) ) µ EFG 1 1 µ λ λ ∂ = f ǫ ǫ IJ CDB I D I D D λ G Γ − h λ λ Γ Γ C C C + B λ µ ) through [ C C ν µν A C µνλ A Γ Γ ˜ µ B must satisfy a ‘fundamental identity’ for real 3-algebras [ F 2 2 C Γ ˜ 2.4 B ǫ ǫ A 2 D µν A ( ( ν ǫ i i ˜ (¯ F ∂ 2 i ν v − +2 − = ABC f ) will follow from ( = = A A µ B µν B ˜ ˜ A F B µ A 2.18 ˜ A ] 2 , δ 1 δ [ For the first term to give the correct gauge transformation we We also see that the third line gives the constraint: ] and as a result ( where in the lastfield-strength line is we defined have from given ( what the first three lines s We continue with theexpressions closure we will on freely the use gauge the constraints fields. found above. In order to mi Given this we see that the second term is a translation provid 4 2.3 Closure on The second line gives the constraint: We will see shortly that which implies that thepersymmetry physics leads is to (largely) five-dimensiona JHEP08(2010)083 is A (2.21) (2.24) (2.22) (2.23) (2.25) CDB D . f E ] D E ] = 0 FG ) al identity: D f ] FG A A A E | f C [ A ) FG and thus C CDB CDB A [ I f C g DB f A A g A D X DB − formation if C K G   g h. The fourth line will CDB | µνλ B Ψ A g τ A A d-strength. In particular B K X CDB G [ H CBD Γ g J al line. F A D − f X C I D g J X F − DBC A ¯ CDB CDB Ψ K I X G C h g g -equation of motion: X σ τ B ( DBC = X I X D C ρ g B C H D I CDB F ρ D E Ψ ] I A X D C I C ] f τ ) . X λ ( νλ X . . Γ X I D C κ E J C D ρ ) one sees that all the terms quadratic C σ H C B CDB A A Ψ ρ X C B µνλρστ X C ρ ¯ B = 0 C Ψ ] J ǫ f σ , κ ) E B C λ σ i C 1 B 8 Γ 2.13 G C C ǫ D CDB C CDB I ρ − B + I g C f EF Γ ] C A ρ A µνλρστ µνλ B X µνλρσ D = = – 6 – ǫ C Γ IJKLM ρ ] B H f Γ 2 ǫ A A IJKLM ǫ + µνλρστ ) A C E C νλ ǫ (¯ ǫ 1 CDB ) ˜ i ) B ¯ F A ǫ 1 Ψ i µνλρστ 2 f 1 2 ] ǫ ˜ ǫ  Λ ǫ )( I DBC ) D ) CDB + ABC 1 − 1 [ 1 LM IJ h g + ǫ νλρ ǫ X ǫ f Γ Γ LM J I τ K H Γ . Just as with multiple M2-branes, consistency of the gauge Γ Γ µν µν µ Γ D [ [ [ σ µ ρ τ ρ [ I µνλ A C µνλ A Γ µνλ D Γ Γ Γ Γ 2 H 4 X 2 2 2 2 Γ H ǫ ρ ǫ ǫ ǫ ǫ 2 σ (¯ B ρ  (¯ (¯ (¯ (¯ ǫ i ρ i i i i D (¯ C 8 C,D,B D 3 ρ i v 6 6 ρ v v + +3 + +2 − − − µνλ A = = H µνλρστ ǫ implies that the structure constants satisfy the fundament 1 4 A B µνλ A + ˜ Λ H A ] ] 2 , δ νλρ 1 vanish. δ H [ µ λ A The second term of the first line gives the correct gauge trans Demanding that the seventh line vanishes gives the [ C D symmetries where again we have written the required expression in the fin Using this, along with the second condition in ( totally antisymmetric in In addition onevanish sees if that the second and third lines now vanis Given the previous conditions this implies that 2.4 Closure on in We continue with thewe closure find: on the antisymmetric tensor fiel JHEP08(2010)083 (2.31) (2.28) (2.30) (2.29) (2.26) A . We will . . = 0 (2.27) ) CDB . This would A A µν A f = 0 A A B ] D A Ψ ) νλ CDB CDB τ CDB A f f B ation to obtain the Γ , f µ D -equation of motion, D CDB [ C D A f Ψ Ψ ¯ D H Ψ CDB τ I D σ B f Γ Γ EFG ν C D = 3 C CDB , tions that we found above, f ns Γ ¯ f EFG Ψ Ψ I ν ist a suitable F B I f σ B Γ C I X F = 0 C ν I µνλ A J C E µνλρστ X µνλ A Γ A ǫ . H X J