Relation between non trivial M2-branes and D2-branes with fluxes
RRelation between non trivial M2-branes andD2-branes with fluxes
M P Garcia del Moral ,a , C Las Heras ,b Departamento de F´ısica, Universidad de Antofagasta, Antofagasta, Chile.E-mail: a [email protected], b [email protected] Abstract.
We show the relation between three non trivial sectors of M2-brane theoryformulated in the LCG connected among them by canonical transformations. These sectorscorrespond to the supermembrane theory formulated on a M × T on three different constantthree-form backgrounds: M2-brane with constant C − , M2-brane with constant C ± and M2-brane with a generic constant C denoted as CM2-brane. The first two exhibit a purely discretesupersymmetric spectrum once the central charge condition, or equivalently, the correspondingflux condition has been turned on. The CM2-brane is conjectured to share this spectral propertyonce that fluxes C ± are turned on. As shown in [1] they are duals to three inequivalent sectors ofthe D2-branes with specific worldvolume and background RR and NSNS quantization conditionson each case.
1. Introduction
Wrapped M2-brane theories were first considered in [2, 3] as potential interesting well-definedquantum sectors of M-theory. In [4, 5] it was shown that compactification by itself does notremove classical instabilities and the spectrum at quantum level remains continuous. However,a sector of the wrapped M2-brane subject to a topological condition associated to an irreduciblewrapping was introduced in [6] and denoted by supermembrane with central charges. In [7]the discreteness of its supersymmetric spectrum was proved. Besides the deep characterizationof this sector of the theory that describes some of the microscopical degrees of freedom of M-theory, another nontrivial sector was found in [8] associated to a wrapped M2-brane on thesame flat metric and under the presence of C ± fluxes. For the particular case when only the C − flux is present both theories are duals or equivalent. In [1] it was obtained the LCG toroidallywrapped supermembrane with a general constant three form background formulation, namedCM2-branes. This sector becomes also nontrivial under a C ± flux condition. We will brieflyreview these sectors, and make some comments on their principal features and show their explicitrelations among them. Moreover, in [1] we obtained for each sector its D2-brane duals whichcorrespond to D2 branes with specific worldvolume and background RR and NSNS fluxes. Wedenote these sectors as nontrivial D2-branes and we will discuss the relation among them.
2. Non trivial M2 branes
Let us consider the Light-Cone (LC) formulation of a M2-brane theory on M × T on a constantthree-form background C denoted CM2 [1]. We may use the residual freedom of the gaugetransformations associated to the three-form to set C + − a = 0. We will choose a constant a r X i v : . [ h e p - t h ] J a n ackground with non trivial components given by C ± ab and C abc . We will consider a foliationof M2-brane worldvolume, such that, Σ is a Riemann surface of genus one related to the spatialdirections. Moreover, we will consider a non trivial flux condition on T which implies throughits pullback a flux condition on Σ as shown in [1, 8] (cid:90) T (cid:101) F ± = k ± → (cid:90) Σ C ± = k ± , (1)where (cid:101) F ± = C ± rs M rp M sq d (cid:101) X p ∧ d (cid:101) X q and C ± = C ± rs dX r ∧ dX s , being (cid:101) X r the targettorus coordinates which are identified with the minimal maps (cid:101) X r = (cid:98) X r ( σ , σ ) [9]. Dueto the nontrivial worldvolume flux condition, the closed one-forms are decomposed dX r = M rs d (cid:98) X s + dA r , with dA r a dynamical exact one-form and dX rh its harmonic counterpart, being dX rh = M rs d (cid:98) X s and M s + iM s = 2 πR ( l s + m s τ ). The embedding maps dX r of the