Multiple M-wave interaction with eleven dimensional supergravity

1. Multiple M-wave interaction with eleven dimensional supergravity IKERBASQUE, The Basque Foundation for Science and Depto de Física Teórica, Universidad del País Vasco UPV/EHU, Bilbao, Spain Based on: Phys.Lett.B687[arXiv:0912.5125], Phys.Rev.Lett. B105 [arXiv:1003.0399] and paper in preparation Igor A. Bandos • Result: covariant generalization of the Matrix model eqs. for arbitrary super-gravity background – is obtained in the frame of superembedding approach • Outline: - Introduction (to put in a general M-theory context) • Superembedding approach to M0 and multiple M0 system (mM0). • - Equations of motion for mM0 • - BPS equations for susy mM0 and its (1/2 BPS) fuzzy S² solution • - Conclusions. I. Bandos, Multiple M0+ SUGRA

2. (single) M-wave=M0-brane • = massless 11D superparticle [Bergshoeff+Townsend 1996] • Its quantization produces linearized supergravity supermultiplet • → M-wave= M0-brane = ’’supergraviton’’ • Dimensional reduction of M0-brane gives D0-brane (Dirichlet particle or Dirichlet 0-brane). I. Bandos, Multiple M0+ SUGRA

3. 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 for years (84-97) • Multiple p-brane actions (multiple superparticle action for p=0): • Multiple Dp-branes (mDp): (very) low energy limit = U(N) SYM (1995) • In search for a complete (a more complete) supersymmetric, diffeomoprhism and Lorenz invariant action =only a particular progress: -purely bosonic mD9 [Tseytlin]: non-Abelian BI with symm. trace (no susy); - Myers [1999] a `dielectric brane action’= no susy, no Loreentz invartiance(!) - Howe, Lindsrom and Wulff [2005-2007]: the boundary fermion approach. Lorentz and susy inv. action for mDp, but on the ‘minus one quantization level’. - I.B. 2009 - superembedding approach to mD0 (proposed for mDp but presently developed for mD0 only) • Multiple Mp-branes (mMp): even more complicated. - (very) low energy limit = BLG (2007; 3-algebras) ABJM (2008) - purely bosonic mM0= [Janssen & Y. Losano 2002]) - nonlinear generalization of (Lorentzian) BLG =[Iengo & J. Russo 2008] - Superembedding approach to mM0 system [I.B. 2009-10] (multiple M-waves or multiple massless superparticle in 11D; mMp with p=0) (results in this talk). I. Bandos, Multiple M0+ SUGRA

4. Bosonic particles and p-branes. Target space =spacetime M Worldvolume (worldline) W The embedding of Win Σ can be described by coordinate functions These worldvolume fields carrying vector indices of D-dim spacetime coordinates are restricted by the p-brane equations of motion minimal surface equation (Geodesic eq. for p=0) I. Bandos, Multiple M0+ SUGRA

5. Super-p-branes. Target superspaces and worldvolume Target D(=11) SUPERspace Σ Grassmann or fermionic coordinates W Worldvolume (worldline) The embedding of W in Σ can be described by coordinate functions These bosonic and fermionic worldvolume fields, carrying indices of 11D superspace coordinates, are restricted by the super-p-brane equations of motion I. Bandos, Multiple M0+ SUGRA

6. (M)p-branes in superembedding approach. I. Target and worldline superspaces. Target D=11 SUPERspace Σ Worldvolume (worldline) SUPERspace W The embedding of W in Σ can be described by coordinate functions These worldvolume superfields carrying indices of 11D superspace coordinates are restricted by the superembedding equation. I. Bandos, Multiple M0+ SUGRA

7. M0-brane (M-wave) in superembedding approach. I. Superembedding equation Supervielbein of D=11 superspace Σ Supervielbein of worldline superspace W General decomposition of the pull-back of 11D supervielbein Superembedding equation states that the pull-back of bosonic vielbein has vanishing fermionic projection:. I. Bandos, Multiple M0+ SUGRA

8. Superembedding equation Equations of motion of (single) M0-brane (+ conv. constr.) geometry of worldline superspace SO(1,1) curvature of vanishes, 4-form flux of 11D SG= field strength of 3-form Moving frame vectors The 4-form flux superfield enters the solution of the 11D superspace SUGRA constraints [Cremmer & Ferrara 80, Brink & Howe 80] (which results in SUGRA eqs. of motion): I. Bandos, Multiple M0+ SUGRA

9. Moving frame and spinor moving frame variables appear in the conventional constraints determining the induced supervielbein so that Equivalent form of the superembedding eq. + conventional constraints. M0 equations of motion I. Bandos, Multiple M0+ SUGRA

10. Multiple M0 description by d=1, =16 SU(N) SYM on . • We describe the multiple M0 by 1d =16 SYM on • The embedding of into the 11D SUGRA superspace is determined by the superembedding equation ‘center of energy’ motion of the mM0 system is defined by the single M0 eqs 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). I. Bandos, Multiple M0+ SUGRA

11. Multiple M0 description by d=1, =16 SYM on Basic SYM constraints and superembedding-like equation. is an su(N) valued 1-form potential on with the field strength We impose constraint A clear candidate for the description of relative motion of the mM0-constituents! Bianchi identities DG=0 the superembedding-like equation Studying its selfconsistecy conditions, we find the dynamical equations describing the relative motion of mM0 constituents. I. Bandos, Multiple M0+ SUGRA

12. Equations for the relative motion of multiple M0 in an arbitrary supergravity background follow from the constr. 1d Dirac equation Gauss constraint Bosonic equations of motion Coupling to higher form characteristic for the Emparan-Myers dielectric brane effect I. Bandos, Multiple M0+ SUGRA

13. Conclusions and outlook • BPS equations for the SUSY bosonic configurations of mM0 • We have obtained Eqs. for mM0 system (generalization of Matrix model eqs.) in an arbitrary 11D SUGRA background • The ½ BPS equation with nonvanishing 4-form flux has a fuzzy two-sphere solution (M2) • The famous Nahm equations = SO(3) invariant ¼ BPS equation with vanishing 4-form flux Some directions for future study • Search for more supersymmetric solutions of our mM0 Eqs, describing branes and brane intersection (could Basu-Harvey-type equation appear?) • Matrix model equations in a particular interesting backgrounds, e.g. AdS(4)xS(7) and AdS(7)xS(4), and their application, in particular in the frame of AdS/CFT. • 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 w/v superspace of a single brane)? Thanks for your attention! I. Bandos, Multiple M0+ SUGRA

14. I. Bandos, Multiple M0+ SUGRA

15. BPS equations for the supersymmetric pure bosonic solutions of mM0 equations SUSY preservation by relative motion of mM0 constituents SUSY preservation by center of energy motion ½ BPS equation (16 susy’s preserved) has fuzzy S² solution modeling M2 brane by mM0 configuration 1/4 BPS equation (8 susy’s) with SO(3) symmetryand Nahm equation I. Bandos, Multiple M0+ SUGRA

16. Thank you for your attention! I. Bandos, Multiple M0+ SUGRA