X E ν i B 8 H J C ρ C X + + CDB J I µνλρστ X C C D = 0 A ǫ f ρ A C X X A νG Ψ i D C 12 µ C + νG Ψ ν D B I = + CDB C A = 0, along with the CDB such that µ Γ C f ν B A A Ψ f ν I B D (Γ µ C − Γ – 7 – A A K X ν ] D B µν A A − B τ CDB Γ µ CDB C µν B µνλ D σ A f Ψ f D ˜ I C F (Γ CDB A H I D I D C λ )Γ σ CDB [ ρ X f B 1 ) that we obtained from demanding that the selfduality Ψ X X f A ǫ D CDB ˜ ρ D D )Γ Λ τ + σ B D 1 f K ρ C Ψ 2.8 D ǫ D A C + I Ψ Γ D σ C ρ C A I I C CDB σ Γ Ψ A Γ Ψ C Γ ν f µ Γ 2 X I ν Ψ 2 ǫ Γ σ Γ B D ǫ µ = (¯ Γ BCD ν B ν (¯ µ i µνλρστ C ν B 4 f D i A ǫ 4 3 Γ C Γ µνλ D µ ν C D 1 4 ν I B C − v + C H C C X A + λ ¯ C µ CDB C Ψ = C µνλρστ Ψ f i A C + 2 C ¯ ǫ Ψ ] A I D i 6 1 A 2 − − = X ]Ψ Ψ νλρ A ρ I + ν 2 A − A µ B H I D C X B , δ D µ 2 ] [ µ 1 ρ C µν µ X δ is preserved under supersymmetry. ˜ 2 [ D D C F D λρ D B A 0 = 0 = 0 = 0 = 0 = Γ 0 = B µνλ A We can also take a supersymmetry variation of the Fermion equ One could try to introduce a field µν H [ ˜ F We see that the Bianchi identity scalar equation of motion: of but this over-constrains the fields and hence there cannot ex Here we achieve closure with the Fermion equation of motion 2.5 Closure on verify this in the next section. 2.6 Summary Let us summarise the results of our computation. The equatio lead to the algebraic constraint Finally we look atone the gets closure on the fermions. Using the rela implies that which agrees with the condition ( JHEP08(2010)083 = with (3.1) (3.2) (3.3) (2.33) (2.32) ]. Fur- = 0. . , raising AB 4 ABCD – h G f 1 , as well as ǫ 4 A EF ] A ) = D ) by expanding B f , a, b, . . . CDB N E ,T − ( f A, B, C, D , A J D su T ABC X [ = I = + C f 3-algebra of [ Tr ( G 0) supersymmetry trans- X A , λ B A C e metric in group space. In , IJ n turn implies that ] by starting with an ordinary CDB . Γ but one can also consider three- f 16 abc λ and all remaining components of f D Γ         ]. For a recent review of 3-algebras such that 2 1 Ψ G 0 0 = I C 25 ± , − G – − X T ǫ h ...... + A κ B 23 abc δ 0 0 C λ 5 µνλ A 1 . . . gδ H ]. In this section we move on to investigate the − µνλκ = – 8 – 1 0 17 Γ , f µνλ . . . 0 0 0 0 0 i – I − Γ c λ A Γ 1 2 A 15 ¯ ab ǫ C         i and hence antisymmetric in all of h f 1 ] Euclidean four-dimensional 3-algebra, 3! = + = CDB 22 + obeys the fundamental identity: A c , f ǫ C,D AB Ψ D ab I A D ] 21 h ] λ + Ψ X ) + 2. The structure constants are given by f λ C D µ G A ( C ABC D µν [ [ Ψ I f I µλ Γ Γ ¯ ǫ , Γ Γ dim µ i = ¯ ¯ ǫ ǫ i i D = 3 = = 0 = Γ = ]. I µ A A A ABC B 26 µ A Ψ f and adding two lightlike generators ˜ δC δ δX A selfdual, are invariant under the six-dimensional (2 µνλ A is antisymmetric in δ G the structure constants of the Lie algebra E δH c vanishing. The metric is given by µνλ A ab ABC D f H f The above are precisely the structure constants for the real We next look for vacua of this theory in the particular case of ABC DE f an infinite set of Lorentzian 3-algebras [ with provided that the total dimension to formations with Lie algebra vacuum solutions of our theoryalgebras for with these more two than possibilities, onein timelike physics directions see [ [ thermore we need to endow the 3-algebra with an inner product The Lorentzian 3-algebras can be constructed e.g. as in [ 3.1 Lorentzian case around a particular point 3 Relation to five-dimensional3-algebras SYM can be classifiedparticular there according is to exactly one the [ signature of th which one can construct gauge-invariant quantities. This i h JHEP08(2010)083 has (3.7) (3.6) (3.9) (3.4) (3.5) (3.8) g ) that 2.31 ǫ a a cd cd a f f J cb d d f X ef I c ersymmetry transforma- f α c ǫ mations and the fifth line I f rth line of ( X ) A ) that d X IJ µνλ ± J e = Ψ Γ H y for the structure constants of β 5 2.32 X b Γ Γ J c α a c ˜ 1 2 µνλ A X ¯ , Ψ Γ 2 i 2 a − g 2 1 ǫ db 1 5 + , zero. As a result the latter correspond − A 3! f , ] . I d d a I ± A αβ a A + 5 ± cd X a a Ψ B X νλρ αβ ǫ H , β f µ µ Ψ αβ 5 I F cd ± ] d H µν ˜ ∂ ∂ D Γ λ f ˜ µ µ F µ X Ψ I [ c d ∂ gH 1 – 9 – I µ 2 ∂ ∂ YM αβ ± constant. µνλ d X Ψ ∂ g Γ µν also has the correct scaling dimension. Further- Γ ( [ = I Ψ I 5 g H a I 2 1 Γ Γ Γ a Γ 5 = . ¯ ǫ 5 b Γ c cd 0 = 0 = 0 = Γ µ ∂ i ) super-Yang-Mills theory. In particular since ¯ ǫ 5 YM + a Γ ¯ i Ψ g αβ I ǫ c gf N db = a i αβ 2 ˜ I a F 1 2 f = 3 = = Γ 1) Lorentz and conformal symmetries are spontaneously d g d , , while one also has from ( X gX Ψ I I + ± ± ± b α − Ψ 5 a + a 5 a ˜ Ψ I ∂ a X D ]5 a Γ δ 5 δX Ψ a I µνλ X b I α ,H = Γ Ψ βγ αβ α α Γ Γ µ α I d ˜ ¯ ¯ ǫ ǫ ˜ δH H = 0, rendering A H D ˜ i i D 2 YM α X α α [ g µ g 5 − ˜ ˜ ˜ µ = = Γ = ) become: D D ∂ D 1, we see that I a ∂ = 1) Lorentz invariance. I a a − , X b g α a Ψ α 2.31 ˜ δ 0 = 0 = 0 = 0 = Γ 0 = δX A ∂ δ = 5 and all other components of I a α, X ) of five-dimensional SU( α ) reduces to = ˜ 2.5 D µ ). Hence the off-shell SO(5 The rest of ( However we also have the additional equations 2.31 N ( scaling dimension more the fundamental identity reducessu to the Jacobi identit to flat connections thatof can ( be set to zero up to gauge transfor while all other fields are set to zero. One then has from the fou tions ( we recover the equations of motion, Bianchi identity and sup We immediately see that with the identifications with where broken to an SO(4 with transformations JHEP08(2010)083 = i and = 0, µ A ) only (3.11) (3.10) c C α b h -algebra ˜ αβ a A ably one 4 but using . Singling 2.27 B a A a β c 5 ˜ db A , and use the D 3-algebra the ) super-Yang- f A + d ABCD 4 B N ε ef A c = 0 and hence we β b µν f cua of the theory = ˜ ˜ F A and the gauge field ± 0) tensor multiplet, . Thus we conclude 1)-covariantising the ] , a uations with positive , a 5 = 2 + α c db ˜ a νλρ C A 4). Thus the f ABCD d fc H dimensional SU(2) super- − al (2 ≡ f µ satz for the six-dimensional f [ ef new direction), add the free g the fields six-dimensional c ations of motion and super- . uper-Yang-Mills along with a ∂ f β b Lorentzian result, in contrast ric 3-algebra expressions one eb ± → a ˜ 2 YM supercharges. lar: For the A f = cb g α O(4), f − ∂ a ) we have, assuming ) . To start we note that ( a c fc − . ec 5 f 3.4 α f a c a µν A c α b B cb eb ˜ B A fb β f f ]: Starting with the SU( β c f ˜ D ∂ 5 18 − α − a = c ec gB 5 f a – 10 – are now the structure constants of SU(2) and we β − c αβ b 4 and expanding the theory around a vev B ˜ fb F = α abc f a, . b ˜ f = 0. This provides the off-shell conformal invariance. D 0) multiplets in six dimensions. a ± ( , = α ] 5 g + ˜ . We then promote the coupling into a field, while im- A νλ C A = . Examining the derivative terms leads to b α B α a b a ∂ ˜ µ b [ 5 A β ), where ∂ αβ ≡ B ˜ ). ) by considering the replacement ( F a 3.6 a b = 3 α cb 3.9 ± in general. ˜ 2.32 f A 0) multiplets which are genuinely six-dimensional. Presum , − α c µνλ ) and ( a µν A A H 5 , ) and ( B 3.5 β 5 ) and ( B 3.8 a ) between the Lie and 3-algebra generators. By SO(5 αβ α ∂ H 2.31 0) tensor multiplets and the flat gauge fields that complete 3.1 , = 1 2 YM g a leads to ( 5 β ≡ 4 A Next, let us look at the nonabelian fields. From ( To summarise, for the choice of a Lorentzian 3-algebra the va Finally we return to the existence of a 2-form In fact it would have been possible to arrive at our initial an δ B µ 5 α a αβ ˜ D If we now look at the nonlinear terms we require However we should compare this using the Jacobi identity one finds instead can locally write but imposing the external constraintabelian that (2 none depend on the Mills theory in fivesymmetry dimensions and transformations, considering we the rename set the of YM equ coupling Finally we perform a trivial lift to six dimensions (by makin two free, abelian (2 interestingly correspond to the ones for five-dimensional s These comprise two free, abelian (2 that there is no does not exhibit any qualitativeto differences the compared case to of the three-dimensional 3-algebra theories with 16 theory by working backwards in the spirit of [ resulting equations and writingarrives everything at ( in terms of gene Yang-Mills. In thisobtained by case ( one has only a single six-dimension can once again identify the theory around this vacuum as five- acts on the nonabelian fields. For the abelian sector we have relations ( posing the external constraint F Using the Euclidean 3-algebra is qualitatively rather simi must be gauged awaydefinite in energy. order to have a3.2 well-defined system of The eq Euclidean case and ‘D4 to D4’ out one of the SO(4) directions, vδ structure constants coincide with the invariant tensor of S JHEP08(2010)083 . ]. = µ A 3.1 27 C u (4.1) (4.2) . In fact, ǫ ) where = i a ǫ cd f 1) and conformal d , 1234 u, v, x Ψ uv τ . The abelian sector Γ calar fields vanishes. It + A c = ( δ ian sector of the theory metry. The equations of ¯ Ψ µ v ection of M2-branes sus- µ to go from a conventional ual strings as well as their x gδ ctively lives in 4 space and = 0. n of the selfdual string [ shell SO(5 model or lightcone description v = µνλρτv a slightly different choice for D ǫ i -energy theory on the D4-brane µ A 8 efore more appropriately describes ig 0) tensor multiplets. However the C h , − in addition to Γ 0) theory derived above can have any a , ǫ ]. cd f ]. Thus let us set all the abelian fields to 18 = I d 4. Following the conventions of section ǫ 28 , X 6 3 τ a Γ , D 2 a cd – 11 – uv I vanishes when acting on all