SU(N) membrane B∧F theory with dual-pairs

arXiv:0811.1690v3 [hep-th] 4 Dec 2008 hep-th/yymm.nnnn November 3, 2018 SU (N ) membrane B ∧ F theory with dual-pairs Harvendra Singh Saha Institute of Nuclear Physics 1/AF, Bidhannagar, Kolkata 700064, India E-mail: h.singh [AT] saha.ac.in Abstract We construct a SU(N) membrane B ∧ F theory with dual pairs of scalar and tensor fields. The moduli space of the theory is precisely that of N M2-branes on the noncompact flat space. The theory possesses explicit SO(8) invariance and is an extension of the maximal SU(N) super-Yang-Mills theory. 1 Interestingly, recently certain type of matter Chern-Simons field theories in (1+2) dimensions have been proposed to be the low energy theories describing supermembranes. Amongst these, the originally proposed Bagger-Lambert-Gustavsson (BLG) theory has N = 8 SO(8) superconformal invariance but the theory is known only for SO(4) tri-algebra [1, 2]. Although for noncompact case of tri-algebras, BLG theory can be extended to admit SU(N) symmetry [3]. But these theories have ghost fields in the spectrum and once these are gauge-fixed the theory eventually reduces to the SU(N) super Yang-mills [4]. Another interesting class of matterChern-Simons theories proposed by Aharony-Bergman-Jafferis-Maldacena (ABJM) [5], however have ordinary Lie-algebra structure. These theories admit N = 6 SU(N)k × SU(N)−k superconformal symmetry, and is conjectured to be dual to M-theory on AdS4 × S 7 /Zk with the level k > 2. For k = 1, 2 the theory supposedly becomes maximally supersymmetric BLG theory. It is now clear that the understanding of Chern-Simons theories is essential to know the M-theory origin of the SU(N) Yang-Mills theory which describes N D2branes on R7 , and vice-versa. In particular, in the works [6, 7] the authors have attempted to understand this link to some extent. The work [6] is of particular importance to us in this paper. We take a parallel but rather distinct approach I where we augment the B-F theory with scalars and dual 2-rank tensor fields, C(2) . This leads us to a membrane B-F theory which has SU(N) gauge symmetry and has SO(8) R-invariance as well as the scale invariance. The theory does not have any ghost degrees of freedom and also has no tri-algebras. It presumably also has maximal supersymmetry as it is simply the topological extension of the 3D super Yang-Mills theory. The low energy SU(N) super Yang-Mills theory with maximal supersymmetry is written as SSY M = 1 gY2 M ij 2 1 µν i µ i F F − D X D X − (X ) µν µ 4gY2 M 2 4 i i + Ψ̄γ µ Dµ Ψ + Ψ̄Γi [X i , Ψ]) 2 2 Z d3 x T r(− (1) where we defined X ij = [X i , X j ]. The field strengths are Dµ X i = ∂µ X i − [Aµ , X i ] Fµν = ∂[µ Aν] − [Aµ , Aν ], (2) The bosonic fields (Aµ ; X 1 , ..., X 7 ) are all in the adjoint of SU(N) and the fermions ΨA α form 2-compt spinor of 3D and 8-compt spinor of SO(7). The theory has an explicit SO(7) R-symmetry under which supercharges get rotated. The YM theory actually lives on the boundary of AdS4 × S 6 (with varying string coupling), which is the near horizon geometry of N D2-branes. 2 The scale (mass) dimensions are 1 [X i ] = , [Aµ ] = 1, [Ψ] = 1, [gY2 M ] = 1 2 Notice that the Yang-Mills coupling constant is dimensionful in 3D ! So the super Yang-Mills can hardly be a conformal theory. In fact the YM coupling has a flow. Although the theory has good high energy behaviour where it becomes a free theory in UV regime, but in IR it is known to flow to a strongly coupled superconformal fixed point. The conformal nature of the YM theory at the IR fixed point has remained illusive though. Whether it describes M2-brane theory has not been quite clear? For several reasons it is expected that a theory of multiple M2-branes in flat space should have maximal supersymmetry, should be conformal, should have SO(8) Rsymmetry and possibly a gauge symmetry if it were an interacting theory. But the actual content of the theory has remained illusive so far. A way ahead was suggested by the authors [6] where one can make use of de-Wit-Nicolai-Samtleben duality transformations [8]. The dNS proposal is based on the fact that a propagating vector field in 3D contributes one degree of freedom. It is a familiar kind of Poincare duality between gauge field and a scalar field ( 2!1 ǫµνλ Fνλ = ∂ µ φ). Instead in a non-Abelian situation we can define 1 ǫµνλ Fνλ = D µ φ − gY M B µ 2!gY M (3) We can easily see that − 1 1 1 T r(Fµν )2 ≡ − T r(D µ φ − gY M B µ )2 + T r( ǫµνλ Bµ Fνλ ) 2 4gY M 2 2 (4) Thus the duality introduces a pair of adjoint fields Bµ and φ. This duality has been used in going from SO(7) super-Yang-Mills to the BLG theory which has SO(8) R-symmetry [6]. Actually after incorporating the dNS transformation the SYM Lagrangian takes the form of a matter B-F (BF) Lagrangian 1 1 1 d3 x T r(− (D µ φ − gY M B µ )2 + ǫµνλ Bµ Fνλ − Dµ X i D µ X i 2 2 2 2 i i g − Y M (X ij )2 + Ψ̄γ µ Dµ Ψ + Ψ̄Γi [X i , Ψ]) 4 2 2 SBF = Z (5) where we can now identify φ with X 8 . This field alongwith the rest X i ’s forms an SO(8) vector: X I (1 ≤ I ≤ 8). One then also defines a coupling constant 8-vector: g I = (0, · · · , 0, gY M ). 