SDiff invariant Bagger-Lambert-Gustavsson model and its N=8 superspace formulations Igor A. Bandos Ikerbasque and Dept of Theoretical Physics, Univ.of the Basque Country, Bilbao, Spain and ITP KIPT, Kharkov Ukraine Based on I.B. & P. K. Townsend, JHEP 0902, 013 (2009) [arXiv:0808.1583v2] and I. B., Phys.Lett. B669, 193 (2008) [arXiv:0808.3568] Introduction. 3-algebras and Nambu brackets. BLG model in d=3 spacetime, its relation to M2-brane, and with SDiff3 gauge theories; N=8 superfield formulation. BLG equations of motion in standard N=8 superspace. N=8 superfield action for NB BLG model in pure spinor superspace Conclusion. Talk prepared for Niederlie Fest, Praga, May 5-7, 2008. Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Introduction In the fall of 2007, motivated by a search for a multiple M2-brane model, Bagger, Lambert and Gustavsson proposed a new d=3, N=8 super-symmetric action based on Filippov 3-algebra instead of Lie algebra. An example of an infinite dimensional 3-algebra is defined by the Nambu bracket for functions on a compact 3dim manifold M3 , Another example of finite dimensional 3-algebra, which was present already in the first paper of Bagger and Lambert, is 4 realized by generators related to the ones of the so(4) Lie algebra (=su(2)su(2)) Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

The general Filippov 3-algebra is defined by 3-brackets Lie algebra is defined by anti-symm bracket of two elements The general Filippov 3-algebra is defined by 3-brackets another, non-anti- Symm. 3-alg [Cherkis & Saemann] These are antisymmetric, and obey the fundamental identity These properties are sufficient to construct the BLG field equations. To construct the BLG Lagrangian one needs also the invariant inner product For the metric Filippov 3-algebra the structure constants obey Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Abstract BLG model 8s of SO(8) 8v 3-algebra valued fields Gauge field, in bi-fundamental of the 3-algebra Lagrangian density: Trace of the 3-algebra Covariant derivative constructed with using SO(8) generator in 8s Chern-Simons term for Aμ It possesses d=3 N=8 susy + 8 conformal susy = 32 fermionic generators superconformal symmetry The properties expected for low energy limit of the system of (nearly) coincident M2-branes (11D supermembranes): N M2 ‘s #(N) Ta -s Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

The place of BLG(-like) models in M theory BLG model was assumed to describe low energy dynamics of multiple M2-system M-branes: M5-brane 11D SURGA M2-brane=supermembrane D=11 IIA Superstr. D2-brane Heterotic. E8xE8 D=10 Dp-branes: M-theory IIB Superstr. D3-brane Heterotic.SO(32) Type I Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

