EuroStrings 10, MadridI. Bandos, Multiple M0+ SUGRA Multiple M0 (multiple M-wave) system in 11D supergravity background Igor A. Bandos Based on Phys.Lett

EuroStrings 10, Madrid I. Bandos, Multiple M0+ SUGRA

Multiple M0 (multiple M-wave) system in

11D supergravity background Igor A. Bandos

Based on Phys.Lett. B687 [arXiv:0912.5125], arXiv:1003.0399 and paper in preparation

- Introduction. - Superembedding approach to a single M0-brane. - Worldline superspace , 11D SUGRA superspace , - Superembedding equation and - Equations of motion - Superembedding approach to multiple M0 system. - d=1 =16 SU(N) SYM constraints on the worldline superspace - Equations of relative motion from SYM constraints, coupling to fluxes of 11D supergravity and Emparan-Myers dielectric brane effect for mM0 - Rigid structure of mM0 equations - BPS equation and its fuzzy sphere solution describing M2 as a 1/2BPS state of mM0 - Conclusions.

IKERBASQUE, The Basque Foundation for Science and Depto de Física Teórica, Universidad del País Vasco EHU/UPV, Bilbao, Spain


Introduction• Supersymmetric extended objects - super-p-branes- and multiple brane

systems play important role in String/M-theory, AdS/CFT etc. • Single p-brane actions are known (mainly known) for years: 10D fundamental strings: (1984) 10D type II Dp-branes: (1996 (D0-1982)) 11D Mp-branes. M5-brane (1997) M2-brane (1987) M0-brane (1996)• As far as the action for multiple p-brane system is concerned, • it was appreciated long ago that, in the case of multiple Dp-branes, its

low energy limit is provided by the U(N) SYM action (1995):

• However, in search for a complete (a more complete) supersymmemtric, diffeomoprhism and Lorenz invariant action for multiple Dp (Mp) system, which would be a counterpart of action for single Dp-brane, only a particular progress has been reached.


• For the bosonic limit of the multiple D9-brane system (spacetime filling: p=9=D-1) Tseytlin proposed to use a non-Abelian Born-Infeld action based on symmetric trace prescription.

• Although a search for its SUSY generalization was not successful, it was used by Myers [1999] as a starting point to obtain, by the chain of T-duality transformations a `dielectric brane action’.

• However, although widely accepted, the Meyers action does not possess neither SUSY nor Lorentz (diffeomorphism) symmetry. The question of mere existence of its straightforward Lorentz covariant and supersymmetric generalization remains open.

• In search for Lorentz covariant and supersymmetric action for mDp-brane systems, Howe, Lindsrom and Wulff developed [2005-2007] the boundary fermion approach. This gives the Lorentz covariant and supersymmetric action but on the ‘minus one quantization level’.


Boundary fermion approach by P. Howe, U. Lindstrom and L. Wulff [05-07]

• gives a complete supersymmetric and Lorentz covariant (reparametrization invariant) pre-classical (minus one quantization) description of the multiple D-brane system in terms of a superspace with ‘boundary fermion directions’ .

• To arrive at a description similar to the DBI+WZ description of single Dp-brane, one has to quantize the boundary fermion sector.

• The quantization of the model, if were performed in a complete form (quantizing both coordinate functions and boundary fermions), should describe supergravity (and higher stringy modes) together with D-branes.

• By quantization of the boundary fermion sector only (by the prescription [Marcus+Sagnotti 86]) in pure bosonic case [05], the Myers action was reproduced but the Lorentz invariance was lost.

• In the supersymmetric boundary fermion action [07] the kappa-symmetry and reparametrization symmetry parameters depend on boundary fermions--> non-Abelian kappa symmetry [the previous attempts to construct action with it were not successful].

• Probably the further development of the boundary fermion approach will help to resolve the problems of the non-abelian reparametrization and kappa-symmetry and provide a covariant multiple Dp-brane action of the ‘usual’ type.

• It might also happen that the boundary fermion action is the best what one can write for the description of multiple Dp-brane action, while the covariant quantrization needs to be complete and thus inevitably gives not just ‘classical’ description of multiple Dp-branes, but also the supergravity and/or other stringy degrees of freedom.

• In both cases, in the middle time it looks reasonable to search also for a more straightforward, probably approximate, but covariant and SUSY approach, going beyond U(N) SYM and formulated in terms of variables similar to ones used to write single Dp-brane action.


• The situation with multiple M-branes was even more complicated• The purely bosonic Myers action for D0-brane was lifted to 11D to obtain a purely bosonic

action for multiple M0-brane (‘multiple gravitational waves’ [Janssen & Y. Losano 2002])

• On this line, recently it was proposed [I.B., Phys.Lett. B 2009] to use the superembedding approach to search for a multiple D0-brane equations (and multiple Dp with p>0)

• The idea is to describe the relative motion of mDp constituents by SU(N) SYM on the worldvolume superspace of one Dp-brane.

• Presently this prescription has been elaborated for mD0 only. It reproduces the U(N) SYM model at very low energies and allows to describe the mD0 interaction with IIA supergravity.