compactsector satisfy the standard wrapping condition (cid:72) C S dX = 2 πR ( l s + m s τ ) with X = X + iX .It is worth to notice that when C ± rs = (cid:15) rs , the flux condition induced on Σ is in one to onecorrespondence with the the so-called central charge condition [6] (cid:90) Σ dX r ∧ dX s = (cid:15) rs det ( W ) , det ( W ) = n (cid:54) = 0 , (2)with W the winding matrix. Therefore, CM2-brane supersymmetric LCG Hamiltonian, as ageneralization of the bosonic one obtained in [1], is given by H CM = T (cid:90) Σ √ W d σ (cid:32) P m − C (2) m √ W (cid:33) + 12 (cid:32) P r − C (2) r √ W (cid:33) + 14 { X m , X n } + 12 ( D r X m ) , + 12 ( ∗ (cid:98) F ) + 14 ( F rs ) − ¯ θ Γ − Γ r D r θ − ¯ θ Γ − Γ m { X m , θ } (cid:27) − (cid:90) Σ d σC + , (3)subject to the residual symmetry associated to the local and global area preservingdiffeomorphisms (APD) constraints, (cid:15) uv ∂ u (cid:20) P m ∂ v X m √ W + P r ∂ v X r √ W + ¯ S∂ v θ √ W (cid:21) ≈ , (cid:73) C S (cid:20) P m dX m √ W + P r dX r √ W + ¯ Sdθ √ W (cid:21) ≈ , (4)where m = 3 , . . . , r, s = 1 , θ is a Majorana spinor of 32 components. The background terms are C (2) m = 12 (cid:15) uv ∂ u X ¯ n ∂ v X n C m ¯ nn + (cid:15) uv ∂ u X n ∂ v X r C mnr + 12 (cid:15) uv ∂ u X r ∂ v X s C mrs , (5) C (2) r = 12 (cid:15) uv ∂ u X m ∂ v X n C rmn + (cid:15) uv ∂ u X m ∂ v X s C rms , (6) C ± = 12 (cid:15) uv ∂ u X m ∂ v X n C ± mn + (cid:15) uv ∂ u X n ∂ v X r C ± mr + 12 (cid:15) uv ∂ u X r ∂ v X s C + rs , (7)The gauge symplectic curvature and the Hodge dual of the flux curvature are respectively givenby F rs = D r A s − D s A r + {A r , A s } , ∗ (cid:98) F = (cid:15) uv (cid:98) F uv √ W = 12 (cid:15) rs { X rh , X sh } , (8)ith D r · = D r · + {A r , ·} . In order to relate the CM2-brane with the M2-brane with C ± fluxesdescribed in (3), we find as a new result that there exists a canonical transformation (cid:98) P m = P m − C (2) m , (cid:98) P r = P r − C (2) r , (9)that preserves all brackets of the theory and the kinematic term (cid:90) Σ (cid:16) (cid:98) P m ˙ X m + (cid:98) P r ˙ X r + ¯ S ˙ θ (cid:17) = (cid:90) Σ (cid:16) P m ˙ X m + P r ˙ X r + ¯ S ˙ θ (cid:17) . (10)In fact, it can be checked that this relation holds before imposing the flux conditions. Theresultant Hamiltonian is given by H C ± = (cid:90) Σ √ W d σ (cid:32) (cid:98) P m √ W (cid:33) + 12 (cid:32) (cid:98) P r √ W (cid:33) + 14 { X m , X n } + 12 ( D r X m ) , + 12 ( ∗ (cid:98) F ) + 14 ( F rs ) − ¯ θ Γ − Γ r D r θ − ¯ θ Γ − Γ m { X m , θ } (cid:27) − (cid:90) Σ d σC + , (11)subject to (4). Therefore, the non trivial CM2-brane Hamiltonian [1] is equivalent through acanonical transformation to the M2 brane with C ± fluxes. If a matrix regularization is provided,the spectrum must share the same discreteness properties. One can realize that there is not aflux condition on the constant C abc three-form, however it is only due to the particularity ofthe background considered. For more general toroidal backgrounds, an analogous flux conditionshould be imposed. On the other hand in [8] the authors showed that the M2-brane with C ± fluxes is equivalent (or dual) to the M2-brane with central charge [10] when we set a backgroundwith C + rs = 0 and a flux condition over C − is imposed. One could also consider that only aquantized constant C + rs is present -by imposing C − rs = 0 through a gauge fixing-, in whichcase the theory corresponds to the M2-brane with central charge but with a constant shift inthe Hamiltonian and on its spectrum [8].In consequence, there are at least three non trivial sectors of the toroidally compactified M2-brane on a flat superspace and constant three form background with good quantum properties,i.e. the discreteness of their spectrum, related among them by canonical transformations. Inthe next section we will discuss their non trivial D2-brane duals.