the nonabelian fields. , c f cd d v X f d Ψ D = 1 I -axis, we can use the above equations to describe the a Ψ i Γ v I M5 : 0 1M2 2 : 3 0 4 5 5 6 db v Γ f µνλρτv Γ v ǫ I c Γ 4 g c ) and ¯ µνv d Ψ gX 5 − x 2 a ig + ] gH + a − − 0 Ψ νλρ I a x a µ ( b H X D 2 µ 2 [ 1 µν µ √ ˜ F D D to point along the = µ A v 0 = 0 = Γ 0 = 0 = C ), 5 x vanishing on all fields. Note that the potential term for the s v − 0 D The BPS solitons for the abelian fields will comprise of selfd Hence, with the use of Lorentzian 3-algebras, it is possible These coordinates are well suited for describing the inters In particular let us consider six-dimensional coordinates x ( 2 1 √ by choosing right-moving modes of the selfdual string i.e. modes with ‘neutral string’ generalisations, as studied in [ with we choose any Lorentzian 3-algebra by having that zero here. The constraints imply that The preserved supersymmetries satisfy Γ It is of interest torelevance investigate to whether multiple or not M5-branes.is the essentially (2 five-dimensional As super-Yang-Mills and we ther multiple have D4-branes. seen However in this the section we nonabel will discuss description of five-dimensional super-Yang-Mills,worldvolume, the to low an equivalent 3-algebraic version with off- 4 Null reduction and BPS states The resulting solution should appear as a nonabelian versio would be interesting to tryof to M5-branes. relate this system to a matrix- pended between parallel M5-branes: symmetries, as was also the case for D2-branes [ 1 null dimensions withmotion 16 for supersymmetries the and nonabelian an fields SO(5) are R-sym of the theory againnonabelian consists sector of is a free novel 6-dimensional supersymmetric (2 system that effe JHEP08(2010)083 , . 6 a = a ng cb ǫ was X uij a i f (4.6) (4.5) (4.3) (4.7) (4.4) uv H D vij c = µνλ a since it is H ], the only H 19 = uvi a to impose the 0) tensor mul- , a ijk a , b H ij H ˜ F = 0 ǫ ǫ = 0, one has Γ v u 0) tensor multiplet is ǫ Γ 6, and Γ 6 , . ) now becomes ij ij Γ , b is antiselfdual and = 0, which becomes: Γ ton string” [ Γ a a ns from uv see that the right-moving I > 4.3 A ij anishing ( ˜ F Ψ vkl vij a our (2 ere the gauge structure arises vij a uij a dimensional (2 δ ian four-dimensional instanton H H = H nce with dyonic instantons. We H 4 1 b 4 1 ghtlike. They have smooth finite- = 0. ijkl ns with ‘W-Bosons’ of ca = 0, for ǫ + e they have the expected properties + f 2 1 ǫ I a ǫ u uij a u − X Γ i Γ vij c , H ij I = D ǫ Γ Γ = u I a = 0 I Γ a X I vij a 6 a uij a X u ǫ . Γ ,H u X I u a H D i – 12 – b Γ D 4 1 X a D − i u IJ ui + ǫ ˜ Γ D D 6 ,H F ǫ a Γ 6 a − i − uv cd ǫ Γ X Γ f = i I i J d b Γ Γ D i . In addition one finds that X ca uvi a Γ I c = f I a H uvi a X = 0. In fact this projector breaks another half of the remaini uvi a X 2 g − i ǫ H -BPS equations for our null-reduced theory: ]. -equation of motion. H uvi c v ǫ uvi a 1 4 D − + I H 19 H H Γ i Γ 0 = I a X i D ) to be satisfied and the solution to be supersymmetric we need along with a solution to 4.3 . The first two terms vanish by having that These are essentially the BPS equations of a “dyonic-instan We summarise