sc4, moscow 18/05/09i. bandos, nb blg in n=81 sdiff invariant bagger-lambert-gustavsson igor a....

Sc4, Moscow 18/05/09 I. Bandos, NB BLG in N=8 1

SDiff invariant Bagger-Lambert-Gustavsson

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.

model and its N=8 superspace formulations


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))


The general Filippov 3-algebra is defined by 3-brackets

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

Lie algebra is defined by anti-symm bracket of two elements

another, non-anti-Symm. 3-alg [Cherkis & Saemann]


Abstract BLG model 3-algebra valued fields

Lagrangian density:

8v 8s of SO(8)

Chern-Simons term for Aμ

Gauge field, in bi-fundamentalof the 3-algebra

Trace of the 3-algebra

SO(8) generator in 8s Covariant derivativeconstructed with using

It possesses d=3 N=8 susy

superconformal symmetry

+ 8 conformal susy = 32 fermionic generators

The properties expected for low energy limit of the system of (nearly) coincident M2-branes(11D supermembranes): N M2 ‘s #(N) Ta -s


11D SURGA

IIA Superstr.

IIB Superstr.

Type I

Heterotic. E8xE8

Heterotic.SO(32)

M-theory

The place of BLG(-like) models in M theory

D=11

D=10

M-branes:

M2-brane=supermembrane

M5-brane

Dp-branes:

D3-brane

D2-brane

BLG model was assumed todescribe low energy dynamics

of multiple M2-system


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)

=

• 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

• 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

≈

• A candidate nonlinear multiple (bosonic) M2-brane action [Iengo & Russo 08]

d=3 duality:

[P.K. Townsend 95]

[Bergshoeff, Sezgin, Townsend 87]


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(1,9) SO(1,2) SO(7)

SO(7) corresponds to D2.


Abstract BLG [Bagger & Lambert 07, Gustavsson 07]

SDiff3 inv. BLG model =NB BLG model

3-algebra valued fields

8v 8s of SO(8)

Trace of the 3-algebra

d=3 fields dependent on M3 coordinates

8v 8s of SO(8)

Gauge prepotential

CS-like term for the gauge

Prepotential Aμi

Integral over M3

3-brackets

Nambu brackets

Gauge potential for SDiff3 The model possesses localgauge SDiff3 invariance

[Ho & Matsuo 08, I.B. & Townsend 08]


SDiff3 (SDiff(M3)) gauge fieldsglobal SDiff symm

local SDiff symm

Covariant derivative

Gauge prepotential

Gauge field

Field strength:

also obeys

Pre-field strength

Contains both potential s and pre-potential A

locally on M3

and, in its explicit form,

Chern-Simons like term

Gauge potential


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)

• Bianchi identities

where a fermionic SDiff3 connection (8c)

obey

Basic field strength 28 of SO(8)

• In addition to vector, fermionic spinor and scalar there are many others component fields, but these become dependent on the mass shell

8v

8c8s

Generalized Paulimatrices of SO(8) =

Klebsh-Gordan coeff-s


NB BLG in N=8 superspace (2)

• Hence the name superembedding –like equation

• This relates SDiff gauge field strength with matter and is solved by

lead (in particular) to

Super-Chern-Simons equation

is the local SDiff3 covariantization of thed=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’).

• Selfconsistency conditions for the superembedding –like equations with


NB BLG in N=8 superspace (3)

and

• Reduce the number of fields in the superfields to the fields of NB BLG model

Super CS equation

Superembedding-like equation

• Produce the BLG equations of motion for these fields

• and thus provide the complete on-shell superfield description of the NB BLG model


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

• The “d=3, N=8 pure spinor constraint’’ reads

Complex bosonic

SO(1,2) spinor

8c spinor of SO(8)

• 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]


Properties of d=3, N=8 pure spinors

• As a result of pure spinor constraints,

the only non-vanishing analytical bilinear are

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.

(0,28) and (3,35)


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. givesthe free limit of the superembedding- like eq.


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

The off-shell BLG action is

• We must assume (for consistency) that gauge potential and pre-potential have

‘ghost number 1’, i.e. that with some


NB BLG action in pure spinor superspace is

CS-like term for SDiff3 potential and pre-potential.

It can be obtained as

This CS-like term reads

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


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.


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.


Thank you for your

attention!

sc4, moscow 18/05/09i. bandos, nb blg in n=81 sdiff invariant bagger-lambert-gustavsson igor a....

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