Multiple Membrane Dynamics

Transcription

1 Multiple Membrane Dynamics Sunil Mukhi Tata Institute of Fundamental Research, Mumbai Strings 2008, Geneva, August 19, 2008

2 Based on: M2 to D2, SM and Costis Papageorgakis, arxiv: [hep-th], JHEP 0805:085 (2008). M2-branes on M-folds, Jacques Distler, SM, Costis Papageorgakis and Mark van Raamsdonk, arxiv: [hep-th], JHEP 0805:038 (2008). D2 to D2, Bobby Ezhuthachan, SM and Costis Papageorgakis, arxiv: [hep-th], JHEP 0807:041, (2008). Mohsen Alishahiha and SM, to appear

3 Outline Motivation and background The Higgs mechanism Lorentzian 3-algebras Higher-order corrections for Lorentzian 3-algebras Conclusions

4 Motivation We understand the field theory on multiple D-branes rather well, but the one on multiple M-branes not so well.

5 Motivation We understand the field theory on multiple D-branes rather well, but the one on multiple M-branes not so well. The latter should hold the key to M-theory:

6 Motivation We understand the field theory on multiple D-branes rather well, but the one on multiple M-branes not so well. The latter should hold the key to M-theory: While there is an obstacle (due to (anti) self-dual 2-forms) to writing the M5-brane field theory, there is no obstacle for M2-branes as far as we know.

7 Motivation We understand the field theory on multiple D-branes rather well, but the one on multiple M-branes not so well. The latter should hold the key to M-theory: While there is an obstacle (due to (anti) self-dual 2-forms) to writing the M5-brane field theory, there is no obstacle for M2-branes as far as we know. And yet, despite 200 recent papers and two Strings 2008 talks on the subject, we don t exactly know what the multiple membrane theory is.

8 Motivation We understand the field theory on multiple D-branes rather well, but the one on multiple M-branes not so well. The latter should hold the key to M-theory: While there is an obstacle (due to (anti) self-dual 2-forms) to writing the M5-brane field theory, there is no obstacle for M2-branes as far as we know. And yet, despite 200 recent papers and two Strings 2008 talks on the subject, we don t exactly know what the multiple membrane theory is. Even the French aristocracy doesn t seem to know...

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10 Of course, there is one description that is clearly right and has manifest N = 8 supersymmetry (but not manifest conformal symmetry): 1 lim 2 LSY M gym gym

11 Of course, there is one description that is clearly right and has manifest N =8supersymmetry (but not manifest conformal symmetry): lim g YM 1 g 2 L SY M YM The question is whether this conformal IR fixed point has an explicit Lagrangian description wherein all the symmetries are manifest.

12 Of course, there is one description that is clearly right and has manifest N =8supersymmetry (but not manifest conformal symmetry): lim g YM 1 g 2 L SY M YM The question is whether this conformal IR fixed point has an explicit Lagrangian description wherein all the symmetries are manifest. This includes a global SO(8) R symmetry describing rotations of the space transverse to the membranes enhanced from the SO(7) of SYM.

13 Of course, there is one description that is clearly right and has manifest N =8supersymmetry (but not manifest conformal symmetry): lim g YM 1 g 2 L SY M YM The question is whether this conformal IR fixed point has an explicit Lagrangian description wherein all the symmetries are manifest. This includes a global SO(8) R symmetry describing rotations of the space transverse to the membranes enhanced from the SO(7) of SYM. Let us look at the Lagrangians that have been proposed to describe this limit.

14 Euclidean 3-algebra [Bagger-Lambert, Gustavsson]: Labelled by integer k. Algebra is SU(2) SU(2). Argued to describe a pair of M2 branes at Z k singularity. But no generalisation to > 2 branes.

15 Euclidean 3-algebra [Bagger-Lambert, Gustavsson]: Labelled by integer k. Algebra is SU(2) SU(2). Argued to describe a pair of M2 branes at Z k singularity. But no generalisation to > 2 branes. Lorentzian 3-algebra [Gomis-Milanesi-Russo, Benvenuti-Rodriguez-Gomez-Tonni-Verlinde, Bandres-Lipstein-Schwarz, Gomis-Rodriguez-Gomez-van Raamsdonk-Verlinde]: Based on arbitrary Lie algebras, have N =8superconformal invariance. Certainly correspond to D2-branes, and perhaps to M 2-branes. Status of latter unclear at the moment.