The rôle the BLG model was assumed to play Action for a single Dp-brane (D2-brane) Action for a single M2-brane (11D supermembrane) [P.K. Townsend 95] = d=3 duality: [Bergshoeff, Sezgin, Townsend 87] Multiple Dp-branes = non-Abelian DBI action (wanted!= still in the search) A (commonly accepted) candidate was proposed by Myers [98], but this does not possess neither SUSY nor SO(1,9) (Recent work by P. Horava  Is SUSY just an ‘occasional IR symmetry’ of a Myers action?) HOWEVER, the low energy limit of such a hypothetic action IS known: it is the maximally susy gauge theory, N=4 d=4 SYM in the case of D3-brane A candidate nonlinear multiple (bosonic) M2-brane action [Iengo & Russo 08] Multiple M2-branes = ? Properties were resumed by J. Schwarz [2004]. A search for such an action was the motivation for the study of Bagger, Lambert and Gustavsson The BLG model was assumed to provide the low energy limit for the (hypothetical) action of near-coincident multiple M2-brane system ≈ Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Thus the BLG action was proposed to describe low energy dynamics of N near-coincident M2-branes. But N Dp-branes: Low energy dynamics is described by SU(N) SYM, (#=(N²-1) generators) Low energy dynamics of N M2-branes system might be described by BLG model with some # of 3-algebra generators =#(N). PROBLEM: as it was known long ago (in particular to people studying quantization of Nambu bracket problem [Takhtajan, J.A. de Azcárraga, Perelomov, …]) the only 3-algebras with positively definite metric are 4 or  of some number of 4 with trivial commutative 3-alg. 4 model describes 2 M2-s on an orbifold [Lambert + Tong, 08]. But what to do with N>2 M2-s? The set of not positively definite metric 3-algebras are richer, but the corresponding BLG model contains ghosts and/or breaks (spontaneously) SO(8) symmetry (charsacteristic for M2) down to SO(7) [Jaume Gomis, Jorge Russo, Iengo, Milanezi, 08, Gomis, Van Raamsdonk, Rodriguez-Gomes, Verlinde and others, 08]. Furthermore, a Lorentz 3-algebra can be associated with a Lie algebra. Alternative model – SU(N)xSU(N) susy CS [Aharony, Bergman, Jafferis, Maldacena 08] possesses only =6 susy. BUT there exists an infinite dim 3-algebra of the function on compact 3dim manifold 3 with 3-bracket given by Nambu brackets. NB BLG model uses this 3-algebra It describes a condensate of M2-branes Why SO(8)? Static gauge for M2 SO(1,10) SO(1,2) SO(8) SO(7) corresponds to D2. SO(1,9) SO(1,2) SO(7) Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Abstract BLG [Bagger & Lambert 07, Gustavsson 07] 8s of SO(8) 3-algebra valued fields 3-brackets Trace of the 3-algebra SDiff3 inv. BLG model =NB BLG model [Ho & Matsuo 08, I.B. & Townsend 08] Integral over M3 CS-like term for the gauge Prepotential Aμi Nambu brackets d=3 fields dependent on M3 coordinates 8v 8s of SO(8) Gauge prepotential Gauge potential for SDiff3 The model possesses local gauge SDiff3 invariance Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

SDiff3 (SDiff(M3)) gauge fields global SDiff symm local SDiff symm Gauge potential Gauge field Covariant derivative Gauge prepotential locally on M3 Field strength: also obeys Pre-field strength Chern-Simons like term and, in its explicit form, Contains both potential s and pre-potential A Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Klebsh-Gordan coeff-s NB BLG in N=8 superspace The complete on-shell N=8 superfield description of the NB BLG model is provided by octet (8v) of scalar d=3, N=8 superfields Which obey the superembedding—like equation (see below on the name) Generalized Pauli matrices of SO(8) = Klebsh-Gordan coeff-s 8v 8s 8c a fermionic SDiff3 connection (8c) where obey Basic field strength 28 of SO(8) Bianchi identities In addition to vector, fermionic spinor and scalar there are many others component fields, but these become dependent on the mass shell Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

NB BLG in N=8 superspace (2) is the local SDiff3 covariantization of the d=3, N=8 scalar multiplet superfield eq. and this appears as a linearized limit of the superembedding equation for D=11 supermembrane (in the ‘static gauge’). Hence the name superembedding –like equation Selfconsistency conditions for the superembedding –like equations with lead (in particular) to This relates SDiff gauge field strength with matter and is solved by Super-Chern-Simons equation Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

NB BLG in N=8 superspace (3) Superembedding-like equation Super CS equation and Reduce the number of fields in the superfields to the fields of NB BLG model Produce the BLG equations of motion for these fields and thus provide the complete on-shell superfield description of the NB BLG model Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