the After setting the fermions to zero the BPS condition is The interesting nonabelian solutions should involve a nonv ǫ b 6 a Γ ij ˜ F while the remaining ones after imposing The solutions to these equations consist of taking a nonabel 5 Conclusions In this paper we havetiplet. constructed The a result nonabelian was on-shell an six- interacting system of equations wh energy solutions. Althoughunclear, the it is M-theory interesting interpretation toof see of these a solutions arise string-like sinc modes defect of between the self-dual parallel stringalso M5-branes. are note in that one-to-one Here the corresponde possibility we of relating dyonic-instanto difference being that here the dyonic-instanton profile is li left-moving projector: Γ with Note that here we have not directly included the contributio in order to satisfy the For ( supercharges bringing the number of preserved ones to 4. Eq. is selfdual in the transverse space. related by selfduality to where we have used the fact that from the second projection, Γ − already mentioned in [ JHEP08(2010)083 . . G µν A × which B G copy of µ A tions on C y with conformal never appears and choice of viewed as exploratory. µν A , Sunil Mukhi, Savdeep ich is expected to contain nimal couplings to B y dimensionally reducing ly six-dimensional system at structures might be at ymmetry. Finally we note theory without also giving Majorana with the correct ection on the spectrum or litonic D-brane states even ates in a local manner by se of this work, while N.L. ensional object in 10 spatial t apparently obtained by di- symmetries and an SO(5) R- . This leads to a nonabelian ents. C.P. would like to thank uge group of the form on of the M5-brane. We found oving BPS states of M2-branes µ A metric Lie 3-algebras, was not ations are essentially just those five dimensions. However in our ing. This problem is reminiscent tive however, and for a spacelike C . It is tempting to speculate that = 0. Here we obtain multiple non- i 0) tensor multiplet. Although the gauge , µ A C h – 13 – gauge theory and then identifying the electric states of one is now constrained to vanish one could consider compactifica G B × A µν ˜ F G we essentially obtain a reformulation of the D4-brane theor µ A 1) Lorentz invariance. We additionally investigated a null , C We could also have considered a timelike choice for Another possible case to study is to set We note that in our construction the nonabelian two-form In terms of applications to M5-branes, our results should be with magnetic states of the other, either as an explicit proj and SO(5 choice of from a 3-algebra. The on-shell conditions are quite restric that this system had dyonicsuspended instanton strings between as parallel the M5-branes. right-m led to a novel supersymmetricmensional system. reduction These of equations ten-dimensional are super-Yang-Mills they no can be related to a lightcone or matrix-model formulati interacting copies of the six-dimensional abelianfield (2 strength of a Euclideanten-dimensional D4-brane Wick-rotated in super-Yang-Mills theory 10 to case, spatial since dimensions, we