16 Euclidean 3-algebra [Bagger-Lambert, Gustavsson]: Labelled by integer k. Algebra is SU(2) SU(2). Argued to describe a pair of M2 branes at Z k singularity. But no generalisation to > 2 branes. Lorentzian 3-algebra [Gomis-Milanesi-Russo, Benvenuti-Rodriguez-Gomez-Tonni-Verlinde, Bandres-Lipstein-Schwarz, Gomis-Rodriguez-Gomez-van Raamsdonk-Verlinde]: Based on arbitrary Lie algebras, have N =8superconformal invariance. Certainly correspond to D2-branes, and perhaps to M 2-branes. Status of latter unclear at the moment. ABJM theories [Aharony-Bergman-Jafferis-Maldacena]: Labelled by algebra G G and integer k, with N =6superconformal invariance. Is actually a relaxed 3-algebra. Describe multiple M2-branes at orbifold singularities. But the k =1theory is missing two manifest supersymmetries and decoupling of CM mode not visible.

17 These theories all have 8 scalars and 8 fermions.

18 These theories all have 8 scalars and 8 fermions. And they have non-dynamical (Chern-Simons-like) gauge fields.

19 These theories all have 8 scalars and 8 fermions. And they have non-dynamical (Chern-Simons-like) gauge fields. Thus the basic classification is: (i) Euclidean signature 3-algebras, which are G G Chern-Simons theories: k tr A da A A A Ã dã 2 3Ã Ã Ã BLG : G = SU(2) ABJM : G = SU(N) or U(N), any N (+ other choices) both : scalars, fermions are bi-fundamental, e.g. X I aȧ (ii) Lorentzian signature 3-algebras, which are B F theories based on any Lie algebra. scalars, fermions are singlet + adjoint, e.g. X I +, X I

20 Both classes make use of the triple product X IJK : Euclidean : X IJK X I X J X K, X I bi-fundamental Lorentzian : X IJK X+[X I J, X K ]+cyclic X+ I = singlet, X J = adjoint

21 Both classes make use of the triple product X IJK : Euclidean : X IJK X I X J X K, X I bi-fundamental Lorentzian : X IJK X+[X I J, X K ]+cyclic X+ I = singlet, X J = adjoint The potential is: V (X) (X IJK ) 2 therefore sextic.

22 One might have expected a simple and unique description for the theory on N M2-branes in flat spacetime.

23 One might have expected a simple and unique description for the theory on N M2-branes in flat spacetime. It is basically an analogue of 4d N =4super-Yang-Mills!

24 One might have expected a simple and unique description for the theory on N M2-branes in flat spacetime. It is basically an analogue of 4d N =4super-Yang-Mills! The only excuse we have for not doing better is that the theory we seek will be strongly coupled. So it s not even clear what the classical action means.

25 One might have expected a simple and unique description for the theory on N M2-branes in flat spacetime. It is basically an analogue of 4d N =4super-Yang-Mills! The only excuse we have for not doing better is that the theory we seek will be strongly coupled. So it s not even clear what the classical action means. However it s also maximally superconformal, which should give us a lot of power in dealing with it.

26 One might have expected a simple and unique description for the theory on N M2-branes in flat spacetime. It is basically an analogue of 4d N =4super-Yang-Mills! The only excuse we have for not doing better is that the theory we seek will be strongly coupled. So it s not even clear what the classical action means. However it s also maximally superconformal, which should give us a lot of power in dealing with it. In this talk I ll deal with some things we have understood about the desired theory.

27 Outline Motivation and background The Higgs mechanism Lorentzian 3-algebras Higher-order corrections for Lorentzian 3-algebras Conclusions

28 The Higgs mechanism For the G G Chern-Simons class of theories, the following holds true [SM-Papageorgakis].

29 The Higgs mechanism For the G G Chern-Simons class of theories, the following holds true [SM-Papageorgakis]. If we give a vev v to one component of the bi-fundamental fields, then at energies below this vev, the Lagrangian becomes: L (G G) CS = 1 1 vev v v 2 L(G) SY M + O v 3 and the G gauge field has become dynamical!