3. Non Trivial D2-branes
Let us consider the Dirac Born Infeld (DBI) LC formulation of a D2-brane coupled to a constantRR and NSNS flux background. The LCG formulation without the coupling to the backgroundfields, was known from the works of [11, 12, 13]. We generalized this result in [1] by considering aD2-brane on M × T on a constant C background. The physical Hamiltonian was obtained aftera proper elimination of the non physical degrees of freedom through a canonical transformation.In fact, as the coupling with background three-form in eleven and ten dimensions, respectively,is given by, essentially, a Wess Zumino term, a similar structure arise when the LCG formulationis considered. Therefore, the previous M2-brane LCG Hamiltonian formulation experience on M × T helps to obtain the corresponding one associated to a D2-brane coupled to RR andNSNS background fields H = 12 √ W [( P M + B M ) + Π u Π v γ uv + G ] − C (10)+ − B + (12)where G = γ + F with F = det ( F uv + B uv ). Moreover B M = Π u ∂ u X N B MN and B + =Π u ∂ u X N B + N as in [1]. This Hamiltonian is subject to: a residual constraint φ related with the2-brane APD and a first class Gauss constraint χ associated to the BI U(1) symmetry overthe D2 worldvolume (cid:101) Σ, φ = (cid:15) uv ∂ u (cid:20) P M ∂ v X M √ W + Π u F vu √ W (cid:21) ; χ = ∂ u Π u , (13)where M = 1 , . . . , M × S . If a toroidally compactifiedbackground target space, like M × T , is considered and a quantization condition on RR andNSNS background fields is imposed, they imply flux conditions on T and (cid:101) Σ. In order to seethis, let us discuss the next scenarios.
Let us consider the LCG formulation of D2-branes on M × T coupled to RR and NSNSbackground fields, in such a way that C (10)+ − M , B + − and B − M has been set to zero fixing the RRthree-form and NSNS two-form gauge invariance, respectively. Moreover, the only non trivialbut constant components that we will consider by fixing the background are C ± rs , B rs and B + r with r = 1 , (cid:101) F ± and (cid:101) B as (cid:90) T (cid:101) F ± = k ± , (cid:90) T (cid:101) B = k B (14)where k B , k ± ∈ Z / { } , (cid:101) F ± = C (10) ± rs M rp M sq d (cid:101) X p ∧ d (cid:101) X q , (cid:101) B = B rs M rp M sq d (cid:101) X p ∧ d (cid:101) X q , B rs = b(cid:15) rs and C (10) ± rs being defined as in [1]. It can be checked that these flux conditions on T implies thefollowing D2-brane worldvolume flux conditions (cid:90) Σ C (10) ± = k ± , (cid:90) (cid:101) Σ B = k B , (15)with C (10) ± = C (10) ± rs dX r ∧ dX s and B = B rs dX r ∧ dX s where, as before, the one-formsdecomposed on its exact and harmonic part, and we have also considered an identification (cid:101) X = (cid:98) X ( σ , σ ). The LCG Hamiltonian of a D2-brane on M × T on a constant RR and NSNSbackground in [1] was shown to be H D = (cid:90) d σ (cid:40)
12 ( P α ) √ W + 12 ( P r − B r ) √ W + 12 Π u Π v γ uv √ W + 12 (cid:101) G √ W , + 12 √ W ( D r X α ) + √ W (cid:20)
12 ( ∗ (cid:98) F ) + 14 ( F rs ) (cid:21) − C (10)+ − B + (cid:27) (16)subject to the residual constraints associated to Gauss law and the local and global APD ∂ u Π u ≈ (cid:15) uv ∂ u (cid:20) P α ∂ v X α √ W + P r ∂ v X r √ W + Π w F vw √ W (cid:21) ≈ (cid:73) C S (cid:20) P α ∂ v X α √ W + P r ∂ v X r √ W + Π w F vw √ W (cid:21) d v σ ≈ (cid:101) G = (cid:101) γ + F , (cid:101) γ = det ( ∂ u X α ∂ v X α ), F = det (cid:104) F uv + √ W k B (cid:15) uv (cid:16) ∗ (cid:98) F + (cid:15) rs F rs (cid:17)(cid:105) and F = dA .We have shown in [1] the Hamiltonian (16) is dual in the sense of [14, 15] to CM2 brane theoryon a constant flux background (3) by a proper fixing of the corresponding background.e can mention that the worldvolume flux condition can also be interpreted in terms of theirreducible wrapping condition -found in [6] in the context of wrapped M2-branes- of D2-branes[1]. It is clear that if C ± rs = (cid:15) rs and B rs = (cid:15) rs , then both quantization conditions (15) on (cid:101) Σ arein one to one correspondence with this last one.