• In this talk we describe the generalization of this superembedding description of D=10 D0-brane for the case of D=11 M0-brane.

• Till 2007 it was even unclear what is the very low energy description of the multiple M2 system (analyses by J.H. Schwarz [2004] gave some no-go type statements).

• In 2007 Bagger and Lambert and, independently, Gustavsson proposed on this role a model (BLG model) based on the Filippov 3-algebra instead of Lie algebra.

• However, now the commonly accepted candidate on the very low energy description of mM2 is the SU(N) X SU(N) invariant Chern-Simons plus matter model (ABJM model) was proposed by Aharony, Bergman, Jafferis and Maldacena (only N=6, i.e. 24 of 32 SUSY is manifest).

• The purely bosonic non-covariant nonlinear generalization of the BLG action (generalizing Myers action for mD2 for the case of mM2) was constructed in [Inego & J. Russo 2008])


• As far as single D0 -brane (i.e. massive 10D type IIA (=2) superparticle) can be obtained by dimensional reduction of M0-brane (i.e. of massless 11D superparticle [Bergshoeff & Townsend 96]), and, as it is known from the classical Matrix model paper [Banks, Fischler, Shenker & Susskind 97], the D0-brane equation in flat space time show the hidden 11D symmetry, one can expect that the construction of mM0-model is equivalent to revealing 11D symmetry in mD0 model.

• Such a close relation of mM0 with mD0 together with our previous derivation of the mD0 equations from superembedding approach, suggests the to construct the superembedding approach to mM0 on the basis of the SU(N) SYM on the worldline superspace of a single M0-brane.

• In this talk we will report on the result of developing superembedding approach to multiple M0 system (mM0)

• The valididy of such type construction for mM2 is not so evident (although worth trying) as far as there one might expect a 3-algebra structure, like in BLG model or/and not all supersymmetries being manifest, like in ABJM model.

• This gives us the covariant generalization of Matrix model equation for the case of nontrivial supergravity background.

• Let us begin by describing superembedding approach to single M0-brane and the derivation of the dynamical equations for M0-brane in this framework.


• It had shown its efficiency in search for the single D-brane and M5-brane equations [Howe + Sezgin 96]. In particular, Howe and Sezgin derived the M5-brane equations of motion from the superembedding approach [1996] some months before the covariant supersymmetric and kappa-symmetric action was constructed in [I.B., Lechner, Nurmagambetov, Pasti, Sorokin, Tonin PRL 97 and Aganagic, Popescu, Schwarz NPB 97].

• The superembedding approach [I.B. , Pasti, Sorokin, Tonin, Volkov, 1995], following the so-called STV [Sorokin, Tkach, Volkov 1988] approach to D=3,4 superparticle (generalized to superstrings in D=3,4,6,10 in the works by Berkovits, Tonin, Sorokin, Pashnev, Pasti, Ivanov, Sokatchev, Howe, Delduc and others) describes p-branes in terms of worldvolume superfields.

So it looks natural to apply it in the search for multiple brane equations


Bosonic string (p-brane) NSR (spinning) string

Worldsheet susy

STV approach and STVZ , Superembedding approach

Double supersymmetry

(w-s+ target space)

Target space

susy

Target space

susy

Worldsheet susy

Green-Schwarz superstring

Superembedding approach to D-dim. super-p-branes.

Also ‘’Spinning superpart.’’S.J.Gates, H. Nishino, J. Kovalski-Gliktman,J. Lukierski, J. van Holten,R. Mktrchan, A.Kavalov,…

But preserving equivalence to GS superstrings and BS superparticles!

Superembedding approach and its place w/r to other ones


M0-brane in superembedding approach. I. Target and worldline superspaces.

W

Σ Target D=11 superspace

Worldvolume (worldline) superspace

The embedding of W in Σ can be described by coordinate functions

These worldvolume superfields carrying indices of 11D superspace coordinatesare restricted by the superembedding equation.


M0-brane in superembedding approach. I. Superembedding equation

W

Σ Supervielbein of D=11 superspace

Superembedding equation states that the pull-back of bosonic vielbein has vanishing fermionic projection:.

Supervielbein of worldline superspace

General decomposition of the pull-back of type IIA supervielbein


Superembedding equation

Contains M0-brane equations among their consequences. Furthermore,

together with some conventional constraints it determines the geometry of worldline superspace and of the normal bundle over it:

SO(1,1) curvature of vanishes,

4-form flux of 11D SG superfieldMoving frame vectors

The 4-form flux superfield enters the solution of the 11D superspace SUGRA constraints [Cremmer & Ferrara 80, Brink & Howe 80]:


fermionic flux of 11D SG Spinor moving frame variable

The SO(9) curvature of the normal bundle over the worldline superspace

Involves, in addition to

, also

Moving frame variables

(Auxiliary) moving frame superfields are elements of the Lorentz group valued matrix.

This is to say they obey


Spinor moving frame superfields, entering are elements of the Spin group valued matrix

`covering’ the moving frame matrix as an SO(1.10) group element

This is to say they are `square roots’ of the light-like moving frame variables, e.g.:

One might wander whether these spinor moving frame variables come from?