30 The Higgs mechanism For the G G Chern-Simons class of theories, the following holds true [SM-Papageorgakis]. If we give a vev v to one component of the bi-fundamental fields, then at energies below this vev, the Lagrangian becomes: L (G G) CS = 1 1 vev v v 2 L(G) SY M + O v 3 and the G gauge field has become dynamical! This is an unusual result. In SYM with gauge group G, when we give a vev to one component of an adjoint scalar, at low energy the Lagrangian becomes: 1 g 2 L (G) SY M YM = 1 vev v g 2 YM L (G G) SY M where G is the subgroup that commutes with the vev.

31 Let s give a quick derivation of this novel Higgs mechanism, first for k =1: L CS = tr A da A A A Ã dã 2 3Ã Ã Ã = tr A F A A A where A ± = A ± Ã, F + = da A + A +.

32 Let s give a quick derivation of this novel Higgs mechanism, first for k =1: L CS = tr A da A A A à dã 2 3à à à = tr A F A A A where A ± = A ± Ã, F + = da A + A +. Also the covariant derivative on a scalar field is: D µ X = µ X A µ X + Xõ

33 Let s give a quick derivation of this novel Higgs mechanism, first for k =1: L CS = tr A da A A A à dã 2 3à à à = tr A F A A A where A ± = A ± Ã, F + = da A + A +. Also the covariant derivative on a scalar field is: If X = v 1 then: D µ X = µ X A µ X + Xõ (D µ X) 2 v 2 (A ) µ (A ) µ +

34 Let s give a quick derivation of this novel Higgs mechanism, first for k =1: L CS = tr A da A A A à dã 2 3à à à = tr A F A A A where A ± = A ± Ã, F + = da A + A +. Also the covariant derivative on a scalar field is: If X = v 1 then: D µ X = µ X A µ X + Xõ (D µ X) 2 v 2 (A ) µ (A ) µ + Thus, A is massive but not dynamical. Integrating it out gives us: 1 1 4v 2 (F +) µν (F + ) µν + O v 3 so A + becomes dynamical.

35 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly.

36 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly. But how should we physically interpret this? L G G CS = 1 vev v v 2 LG SY M + O 1 v 3

37 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly. But how should we physically interpret this? L G G CS = 1 vev v v 2 LG SY M + O 1 v 3 It seems like the M2 is becoming a D2 with YM coupling v.

38 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly. But how should we physically interpret this? L G G CS = 1 vev v v 2 LG SY M + O 1 v 3 It seems like the M2 is becoming a D2 with YM coupling v. Have we somehow compactified the theory? No.

39 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly. But how should we physically interpret this? L G G CS = 1 vev v v 2 LG SY M + O 1 v 3 It seems like the M2 is becoming a D2 with YM coupling v. Have we somehow compactified the theory? No. For any finite v, there are corrections to the SYM. These decouple only as v. So at best we can say that: L G G CS = lim vev v v 1 v 2 LG SY M

40 One can check that the bi-fundamental X I reduces to an adjoint under A +. The rest of N =8SYM assembles itself correctly. But how should we physically interpret this? L G G CS = 1 vev v v 2 LG SY M + O 1 v 3 It seems like the M2 is becoming a D2 with YM coupling v. Have we somehow compactified the theory? No. For any finite v, there are corrections to the SYM. These decouple only as v. So at best we can say that: L G G CS = lim vev v v 1 v 2 LG SY M The RHS is by definition the theory on M2-branes! So this is more like a proof that the original Chern-Simons theory really is the theory on M2-branes.

41 However once we introduce the Chern-Simons level k then the analysis is different [Distler-SM-Papageorgakis-van Raamsdonk]: L G G CS = k k vev v v 2 LG SY M + O v 3

42 However once we introduce the Chern-Simons level k then the analysis is different [Distler-SM-Papageorgakis-van Raamsdonk]: L G G CS = k k vev v v 2 LG SY M + O v 3 If we take k,v with v 2 /k = g YM fixed, then in this limit the RHS actually becomes: 1 g 2 L G SY M YM and this is definitely the Lagrangian for D2 branes at finite coupling.

43 However once we introduce the Chern-Simons level k then the analysis is different [Distler-SM-Papageorgakis-van Raamsdonk]: L G G CS = k k vev v v 2 LG SY M + O v 3 If we take k,v with v 2 /k = g YM fixed, then in this limit the RHS actually becomes: 1 g 2 L G SY M YM and this is definitely the Lagrangian for D2 branes at finite coupling. So this time we have compactified the theory! How can that be?