do obtained notnumber need b of to components. Wick rotate, the fermions remain supersymmetric system in 5symmetry. spatial This directions is with the 16 correctdimensions, symmetry super e.g. to static describe 5-branes a in five-dim 11-dimensions. The equ manifolds which admit non-trivial flat connections. indeed does not seem to exist. Thus we cannot write down any mi Euclidean 3-algebras are often associated with a product ga Even if we hadof achieved equations complete it success would in stilla writing not down Lagrangian. be a enough to ful Neverthelessplay. define it the The is quantum role ofan of interest assumption 3-algebras, to but in try rather particular and arose totally through see antisym the wh demands of supers This may be problematic instates the which quantum are theory minimally of coupled, M5-branes suchof wh as Ramond-Ramond the charges selfdual str in supergravitythough which the appear supergravity as fields so do not couple minimally. Acknowledgments We would like toSethi thank and Jon Kostas Skenderis Bagger, for Dario variousRutgers discussions Martelli, University and and comm Greg CERN Moore for hospitality during the cour This suggests aconsidering method a of realising electric and magneticthrough st an on-shell relation. G JHEP08(2010)083 C , − χ  have (A.5) (A.3) (A.2) (A.1) (A.4) (A.6) χ = 2 ǫ mnp T )Γ C (including e following 1 mnpqr ǫ and 1 )Γ ǫ 1 , mnp ǫ Γ  2 ǫ χ (¯ mnpqr 1 3! Γ 2 acting on the latter are mnpqr − ǫ rived from the 11d Fierz . (¯ χ )Γ 1 5! 1 e partially supported by the ǫ mn = 0 + } χ )Γ ion I 1 the opposite) the surviving terms ǫ Γ mnpqr mn 2 Γ ǫ mn C, )Γ 2 , 1 ǫ { Γ (¯ ǫ 2 . The fermions are Goldstinos of the Ψ ǫ 0 1 SO(5) one has that since and ¯ 5! (¯ mn = 0 = 0 − ǫ . . This makes it antisymmetric 1 1 × 0 C Γ 2! ] = 0 ǫ + } } 2 I T = = Γ I ǫ µ 1) χ − Γ (¯ . One also has that ǫ Γ and , Γ , = Γ Ψ = SO(5) and, by defining the chirality matrix of , , C χ 1 2! C µ – 14 – m C × − Γ mnpq (while ¯ − ¯ Ψ = Ψ { SO(5 )Γ χ 012345 1) = 012345 )Γ T µ 012345 1 012345 , Γ 1 m ǫ Γ 1 Γ ǫ [Γ Γ → m − { − )Γ , where 012345 Γ 1 1) C 1 ǫ 2 = , SO(5 10. The conjugate spinors are defined with the charge − ǫ mnpq m 1 Γ C Γ − → 2 2 + (¯ m ǫ with ǫ C (¯ ,..., Γ (¯ χ 1) µ ) 1 , C 4! C  Γ 1 ǫ − − + C 1 = 6 2 16 ǫ = (¯ I = − †  ] = C m 5, , they and the unbroken supersymmetry parameters satisfy th 2 10. This is what survives by only keeping symmetric matrices 11 2 [ ǫ − ) 2 χ ,..., 1 012345 ,..., − ǫ (¯ = = 0 = 0 − 1 1 ǫ µ ǫ m ) 1) as Γ ). By doing the split SO(10 ) , χ We make use of the appropriate Fierz identities. These are de χ 2 C 2 ǫ ǫ (¯ (¯ the following combination in eleven-dimensions is and for our representation we can choose conjugation matrix SO(5 real and satisfy, Γ STFC rolling grant ST/G000395/1. A Notation, conventions and usefulWe relations work with 32-component Majorana spinors. The Γ-matrices would like to thank Johns Hopkins University. The authors ar and antihermitian chirality conditions where Their (anti)commutation relations are symmetry breaking SO(10 identities by reduction. Starting from the standard expans where the the same chirality with respect to Γ JHEP08(2010)083 χ χ , IJ (A.8) (A.7) ]. Γ ommons . , µνλ  -branes )Γ . (2009) 66 2 χ 1 SPIRES ǫ (2008) 091  M , ][ χ IJ 10 Γ (2008) 105 µνλρσ IJ χ , B 811 Γ I )Γ (2008) 89 (2007) 045020 02 µνλ 1 Γ five-brane with the ǫ Γ µ ]. µνλ 2 JHEP ǫ 456 )Γ )Γ superconformal (¯ , 1 1 D 75 ǫ 1 ǫ = 11 2! µνλρσ JHEP mmercial use, distribution, I r(s) and source are credited. , Γ Γ = 6 IJ 1 ]. ]. D 3! 