(Auxiliary) moving frame superfields are elements of the Lorentz group valued matrix.

This is to say they obey

,


Moving frame and spinor moving frame variablesappear in the conventional constraints determining the induced supervielbein so that

Equivivalent form of the superembedding eq.+ conventional constraints.

M0 equations of motion can be written in the form of

As it has been already said, these equations follow from the superembedding eq. so that this determines completely both the M0 dynamics and the geometry of the worldline superspace


Multiple M0 description by d=1, =16 SU(N) SYM on .

Thus no influence of relative motion on the ‘center of energy motion’ (in distinction to the Myers action; as it had been noticed by Sorokin in [2003] such dependence does not look physical and might actually be a problem of Myers action).

• Following the approach to multiple D0 in [I.B. PLB 09], we propose to

describe the multiple M0 by 1d =16 SYM on on the superspace characteristic for a single M0.

• This is to say the embedding of into the 11D SUGRA superspace is determined by the superembedding equation

This implies that the ‘center of mass’ (center of energy) motion of the mM0 system is defined by the equations characteristic for a single M0 (‘center of energy M0´).


Multiple M0 description by d=1, =16 SYM on Basic SYM constraints and superembedding-like equation.

We define on an su(N) valued 1-form potential

with the field strength

restricted by the constraint

A clear candidate for the description of relative motion of M0-brane constituents comes from dim 1 field strength!

Studying Bianchi identities DG=0 one finds that these constraints require the basic nanoplet of matrix superfields to obey the superembedding-like equation

This is very similar to the superembedding-like equation for the multiple D0 system


Equations for the relative motion of multiple M0s.

1d Dirac equation

Gauss constraint

Bosonic equations of motion

Studying the selfconsistency conditions for

on the M0 worldline superspace one finds equations of motion for multiple M0 system as determined by our superembedding approach.

In the case of flat target superspace one first finds

showing that there is no further fields in the basic su(N) matrix superfield and then the equations describing relative motion of the mM0 constituents

These are basically the same as eqs describing relative motion of the constituents of multiple D0 system in flat type IIA SSP, i.e. d=1 reduction of the 10D SYM = Matrix model equations. This implies that our mM0 model shows the restoration of 11D Lorentz invariance (diff invariance) in the multiple D0 model of our PLB09.

Flat 11D target superspace


Coupling to supergravity background. Flux contribution to the eqs of motion, polarization and dielectric coupling.

In general curved 11D SUGRA superspace the 1d Dirac equation reads

Gauss constraint

Bosonic equations of motion acquires the form of

Coupling to higher form characteristic for the Emparan-Myers dielectric brane effect


Rigid structure of the multiple M0 equations:

Then, for instance, the bosonic equations of motion has to have SO(1,1) wait -6

Einstein Eq. RS Equation

Thus the only results of calculations are


BPS equationsfor the supersymmetric pure bosonic solutions of mM0 eqs. can be obtained from

They reads

is a parameter of ½ of the target space SUSY preserved by the single M0(corresponding to the kappa symmetry single M0 action )

½ BPS equation (16 susy’s preserved)

and

SUSY preservation by center of energy motion

SUSY preservation by relative motion of mM0 constituents


½ BPS equation (16 susy’s preserved)

to simplify:

Fuzzy 2-sphere solution


¼ BPS equation (8 susy’s preserved)

which has a(nother) fuzzy 2-sphere-related solution

The famous Nahm equation

appears as an SO(3) inv

with

and with

is obeyed, in particular, for

½ BPS:


Conclusion• We have presented equations of motion for multiple M0-brane system (mM0)

in an arbitrary 11D supergravity background. • These are derived from the superembedding approach to multiple M0 system, but have a

rigid structure: all the interacting terms can be restored just from the SO(1,1)xSO(9) symmetry requirements, and only vanishing of two coefficients for the terms of the second order in fluxes appear as a result of calculation.

• Search for more supersymmetric solutions of our multiple M0 equations, describing branes and brane intersection (could Basu-Harvey-type equation appear?)

• One more interesting question is whether one can construct a counterpart of the boundary fermion formalism for the multiple M0 system?

• Our superembedding description also allowed to find the BPS conditions for the supersymmetric solutions of multiple M0 equations.

• The one-half BPS equation with nonvanishing 4-form flux has a fuzzy two-sphere solution describing M2-brane as a ½ BPS configuration of multiple M0 system.

• The famous Nahm equations appers as an SO(3) invariant ¼ BPS equation in the case of vanishing 4-form flux projection.

Some directions for future study

• Extension of the approach for higher p mDp- and mMp- systems (mM2-?, mM5-?). Is it consistent to use the same construction (SU(N) SYM on worldvolume superspace of a single brane)?

• Relation with giant graviton (like in [Janssen & Y. Losano 2002]) and other solutions?


Thank you for your attention!

Documents

EuroStrings 10, MadridI. Bandos, Multiple M0+ SUGRA Multiple M0 (multiple M-wave) system in 11D supergravity background Igor A. Bandos Based on Phys.Lett