44 We proposed this should be understood as deconstruction for an orbifold C 4 /Z k : 2π k _ v 2 k v

45 We proposed this should be understood as deconstruction for an orbifold C 4 /Z k : 2π k _ v 2 k v In our paper we observed that the orbifold C 4 /Z k has N =6 supersymmetry and SU(4) R-symmetry. We thought this might be enhanced to N =8for some unknown reason.

46 We proposed this should be understood as deconstruction for an orbifold C 4 /Z k : 2π k _ v 2 k v In our paper we observed that the orbifold C 4 /Z k has N =6 supersymmetry and SU(4) R-symmetry. We thought this might be enhanced to N =8for some unknown reason. Instead, as ABJM found, it s the BLG field theory that needs to be modified to have N =6.

47 We proposed this should be understood as deconstruction for an orbifold C 4 /Z k : 2π k _ v 2 k v In our paper we observed that the orbifold C 4 /Z k has N =6 supersymmetry and SU(4) R-symmetry. We thought this might be enhanced to N =8for some unknown reason. Instead, as ABJM found, it s the BLG field theory that needs to be modified to have N =6. One lesson we learn is that for large k we are in the regime of weakly coupled string theory.

48 We proposed this should be understood as deconstruction for an orbifold C 4 /Z k : 2π k _ v 2 k v In our paper we observed that the orbifold C 4 /Z k has N =6 supersymmetry and SU(4) R-symmetry. We thought this might be enhanced to N =8for some unknown reason. Instead, as ABJM found, it s the BLG field theory that needs to be modified to have N =6. One lesson we learn is that for large k we are in the regime of weakly coupled string theory. A lot can be done in that regime, but for understanding the basics of M2-branes, that is not where we want to be.

49 Outline Motivation and background The Higgs mechanism Lorentzian 3-algebras Higher-order corrections for Lorentzian 3-algebras Conclusions

50 Lorentzian 3-algebras The Lorentzian 3-algebra theories have the following Lagrangian: L (G) L3A = tr 1 2 µνλ B µ F νλ 1 2 ˆD µ X I ˆDµ X I 1 12 X I + [X J, X K ]+X J +[X K, X I ]+X K + [X I, X J ] 2 + (C µi µ X I ) µ X I + + L gauge fixing + L fermions where ˆD µ X I µ X I [A µ, X I ] B µ X I +

51 Lorentzian 3-algebras The Lorentzian 3-algebra theories have the following Lagrangian: L (G) L3A = tr 1 2 µνλ B µ F νλ 1 2 ˆD µ X I ˆDµ X I 1 12 X I + [X J, X K ]+X J +[X K, X I ]+X K + [X I, X J ] 2 + (C µi µ X I ) µ X I + + L gauge fixing + L fermions where ˆD µ X I µ X I [A µ, X I ] B µ X I + These theories have no parameter k.

52 Lorentzian 3-algebras The Lorentzian 3-algebra theories have the following Lagrangian: L (G) L3A = tr 1 2 µνλ B µ F νλ 1 2 ˆD µ X I ˆDµ X I 1 12 X I + [X J, X K ]+X J +[X K, X I ]+X K + [X I, X J ] 2 + (C µi µ X I ) µ X I + + L gauge fixing + L fermions where ˆD µ X I µ X I [A µ, X I ] B µ X I + These theories have no parameter k. They have SO(8) global symmetry acting on the indices I,J,K 1, 2,, 8.

53 Lorentzian 3-algebras The Lorentzian 3-algebra theories have the following Lagrangian: L (G) L3A = tr 1 2 µνλ B µ F νλ 1 2 ˆD µ X I ˆDµ X I 1 12 X I + [X J, X K ]+X J +[X K, X I ]+X K + [X I, X J ] 2 + (C µi µ X I ) µ X I + + L gauge fixing + L fermions where ˆD µ X I µ X I [A µ, X I ] B µ X I + These theories have no parameter k. They have SO(8) global symmetry acting on the indices I,J,K 1, 2,, 8. The equation of motion of the auxiliary gauge field C I µ implies that X + = constant.