2 SPIRES er Γ-matrices with the help of ]. µ Γ Nucl. Phys. ǫ (¯ N + Γ ][ , arXiv:0901.1847 2 [ 1 χ 5! ǫ µνλ I -branes Γ Chern-Simons gauge theories 2(¯ Γ Phys. Rept. + 2 2 SPIRES SPIRES µ Phys. Rev. , ǫ χ SPIRES − (¯ , -branes M ][ ][ )Γ 2 ][ = 6 ’s 1 χ 2! 1 2 µ ǫ M N 1 I IJKL 3! M (2009) 097 )Γ Γ Γ 1 + µ µ ǫ Γ µ 04 )Γ 2 hep-th/9701037 [ Γ ǫ – 15 – 1 2 (¯ ǫ . One then gets ǫ 2 Covariant action for a − ǫ 2(¯ χ JHEP -branes and their gravity duals  µ IJKL , 2 hep-th/9701149 arXiv:0807.0163 arXiv:0711.0955 Γ [ 1 [ [ and )Γ 16 M µ 1 (1997) 41 1 ǫ Γ − ǫ ’s. Moreover, the expression is only nonvanishing when ]. ]. ]. ]. µ 2 µ Three-algebras and Gauge symmetry and supersymmetry of multiple Comments on multiple Modeling multiple ǫ ]. ]. Towards the Quantum Geometry of the M5-brane in a Constant Γ = (¯ 2 ǫ 1 2 4! (¯ ǫ B 398 )  + χ SPIRES SPIRES SPIRES SPIRES 1 (1997) 4332 1 16 SPIRES SPIRES ǫ Covariant action for the super-five-brane of M-theory [ ][ ][ ][ ][ (¯ (2009) 025002 (2008) 065008 ][ − 78 Algebraic structures on parallel − Relations between M-brane and D-brane quantum geometries This article is distributed under the terms of the Creative C = M-theory branes and their interactions 1 ǫ 2 ) Phys. Lett. ǫ D 79 D 77 χ ) , 2 χ ǫ 1 (¯ ǫ (¯ − -Field from Multiple Membranes 1 ǫ arXiv:0710.1707 arXiv:0806.1218 arXiv:0712.3738 arXiv:0709.1260 hep-th/0611108 Chern-Simons-matter theories, chiral field Phys. Rev. Lett. C arXiv:1006.5375 [ [ Phys. Rev. Phys. Rev. [ [ [ ) χ 2 [7] D.S. Berman, [8] P. Pasti, D.P. Sorokin and M. Tonin, [9] I.A. Bandos et al., [5] O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, [6] J. Bagger and N. Lambert, [3] J. Bagger and N. Lambert, [4] A. Gustavsson, [2] J. Bagger and N. Lambert, [1] J. Bagger and N. Lambert, ǫ -tensors. The final answer is [10] C.-S. Chu and D.J. Smith, [11] J. Huddleston, (¯ has the opposite chirality from must involve odd powers of Γ References Attribution Noncommercial License whichand permits reproduction any in any nonco medium, provided the original autho Open Access. It is possible toǫ translate the last line above in terms of few JHEP08(2010)083 , ]. , -brane ory and 5 Adv. , M , -Lie algebras , SPIRES 3 (2008) 054 -algebras in ][ 3 ]. 05 ]. ]. Metric (2008) 041 (2008) 003 ]. superconformal gauge , 07 JHEP ]. SPIRES 07 , SPIRES SPIRES = 8 Metric Lie ][ ][ ][ N SPIRES JHEP JHEP ][ ]. arXiv:0902.4674 ]. ]. SPIRES rder solution to the 2, [ , ]. [ Boundary conditions for nde, ]. D ]. Escobar, to bar and P. Ritter, -)algebra and toroidal compactification 2 3 SPIRES More on the Nambu-Poisson SPIRES SPIRES revisited SPIRES ]. D ][ SPIRES ][ ][ 2 ][ SPIRES ]. 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