3 With this we can write BF Lagrangian in an SO(8) covariant Lagrangian form [6] 1 1 d3 x T r(− (D µ X I − g I B µ )2 + ǫµνλ Bµ Fνλ − U(g I , X I ) 2 2 i i + Ψ̄γ µ Dµ Ψ + g I Ψ̄ΓIJ [X J , Ψ]) 2 2 SBF = Z (6) where the potential is defined as U= 1 VIJK V IJK 2.3! (7) with the help of a completely antisymmetrized object VIJK = g[I XJK] ≡ gI XJK + cyclic permutations (8) Specially, we must note that parameters g I are in the 8v while X IJ = [X I , X J ] are in the adjoint of SO(8) group. So as such the antisymmetrization of VIJK should not be confused with any tri-algebra like in BLG theory. However, it can be extended to have a Lorentzian tri-algebra structure [6]. 1 The action (6) has an SO(8) invariance provided the couplings g I transform along with various fields under SO(8) rotations. Thus, although the theory has SO(8) invariance but its action is transitive on the coupling parameters in the theory. After the transformations we get a new theory with a new set of couplings. The N = 8 susy transformations can also be formally written in SO(8) covariant form [6]. It is noteworthy here to mention that such phenomena have also been observed in the case of massive supergravity theory as well, see for an instance [9]. In the present scenario, the legitimate step would be like that in the Romans’ theory in ten dimensions [10]. There we try to lift the mass parameter (cosmological constant) m to the level of a scalar field M(x) which is then Hodge-dualised to a 10-form field strength F10 [11]. This does not introduce any new degree of freedom in the theory. Instead now the values of the mass parameters become localised in the spacetime. We shall like to implement the same idea here for the 3D case. Note that we have couplings g I in the vector representation of SO(8). So we first define correspondingly 8 scalar fields η I (x) such that g I = < η I (x) >, g I g I = (gY M )2 (9) I In the next step, we introduce 2-form potential C(2) , also in the 8v , whose field I strength will be dual to η . We must also make sure that the vacua are such that 1 Here the SO(8) gamma matrices are Γ8 = Γ̃8 , Γi = Γ̃8 Γ̃i . (The matrices with tilde will henceforth will be named as SO(7) matrices.) 4 η I will be constant. This can be done simply by introducing a new topological term in the SO(8) covariant BF action − Z I C(2) ∧ dη I (10) which is SO(8) invariant and has the gauge invariance under I I I C(2) → C(2) + dα(1) Thus the complete membrane action can be written as (11) 2 3 1 1 1 (VIJK )2 ) d3 x T r(− (D µ X I − η I B µ )2 + ǫµνλ Bµ Fνλ − 2 2 2.3! 1 I − ǫµνλ Cµν ∂λ η I + Sf ermions (12) 2 SM BF = Z where D µ X I = ∂ µ X I − [Aµ , X I ], VIJK = η[I XJK] . The equations of motion are now augmented with two new set of equations. Namely the C I equation ∂λ η I = 0 (13) and the η I equation 1 1 I =0 T r((D µX I − η I B µ )Bµ − V IJK XJK ) + ǫµνλ ∂µ Cνλ 2 2 (14) The C I -equation implies that all η I are constant. The second equation only relates I η I with its dual tensor field Cνλ and should be taken as the duality relation. The rest of the field equations remain unchanged. So the net content of the theory remains intact. There are no free parameters in the theory. The action (12) also has scale invariance. The supersymmetry presumably can also be made manifest which we do not work out here. Henceforth we shall refer to the action (12) as membrane B-F (MBF) theory. Thus in bringing the BF theory to the MBF form we have actually introduced dual pairs of fields (C I , η I ). The introduction of these dual pairs has introduced a new paradigm in the MBF theory. The moduli space of vacua in the MBF theory 2 R At this point we may be tempted to add another possible topological term −θ C(3) , as it does not affect any of the dynamical considerations. Although from topological perspectives it will be necessary. 3 The C(3) can also be relevant while quantising the theory in the nontrivial membrane background. I thank S. Mukhi for this useful remark. 