54 Our Higgs mechanism works in these theories, but it works too well! [Ho-Imamura-Matsuo]

55 Our Higgs mechanism works in these theories, but it works too well! [Ho-Imamura-Matsuo] On giving a vev to the singlet field X I +, say: X 8 + = v one finds: L (G) L3A = 1 vev v v 2 L(G) SY M (+ no corrections)

56 Our Higgs mechanism works in these theories, but it works too well! [Ho-Imamura-Matsuo] On giving a vev to the singlet field X I +, say: X 8 + = v one finds: L (G) L3A = 1 vev v v 2 L(G) SY M (+ no corrections) This leads one to suspect that the theory is a re-formulation of SYM.

57 Our Higgs mechanism works in these theories, but it works too well! [Ho-Imamura-Matsuo] On giving a vev to the singlet field X I +, say: X 8 + = v one finds: L (G) L3A = 1 vev v v 2 L(G) SY M (+ no corrections) This leads one to suspect that the theory is a re-formulation of SYM. In fact it can be derived [Ezhuthachan-SM-Papageorgakis] starting from N =8SYM.

58 The procedure involves a non-abelian (dns) duality [dewit-nicolai-samtleben] on the (2+1)d gauge field.

59 The procedure involves a non-abelian (dns) duality [dewit-nicolai-samtleben] on the (2+1)d gauge field. Start with N =8SYM in (2+1)d. Introducing two new adjoint fields B µ, φ, the dns duality transformation is: 1 F µν F 4gYM 2 µν 1 2 µνλ B µ F νλ 1 2 (D µφ g YM B µ ) 2 Note that D µ is the covariant derivative with respect to the original gauge field A.

60 The procedure involves a non-abelian (dns) duality [dewit-nicolai-samtleben] on the (2+1)d gauge field. Start with N =8SYM in (2+1)d. Introducing two new adjoint fields B µ, φ, the dns duality transformation is: 1 F µν F 4gYM 2 µν 1 2 µνλ B µ F νλ 1 2 (D µφ g YM B µ ) 2 Note that D µ is the covariant derivative with respect to the original gauge field A. In addition to the gauge symmetry G, the new action has a noncompact abelian gauge symmetry: δφ = g YM M, δb µ = D µ M where M(x) is an arbitrary matrix in the adjoint of G.

61 The procedure involves a non-abelian (dns) duality [dewit-nicolai-samtleben] on the (2+1)d gauge field. Start with N =8SYM in (2+1)d. Introducing two new adjoint fields B µ, φ, the dns duality transformation is: 1 F µν F 4gYM 2 µν 1 2 µνλ B µ F νλ 1 2 (D µφ g YM B µ ) 2 Note that D µ is the covariant derivative with respect to the original gauge field A. In addition to the gauge symmetry G, the new action has a noncompact abelian gauge symmetry: δφ = g YM M, δb µ = D µ M where M(x) is an arbitrary matrix in the adjoint of G. To prove the duality, use this symmetry to set φ =0. Then integrating out B µ gives the usual YM kinetic term for F µν.

62 The dns-duality transformed N =8SYM is: L = tr 1 2 µνλ B µ F νλ Dµ φ g YM B µ 1 2 D µx i D µ X i g2 YM 4 [X i, X j ] 2 + fermions

63 The dns-duality transformed N =8SYM is: L = tr 1 2 µνλ B µ F νλ Dµ φ g YM B µ 1 2 D µx i D µ X i g2 YM 4 [X i, X j ] 2 + fermions We can now see the SO(8) invariance appearing.

64 The dns-duality transformed N =8SYM is: L = tr 1 2 µνλ B µ F νλ Dµ φ g YM B µ 1 2 D µx i D µ X i g2 YM 4 [X i, X j ] 2 + fermions We can now see the SO(8) invariance appearing. Rename φ X 8. Then the scalar kinetic terms are: 1 2 ˆD µ X I ˆDµ X I = 1 2 µ X I [A µ, X I ] g I YMB µ 2 where g I YM = (0,...,0,g YM ).

65 The dns-duality transformed N =8SYM is: L = tr 1 2 µνλ B µ F νλ Dµ φ g YM B µ 1 2 D µx i D µ X i g2 YM 4 [X i, X j ] 2 + fermions We can now see the SO(8) invariance appearing. Rename φ X 8. Then the scalar kinetic terms are: 1 2 ˆD µ X I ˆDµ X I = 1 2 µ X I [A µ, X I ] g I YMB µ 2 where g I YM = (0,...,0,g YM ). Next, we can allow g I YM to be an arbitrary 8-vector.