NB BLG in pure spinor superspace [abstract BLG: M NB BLG in pure spinor superspace [abstract BLG: M.Cederwall 2008; NB BLG: I.B & P.K. Townsend 2008] It is hardly possible to write N=8 superfield action for BLG model in the standard d=3, N=8. Martin Cederwall proposed a quite nonstandard action (with Grassmann-odd Lagrangian density) in pure spinor superspace i.e. in N=8 d=3 superspace completed by additional constrained bosonic spinor coordinate called pure spinor SO(1,2) spinor Complex bosonic 8c spinor of SO(8) The “d=3, N=8 pure spinor constraint’’ reads Pure spinor superspace in D=10 was introduced by Howe [91], pure spinor auxiliary fields were considered by Nillsson [86]. The construction by Cederwall can also be considered as a realization of the GIKOS harmonic superspace program [GIKOS=Galperin, Ivanov, Kalitzin, Ogievetski and Sokatchev] Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Properties of d=3, N=8 pure spinors As a result of pure spinor constraints, the only non-vanishing analytical bilinear are (0,28) and (3,35) For instance, These obey the identities and Superfields in pure spinor superspace are assumed to be power series in the pure spinor characterized by ghost number [Cederwall] which, in practical terms, is a degree of homogeneity in λ of the first nonvanishing monom in it. Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Searching for a pure spinor superspace description of BLG model it is natural to begin with constructing scalar d=3 N=8 supermultiplet Let us define BRST operator It is nilpotent due to purity constraint Let us introduce 8v-plet of scalar superfields which are SDiff3 scalars, i.e. The Lagrangian density for an action possessing global SDiff3 inv. reads Notice unusual properties: -0 is Grassmann odd; - we also have 1-st order eqs. for bosonic superfield, etc. Equations of motion can be equivalently written as The lowest 1st order term in λ-decomposition of this eq. gives the free limit of the superembedding- like eq. Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

NB BLG in pure spinor superspace As in standard 3d N=8 superspace the BLG equation can be derived by making the scalar multiplet equation covariant under local SDiff3, to find the action for NB BLG, we have to search for local SDiff3 covariantization of the pure spinor superspace action describing scalar supermultiplet First we covariantize the BRST charge by introducing a Grassmann odd scalar zero-form gauge field transforming under the local SDiff3 as and obeying with some, anticommuting, and spacetime scalar, gauge pre-potential We must assume (for consistency) that gauge potential and pre-potential have ‘ghost number 1’, i.e. that with some The off-shell BLG action is Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

NB BLG action in pure spinor superspace is CS-like term for SDiff3 potential and pre-potential. This CS-like term reads It can be obtained as where is pre-gauge field strength superfield and is SDiff3 gauge field strength. The gauge pre-potential equations read These are CS equation in pure spinor superspace and they contain the BLG superfield equations in the lowest, 2nd order in λ Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

To summarize, the SDiff3 inv. pure spinor superspace action : Contains BLG (super)fields inside the pure spinor superfields Produce the BLG equations of motion and superfield BLG equations for these (super)fields Our analysis has not excluded the presence of additional auxiliary, ghost or physical fields. To state definitely whether these are present, one needs to carry out a more detailed study of field content with the use of gauge symmetries However, even if such extra fields are present, they do not enter the BLG equations of motion which follow from the pure spinor action. Thus this possible auxiliary field sector is decoupled and, whether they are present or not, the pure spinor action is the N=8 superfield action for the (NB) BLG model. Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Conclusion We have reviewed the BLG (Bagger-Lambert-Gustavsson) model with emphasis on its SDiff3 invariant version with 3-algebra realized as the algebra of Nambu brackets (NB) (Nambu-Poisson brackets) which is called NB BLG model. We described the d=3 N=8 superfield formulation of the NB BLG model given by the system of superembedding like equation and CS-like equation imposed on 8v-plet of scalar superfields dependent, in addition the ‘usual’ N=8 superspace coordinates, on coordinates of compact 3-dim manifold M3 and on the spinorial SDiff3 pre-potential superfields. We also present the pure spinor superspace action generalizing the one proposed by Cederwall for the case of NB BLG model invariant under symmetry described by infinite dimensional SDiff3 3-algebra. We show how the NB BLG equations of motion follow form this pure spinor superspace action and that the extra fields, if present, do not modify the BLG equations of motion. Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8

Thank you for your attention! Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8