SDiff invariant Bagger-Lambert-Gustavsson

1
SDiff invariant Bagger-Lambert-Gustavsson
model and its N8 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) arXiv0808.1583v2 and I. B.,
Phys.Lett. B669, 193 (2008) arXiv0808.3568

• Introduction. 3-algebras and Nambu brackets.
• BLG model in d3 spacetime, its relation to
  M2-brane, and with SDiff3 gauge theories
• N8 superfield formulation. BLG equations of
  motion in standard N8 superspace.
• N8 superfield action for NB BLG model in pure
  spinor superspace
• Conclusion.

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Introduction

• In the fall of 2007, motivated by a search for a
  multiple M2-brane model, Bagger, Lambert and
  Gustavsson proposed a new d3, N8
  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))

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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
the structure constants obey
For the metric Filippov 3-algebra
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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 d3 N8 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
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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-branesupermembrane
D11
IIA Superstr.
D2-brane
Heterotic. E8xE8
D10
M-theory
Dp-branes
IIB Superstr.
Heterotic.SO(32)
D3-brane
Type I
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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

d3 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, N4 d4 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

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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 Ngt2 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)
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Abstract BLG Bagger Lambert 07, Gustavsson 07
8v
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
d3 fields dependent on M3 coordinates
8v
8s of SO(8)
Gauge prepotential
Gauge potential for SDiff3
The model possesses local gauge SDiff3 invariance
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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
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NB BLG in N8 superspace

• The complete on-shell N8 superfield description
  of the NB BLG model is provided by octet (8v) of
  scalar d3, N8 superfields
• Which obey the superembeddinglike equation (see
  below on the name)

Generalized Pauli matrices of SO(8)
Klebsh-Gordan coeff-s
8v
8c
8s
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

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NB BLG in N8 superspace (2)
is the local SDiff3 covariantization of the d3,
N8 scalar multiplet superfield eq.
and this appears as a linearized limit of the
superembedding equation for D11 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
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NB BLG in N8 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

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NB BLG in pure spinor superspaceabstract BLG
M.Cederwall 2008 NB BLG I.B P.K. Townsend
2008

• It is hardly possible to write N8 superfield
  action for BLG model in the standard d3, N8.
• Martin Cederwall proposed a quite nonstandard
  action (with Grassmann-odd Lagrangian density) in
  pure spinor superspace i.e. in N8 d3
  superspace completed by additional constrained
  bosonic spinor coordinate called pure spinor

SO(1,2) spinor
Complex bosonic
8c spinor of SO(8)

• The d3, N8 pure spinor constraint reads

• Pure spinor superspace in D10 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
  GIKOSGalperin, Ivanov, Kalitzin, Ogievetski and
  Sokatchev

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Properties of d3, N8 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 ? of the first nonvanishing monom in it.
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Searching for a pure spinor superspace
description of BLG model it is natural to begin
with constructing scalar d3 N8 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 ?-decomposition of
  this eq. gives

the free limit of the superembedding- like eq.
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NB BLG in pure spinor superspace

• As in standard 3d N8 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

with some
ghost number 1, i.e. that
The off-shell BLG action is
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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 ?
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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 N8 superfield
  action for the (NB) BLG model.

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Conclusion

• We have reviewed the BLG (Bagger-Lambert-Gustavsso
  n) 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 d3 N8 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
  N8 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.

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• Thank you for your attention!