66 The action is now SO(8)-invariant if we rotate both the fields X I and the coupling-constant vector gym: I L = tr 1 2 µνλ B µ F νλ 1 ˆD 2 µ X I ˆDµ X I g I YM[X J, X K ]+gym[x J K, X I ]+gym[x K I, X J ]

67 The action is now SO(8)-invariant if we rotate both the fields X I and the coupling-constant vector gym: I L = tr 1 2 µνλ B µ F νλ 1 ˆD 2 µ X I ˆDµ X I g I YM[X J, X K ]+gym[x J K, X I ]+gym[x K I, X J ] This is not yet a symmetry, since it rotates the coupling constant.

68 The action is now SO(8)-invariant if we rotate both the fields X I and the coupling-constant vector gym: I L = tr 1 2 µνλ B µ F νλ 1 ˆD 2 µ X I ˆDµ X I g I YM[X J, X K ]+gym[x J K, X I ]+gym[x K I, X J ] This is not yet a symmetry, since it rotates the coupling constant. The final step is to introduce an 8-vector of new (gauge-singlet) scalars X I + and replace: g I YM X I +(x)

69 The action is now SO(8)-invariant if we rotate both the fields X I and the coupling-constant vector gym: I L = tr 1 2 µνλ B µ F νλ 1 ˆD 2 µ X I ˆDµ X I g I YM[X J, X K ]+gym[x J K, X I ]+gym[x K I, X J ] This is not yet a symmetry, since it rotates the coupling constant. The final step is to introduce an 8-vector of new (gauge-singlet) scalars X I + and replace: g I YM X I +(x) This is legitimate if and only if X I +(x) has an equation of motion that renders it constant. Then on-shell we can recover the original theory by writing X I + = g I YM.

70 Constancy of X I + is imposed by introducing a new set of abelian gauge fields and scalars: C I µ,x I and adding the following term: L C =(C µ I XI ) µ X I +

71 Constancy of X I + is imposed by introducing a new set of abelian gauge fields and scalars: C I µ,x I and adding the following term: This has a shift symmetry L C =(C µ I XI ) µ X I + δx I = λ I, δc I µ = µ λ I which will remove the negative-norm states associated to C I µ.

72 Constancy of X I + is imposed by introducing a new set of abelian gauge fields and scalars: C I µ,x I and adding the following term: This has a shift symmetry L C =(C µ I XI ) µ X I + δx I = λ I, δc I µ = µ λ I which will remove the negative-norm states associated to C I µ. We have thus ended up with the Lorentzian 3-algebra action [Bandres-Lipstein-Schwarz, Gomis-Rodriguez-Gomez-van Raamsdonk-Verlinde]: L = tr 1 2 µνλ B µ F νλ 1 ˆD 2 µ X I ˆDµ X I X I + [X J, X K ]+X+[X J K, X I ]+X+ K [X I, X J ] (C µi µ X I ) µ X I + + L gauge fixing + L fermions

73 The final action has some remarkable properties.

74 The final action has some remarkable properties. It has manifest SO(8) invariance as well as N =8 superconformal invariance.

75 The final action has some remarkable properties. It has manifest SO(8) invariance as well as N =8 superconformal invariance. However, both are spontaneously broken by giving a vev X I + = g I YM and the theory reduces to N =8SYM with coupling g YM.

76 The final action has some remarkable properties. It has manifest SO(8) invariance as well as N =8 superconformal invariance. However, both are spontaneously broken by giving a vev X I + = g I YM and the theory reduces to N =8SYM with coupling g YM. It will certainly describe M2-branes if one can find a way to take X I + =. That has not yet been done.

77 Outline Motivation and background The Higgs mechanism Lorentzian 3-algebras Higher-order corrections for Lorentzian 3-algebras Conclusions

78 Higher-order corrections for Lorentzian 3-algebras One might ask if the non-abelian duality that we have just performed works when higher order (in α ) corrections are included.

79 Higher-order corrections for Lorentzian 3-algebras One might ask if the non-abelian duality that we have just performed works when higher order (in α ) corrections are included. For the Abelian case [Duff, Townsend, Schmidhuber] we know that the analogous duality works for the entire DBI action and that fermions and supersymmetry can also be incorporated [Aganagic-Park-Popescu-Schwarz].