Let us consider once again the LCG formulation of a D2-brane on M × T , where C (10)+ − M = B + − = B − M = 0 have been set to zero using RR three-form and NSNS two-form gauge invarianceand the only non trivial but constant components that we will consider by fixing the backgroundare the C ± rs and B + r with r = 1 ,
2, ( B MN = 0 on expression (12)). Therefore, we may considerthe flux condition over (cid:101) F ± on T which implies a flux condition on C (10) ± over the worldvolume.In this case we obtained in [1] H = (cid:90) d σ (cid:40)
12 ( P α ) √ W + 12 ( P r ) √ W + 12 Π u Π v γ uv √ W + 12 (cid:101) G DBI √ W , + 12 √ W ( D r X α ) + √ W (cid:20)
12 ( ∗ (cid:98) F ) + 14 ( F rs ) (cid:21) − C (10)+ − B + (cid:27) (20)subject to (17), (18) and (19) where (cid:101) G DBI = (cid:101) γ + F . As shown in [1], this LCG Hamiltonian fora D2-brane on M × T with C (10) ± fluxes, is dual to the Hamiltonian (11) which corresponds toM2-brane with C ± fluxes [8] and a NSNS field contribution. However, it is worth to mention thatthe fact that there are canonical transformations (9) on D = 11, which allow us to understandCM2 brane with C ± fluxes [1] as equivalent to M2-brane with fluxes C ± [8] and M2-brane withcentral charges [6], does not implies the same relation in D = 10. However, one can relate (16)with (20), by switching off the transverse components of the NSNS background field B MN = 0,before the imposition of fluxes. They correspond to different D2-brane sectors depending on thethe transverse components of NSNS background field. We consider the dual of a M2-brane LCG Hamiltonian with a C − flux condition, which is givenby (11) setting C + = 0. It is an equivalent topological condition to the central charge conditionin the M2-brane theory [6]. Its dual is the standard toroidally wrapped D2-brane subject to anon trivial flux condition on T associated to the C (10) − rs = (cid:15) rs . This condition can also be seenas a topological irreducible wrapping condition on the D2-brane on a flat toroidal background.As shown in [1] its D2-brane action is dual to M2-brane with central charges. We may also consider the D2-branes on M × T only coupled to constant NSNS backgroundfields in such a way that B + − = B ± r = 0 fixing the residual gauge transformation of the two-form. Moreover, we also impose C (10) ± = 0 by fixing the background, such that the only nontrivial components considered are given by B rs = b(cid:15) rs . By imposing a quantization condition on (cid:101) B over T , which for a constant 2-form is equivalent to a 2-form flux condition over (cid:101) Σ, it canbe checked that the Hamiltonian is given by (16) with C + = B + = 0. The B-field quantizationcondition is also equivalent to the D2-brane irreducible charge condition on the D2 worldvolumefor b = 1, however, the Hamiltonians differ.
4. Discussion
We have shown that the three non trivial sectors of M2-brane theory with discrete spectrum arein fact related to each other by canonical transformations. These sectors named as: M2-braneigure 1: Relations between the non trivial M2-brane and D2-branes with RR and NSNS fluxeswith central charge [6, 10] (or equivalently, M2-brane with C − fluxes), M2-brane with C ± fluxes[8] and CM2-brane with C ± fluxes [1] are duals to non trivial sectors of the D2-branes formulatedon the same target space M × T with specific RR and NSNS flux content for each case. Thesecases are: 1) The CM2-brane with C ± fluxes dual to a D2-brane with RR and NSNS fluxes onthe target and on the worldvolume. 2) The M2-brane with C ± fluxes dual to a D2-brane withRR flux on the target but vanishing transverse components of the B-field on the worldvolumeand 3) the M2-brane with C − fluxes dual to a D2-brane with C (10) − flux (also equivalent to anirreducible wrapping condition). The non trivial D2-brane sectors can be related one to anotherby switching off some of the background fields: i.e. from 1) to 2) imposing B rs = 0 and from2) to 3) C (10)+ = B ± r = 0 . We may also consider a specific background with a flux conditionover the only non trivial but constant background field B rs . This is a particular case of 1) withthe D2-brane dual of CM2-brane with C − fluxes. In summary, the discreteness condition [7]over the M2-branes imply non-vanishing worldvolume and background fluxes over its D2-braneduals. Acknowledgements
CLH is supported by CONICYT PFCHA/DOCTORADO BECAS CHILE/2019- 21190263 andby ANT1956 project of Antofagasta U. The authors also thanks to Semillero funding SEM18-02from Antofagasta U. and to the international ICTP Network NT08 for kind support.
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