Transcript M0-brane covariant quantization and intrinsic complexity

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.
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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 d=3, N=8 supersymmetric 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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I. Bandos, NB BLG in N=8
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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-antiSymm. 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
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I. Bandos, NB BLG in N=8
the structure constants obey
3
Abstract BLG model
8v
8s of SO(8)
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
It possesses d=3 N=8 susy
SO(8) generator in 8s
Chern-Simons
term for Aμ
+ 8 conformal susy = 32 fermionic generators
superconformal symmetry
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The properties expected for low energy limit
I. Bandos,
in N=8of (nearly) coincident M2-branes
4
of NB
theBLG
system
(11D supermembranes): N M2 ‘s #(N) Ta -s
The place of BLG(-like) models in
M theory BLG model was assumed to
M-branes:
M5-brane
M2-brane=supermembrane
D2-brane
describe low energy dynamics
of multiple M2-system
11D SURGA
D=11
IIA Superstr.
Heterotic. E8xE8
Dp-branes:
D=10
M-theory
D3-brane IIB Superstr.
Heterotic.SO(32)
Type I
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The rôle the BLG model was assumed to play
•
Action for a single Dp-brane (D2-brane)
•
[P.K. Townsend 95]
Action for a single M2-brane (11D
supermembrane)
=
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 nearcoincident multiple M2-brane system
•
•
≈
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•
I. Bandos, NB BLG in N=8
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Thus the BLG action was proposed to describe low
energy dynamics of N near-coincident M2-branes. But
•
•
•
•
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 3algebras 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 3algebras 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, RodriguezGomes, Verlinde and others, 08].
Furthermore, a Lorentz 3-algebra can be
associated with a Lie algebra.
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•
•
•
N Dp-branes: Low energy dynamics is
described by
SU(N) SYM, (#=(N²-1) generators)
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)?
SO(1,10)
I. Bandos, NB BLG in N=8
Static gauge for M2
SO(1,2) SO(8)
SO(7) corresponds to D2.
SO(1,9)
SO(1,2) SO(7)
7
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]
CS-like term
for the gauge
Prepotential Aμi
Integral over M3
Nambu brackets
d=3 fields dependent on M3 coordinates
8v
8s of SO(8)
Gauge prepotential
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Gauge potential for SDiff3 The model possesses local
I. Bandos, NB BLG in N=8
gauge SDiff3 invariance
8
SDiff3 (SDiff(M3)) gauge fields
global SDiff symm
local SDiff symm
Gauge field
Gauge potential
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 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)
8v
Generalized Pauli
•
8s
8c
matrices of SO(8) =
Klebsh-Gordan coeff-s
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 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
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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
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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
SO(1,2) spinor
bosonic spinor coordinate called pure spinor
• 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]
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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.
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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 λdecomposition of this eq. gives
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the free limit of the superembedding- like eq.
I. Bandos, NB BLG in N=8
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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
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NB BLG action in pure spinor
superspace is
This CS-like term reads
CS-like term for SDiff3 potential
and pre-potential.
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
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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 N=8 superfield action for the (NB)
BLG model.
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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 CSlike 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.
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Thank you
for your
attention!
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