80 Higher-order corrections for Lorentzian 3-algebras One might ask if the non-abelian duality that we have just performed works when higher order (in α ) corrections are included. For the Abelian case [Duff, Townsend, Schmidhuber] we know that the analogous duality works for the entire DBI action and that fermions and supersymmetry can also be incorporated [Aganagic-Park-Popescu-Schwarz]. Recently we have shown [Alishahiha-SM] that to lowest nontrivial order (F 4 -type corrections) one can indeed dualise the non-abelian SYM into an SO(8)-invariant form.

81 Higher-order corrections for Lorentzian 3-algebras One might ask if the non-abelian duality that we have just performed works when higher order (in α ) corrections are included. For the Abelian case [Duff, Townsend, Schmidhuber] we know that the analogous duality works for the entire DBI action and that fermions and supersymmetry can also be incorporated [Aganagic-Park-Popescu-Schwarz]. Recently we have shown [Alishahiha-SM] that to lowest nontrivial order (F 4 -type corrections) one can indeed dualise the non-abelian SYM into an SO(8)-invariant form. Here of course one cannot do all orders in α because a non-abelian analogue of DBI is still not known.

82 Higher-order corrections for Lorentzian 3-algebras One might ask if the non-abelian duality that we have just performed works when higher order (in α ) corrections are included. For the Abelian case [Duff, Townsend, Schmidhuber] we know that the analogous duality works for the entire DBI action and that fermions and supersymmetry can also be incorporated [Aganagic-Park-Popescu-Schwarz]. Recently we have shown [Alishahiha-SM] that to lowest nontrivial order (F 4 -type corrections) one can indeed dualise the non-abelian SYM into an SO(8)-invariant form. Here of course one cannot do all orders in α because a non-abelian analogue of DBI is still not known. However our approach may have a bearing on that unsolved problem.

83 Let us see how this works. In (2+1)d, the lowest correction to SYM for D2-branes is the sum of the following contributions (here X ij =[X i, X j ]): L (4) 1 = 1 12g 4 YM L (4) 2 = 1 12g 2 YM L (4) 3 = 1 6 F µν F ρσ F µρ F νσ F µνf νρ F ρσ F σµ 1 4 F µνf µν F ρσ F ρσ 1 8 F µνf ρσ F µν F ρσ F µν D µ X i F ρν D ρ X i + F µν D ρ X i F µρ D ν X i 2F µρ F ρν D µ X i D ν X i 2F µρ F ρν D ν X i D µ X i F µν F µν D ρ X i D ρ X i 1 2 F µν D ρ X i F µν D ρ X i 1 2 F µν F µν X ij X ij F µν X ij F µν X ij 1 12 D µ X i D ν X j F µν + D ν X j F µν D µ X i + F µν D µ X i D ν X j X ij

84 L (4) 4 = 1 12 D µ X i D ν X j D ν X i D µ X j + D µ X i D ν X j D µ X j D ν X i + D µ X i D ν X i D ν X j D µ X j D µ X i D µ X i D ν X j D ν X j 1 2 D µx i D ν X j D µ X i D ν X j L (4) 5 = g2 YM 12 X kj D µ X k X ij D µ X i + X ij D µ X k X ik D µ X j 2X kj X ik D µ X j D µ X i 2X ki X jk D µ X j D µ X i X ij X ij D µ X k D µ X k 1 2 Xij D µ X k X ij D µ X k L (4) 6 = g4 YM 12 X ij X kl X ik X jl Xij X jk X kl X li 1 4 Xij X ij X kl X kl 1 8 Xij X kl X ij X kl

85 We have been able to show that this is dual, under the dns transformation, to: L = tr 1 2 µνρ B µ F νρ 1 ˆD 2 µ X I ˆDµ X I ˆDµ X I ˆDν X J ˆDν X I ˆDµ X J + ˆD µ X I ˆDν X J ˆDµ X J ˆDν X I ˆD µ X I ˆDν X I ˆDν X J ˆDµ X J ˆD µ X I ˆDµ X I ˆDν X J ˆDν X J 1 ˆD 2 µ X I ˆDν X J ˆDµ X I ˆDν X J 1 2 XLKJ ˆDµ X K X LIJ ˆDµ X I XLIJ ˆDµ X K X LIK ˆDµ X J X LKJ X LIK ˆDµ X J ˆDµ X I X LKI X LJK ˆDµ X J ˆDµ X I 1 3 XLIJ X LIJ ˆDµ X K ˆDµ X K 1 6 XLIJ ˆDµ X K X LIJ ˆDµ X K 1 6 ρµν ˆD ρ X I ˆDµ X J ˆDν X K X IJK V (X)