5 is now larger than the original SYM/BF theory. To know the moduli space of the MBF theory we need to solve 1 µνλ I ǫ ∂µ Cνλ =0 2! I → η [I T r(X JK]XJK ) − ǫµνλ ∂µ Cνλ =0 ∂ηI U(η, X) − (15) and ∂X I U(η, X) = 0 (16) These equations have quite a few interesting possibilities. I Case-1: We take first Cµν = constt. Since the solution of η I (x) = g I , we find that we need to have X IJ = [X I , X J ] = 0. (17) This can happen when all the X I ’s are taken to be diagonal matrices. That means all M2-branes are coincident. Hence the moduli space is exactly that of N coincident M2-branes on noncompact R8 space. However, the special case can arise when we take η 8 = gY M , η i = 0 . (18) X ij = 0 . (19) This will then require In the simplest case all X i can be taken diagonal, but matrices X 8 can still be nontrivial but constant. These presumably will be the desired Goldstone modes corresponding to the spontaneously broken SO(8) invariance. These will be eaten up by Bµ fields and making them heavy which can be integrated out in order to make the Aµ fields dynamical. All this precisely corresponds to the moduli space of N D2-branes on R7 . For both of the above solutions the components VIJK are vanishing hence the scalar potential altogether vanishes. So these would make the maximally supersymmetric vacua in MBF theory. Case-2: Another rather interesting case is of 3D domain-walls. Let us take the I tensor components C01 to be linearly dependent on one of the space coordinates, x2 (say), then dC I ∼ mI dx0 ∧ dx1 ∧ dx2 (20) is nontrivial, the mI being the slope parameters. The two such phases with different slopes can be separated via domain-walls which are just the line defects in 2-dimensional plane. In this situation, we shall have g I and mI related via 1 [I g T r(X JK]XJK ) − mI = 0 2 6 (21) This will describe a noncommuting (fuzzy) configuration of membranes. However, we are not sure if any static nontrivial fuzzy configuration can be found in which eq. (16) will be simultaneously satisfied. In any case, it will be interesting to find nonstatic solutions. Quantisation: At this point, let us also discuss an interesting quantum aspect which follows straightforwardly from action (12). The equation of motion for X I , 1 µνλ ǫ Fνλ = (D µ X I − η I B µ )η I , 2! (22) and the equation (14) can be combined to give 1 1 I T r( ǫµνλ Fνλ Bµ − U) + η I ǫµνλ ∂µ Cνλ =0 2! 2 (23) In the vacuum where U = 0, it has interesting implications. For example, consider an Euclidean monopole configuration where Fµν 6= 0 inside a 3-dimensional volume V 3 , with a boundary ∂V 3 ∼ S 2 . We can have a configuration where Tr 1 4π Z V3 B ∧ F ∼ p(N) , (24) Here we have taken p(N) ∈ Z to depend upon the rank N of the Yang-Mills group. The actual expression of p(N) however will depend upon the details of the monopole configuration. We are taking SO(7) configuration where η 8 = gY M , η i = 0. The equation (23) leads us to the quantization − 1 q 4π lp Z V3 dC(2) = − 1 q 4π lp Z S2 C(2) = k ∈ Z . (25) ) with gY M ∼ (lp(N 1/2 k , and we introduced lp , the 11-dimensional Planck length. That p) is we need to have a nontrivial C(2) flux over S 2 . It does mean that Yang-Mills coupling in a given topological vacuum is controlled by the ratio of p(N) and k. By having large k limit we can accomodate a weak Yang-Mills coupling. This argument appears almost analogous to large k limit in C4 /Zk orbifold models [5]. In summary, we have taken an approach where we augmented the B-F theory I with scalars and dual 2-rank tensor fields, namely (η I , C(2) ). This led us to a membrane B-F theory which has SU(N) gauge symmetry and has SO(8) R-invariance as well as the scale invariance. There are no free parameters in the action. The theory does not have any ghost degrees of freedom and also has no tri-algebras. So in that aspect our theory is distinct from the B-F theory of [6]. Our theory presumably also has maximal supersymmetry as it is simply the topological extension of the 3D 7 super Yang-Mills theory. Interestingly, the moduli space comes out to be that of N coincident M2-branes on transverse R8 . ——————— Acknowledgements: I am very much thankful to S. Roy and specially S. Mukhi for several helpful discussions. References [1] J. Bagger and N. hep-th/0712.3738. Lambert, hep-th/0611108; hep-th/0711.0955; [2] A. Gustavsson, arXiv:0709.1260 [hep-th]; A. Gustavsson, JHEP 0804, 083 (2008) [arXiv:0802.3456 [hep-th]]. [3] J Gomis, G. Milanesi and J.G. Russo, arXiv:0804.1012; S. Benvenuti, D. Rodriguez-Gomez, E. Tonni and H. Verlinde, arXiv:0805.1087 [hepth]. J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, H. Verlinde, arXiv:0806.0738v2 [hep-th] [4] M. A. Bandres, A. E. Lipstein and J. H. Schwarz, arXiv:0806.0054 [hep-th]; M. A. Bandres, A. E. Lipstein and J. H. 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