86 In the previous expression, ˆD µ X I = µ X I [A µ, X I ] B µ X+ I X IJK = X+[X I J, X K ]+X+[X J K, X I ]+X+ K [X I, X J ]

87 In the previous expression, ˆD µ X I = µ X I [A µ, X I ] B µ X+ I X IJK = X+[X I J, X K ]+X+[X J K, X I ]+X+ K [X I, X J ] Here V (X) is the potential: V (X) = 1 12 XIJK X IJK X NIJ X NKL X MIK X MJL XNIJ X MJK X NKL X MLI 1 4 XNIJ X NIJ X MKL X MKL 1 8 XNIJ X MKL X NIJ X MKL

88 In the previous expression, ˆD µ X I = µ X I [A µ, X I ] B µ X+ I X IJK = X+[X I J, X K ]+X+[X J K, X I ]+X+ K [X I, X J ] Here V (X) is the potential: V (X) = 1 12 XIJK X IJK X NIJ X NKL X MIK X MJL XNIJ X MJK X NKL X MLI 1 4 XNIJ X NIJ X MKL X MKL 1 8 XNIJ X MKL X NIJ X MKL We see that the dual Lagrangian is SO(8) invariant.

89 In the previous expression, ˆD µ X I = µ X I [A µ, X I ] B µ X+ I X IJK = X+[X I J, X K ]+X+[X J K, X I ]+X+ K [X I, X J ] Here V (X) is the potential: V (X) = 1 12 XIJK X IJK X NIJ X NKL X MIK X MJL XNIJ X MJK X NKL X MLI 1 4 XNIJ X NIJ X MKL X MKL 1 8 XNIJ X MKL X NIJ X MKL We see that the dual Lagrangian is SO(8) invariant. It s worth noting that this depends crucially on the relative coefficients of various terms in the original Lagrangian.

90 We see from this that the 3-algebra structure remains intact when higher-derivative corrections are taken into account.

91 We see from this that the 3-algebra structure remains intact when higher-derivative corrections are taken into account. We conjecture that SO(8) enhancement holds to all orders in α.

92 We see from this that the 3-algebra structure remains intact when higher-derivative corrections are taken into account. We conjecture that SO(8) enhancement holds to all orders in α. Unfortunately the all-orders corrections are not known for SYM, so we don t have a starting point from which to check this.

93 Outline Motivation and background The Higgs mechanism Lorentzian 3-algebras Higher-order corrections for Lorentzian 3-algebras Conclusions

94 Summary Much progress has been made towards finding the multiple membrane field theory representing the IR fixed point of N =8SYM.

95 Summary Much progress has been made towards finding the multiple membrane field theory representing the IR fixed point of N =8SYM. But we don t seem to be there yet.

96 Summary Much progress has been made towards finding the multiple membrane field theory representing the IR fixed point of N =8SYM. But we don t seem to be there yet. The existence of a large-order orbifold (deconstruction) limit provides a way (the only one so far) to relate the membrane theory to D2-branes. One would like to understand compactification of transverse or longitudinal directions, as we do for D-branes.

97 Summary Much progress has been made towards finding the multiple membrane field theory representing the IR fixed point of N =8SYM. But we don t seem to be there yet. The existence of a large-order orbifold (deconstruction) limit provides a way (the only one so far) to relate the membrane theory to D2-branes. One would like to understand compactification of transverse or longitudinal directions, as we do for D-branes. An interesting mechanism has been identified to dualise the D2-brane action into a superconformal, SO(8) invariant one. The result is a Lorentzian 3-algebra and this structure is preserved by α corrections.

98 A detailed understanding of multiple membranes should open a new window to M-theory and 11 dimensions.

99 A detailed understanding of multiple membranes should open a new window to M-theory and 11 dimensions....if you were as tiny as a graviton You could enter these dimensions and go wandering on And they d find you...

100