Miami 09, 15/12/2009I. Bandos, Superembedding N Dp1 Superembedding approach in search for multiple D-brane equations. D0 story. Igor A. Bandos Based on

Miami 09, 15/12/2009 I. Bandos, Superembedding N Dp

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Superembedding approach in search for multiple D-brane equations. D0 story.

Igor A. Bandos

Based on Phys.Lett. B 680 (2009) [arXiv:0907.4687] + arXiv:0912.2530 [hep-th]

- Introduction. Dp-branes and multi Dp systems in String/M-theory- Superembedding approach. Dp-branes in superembedding approach . - Superembedding equation and worldvolume gauge field constraints.- D0-brane in superembedding approach. Equations of motion from superembedding. - Multiple D0-brane in superembedding approach. - flat type IIA superspace. Relation with 10D SYM and with M(atrix) model - general type IIB background with fluxes. Myers ‘dielectric brane effect’ from superembedding approach. - Conclusion.

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


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Introduction. Dp-branes and multiple Dp-branes• Dp-brane – the p-branes when open string can have its ends [A.Sagnotti 87,

P.Horava 89, J. Polchinski, J. Dai, R. G. Leigh 89]• special role in M-theory – appreciated in 90th. Carrying RR (Ramond-Ramond)

charges [Polchinski 95]

• Described by the sum of DBI and WZ actions

RR gauge forms

NS-NS gauge 2-formfermion contributions[Townsend 95, Nelsson, Cederwall, Westenberg, Gussich, Sundell 96; Aganagic, Popescu, Schwarz 96, Bergshoeff, Townsend 96]

• Even before the above action was known, Howe and Sezgin [1996] had generalized the superembedding approach [I.B., Pasti, Sorokin, Tonin, Volkov 1995] for Dp-branes and shown that the equations of motion can be obtained from the superembedding equation. [The complete from of the D-brane equations was presented in I.B., Sorokin, Tonin 97.]

• In our discussion it wll not be excessive to notice that similar story happened for M5-brane, Howe and Sezgin derived 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]


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• The gauge fixed form of the low energy approximation is given by maximal U(1) SYM action = dim reduction of 10D SYM action down to d=p+1

• [Witten 95] The low energy approximation to the MULTIPLE (N) Dp-brane system is given by maximally SUSYc U(N) SYM action = dim reduction of 10D SYM action down to d=p+1

Single Dp-brane:

• The application of this low energy desription was quite productive. In particular it allawed for the hypothethis of M(atrix) theory [Banks, Fischler, Shenker, Susskind,96] that the matrix model considered as a theory of nearly coincident D0-branes gives a non-perturbative description of the hypothetical underlying M-theory.

• But what is the complete action for multiple Dp-brane system?

• 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, this widely accepted action does not possess neither SUSY nor Lorentz (diffeomorphism) symmetry. The problem of (the existence of) its SUSY and Lorentz covariant generalization remains open. [see notes on boundary fermion approach]

• As the superembedding approach had shown its efficiency in search for the D-brane and M5-brane equations [Howe + Sezgin 96], it is natural to apply it in searching for multiple Dp eqs.

This is the main goal of present study.


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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 quantum 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 failed].

• Even if the separate quantization of boundary fermions were found to fail to provide a covariant multiple Dp-brane action of the ‘usual’ type, the approach by Howe, Lindstrom & Wulff would give a correct,- but pre-quantized or true classical description of the system.


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Superembedding approach to superstrings and super-p-branes

• Bosonic (Nambu-Goto) string model is formulated in terms of D(=26) bosonic coordinate functions

• Green-Schwarz (GS) superstring is formulated in terms of coordinate functions describing embedding of bosonic surface into superspace

describing embedding of 2-dim worldsheet W² into the D(=26) dim spacetime

• Superembedding approach, following the so-called doubly supersymmetric twistor-like approach to superparticles and superstrings [pioneered in Sorokin, Tkach, Volkov, Mod. Phys. Lett. 89 and often called STV] describes the Brink-Schwarz superparticle and GS superstring in terms of embedding of worldsheet superspace in the target superspace .

• NSR or spinning string (more ancient, although still used) can be formulated in terms of bosonic coordinate superfields


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Superspace –what is it?• Standard (bosonic) spacetime coordinates are numbers (c-numbers according to

Dirac). They take numerical values and are commutative x y = + y x• One can also introduce Grassmann or fermionic coordinates: they are anti-

commutative (a-numbers in Dirac terminology) θζ= - ζθ and nilpotent: in distinction to fermionic operators (q-numbers according to Dirac) θζ= - ζθ is valid also for ζ= θ; hence ζ ζ=0.

• The classical limit of the bosonic creation and annihilation operator is given by usual c-numbers while the (quasi-)classical limit of fermionic creation and annihilation operators is described by Grassmann algebra elements= a-numbers = fermionic numbers.

• The Grassmann coordinates are useful for the (quasi-classical Lagrangian) description of particle spin

• and also for description of supersymmetric theories because the parameter of supersymmetry is a Grassmann number

• This is the case because supersymmetry can be (roughly) defined as symmetry mixing bosonic and fermionic fields.

• Superspace is such a generalization of spacetime that the set of its local coordinates includes, besides usual bosonic spacetime coordinates , also fermionic (Grassmann, anticommuting, nilpotent, a-number) spinor coordinates .

• When theory is formulated in superspace, its supersymmetry becomes manifest.


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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 equivalent to GS!

Let us present the place of superembedding approach as a scheme


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Dp-branes in superembedding approach. 0. Notions and notations.

W

ΣTarget D=10 type II superspace

Worldvolume superspace

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

These worldvolume superfields carrying indices of 10D type II superspace coordinatesare restricted by the superembedding equation.


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Dp-branes in superembedding approach. I. Superembedding equation

W

Σ Supervielbein of D=10 type II superspace

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

Supervielbein of worldvolume superspace

General decomposition of the pull-back of type IIA supervielbein


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Dp-branes in superembedding approach. I’. History and equivalent forms of the superembedding eq.

Sorokin, Tkach, Volkov (STV) 1988- Derived from superfield action (D=3,4, N=1)

Volkov, Zheltukhin 1988- used it to establish equivalence of D=3,4 spinning particle and superparticle

The STV action was generalized for higher D/N superparticles and for superstrings [Howe, Townsend, Delduc, Sokatchev, Pashnev, Galperin, Ivanov. Kapustnikov, Tonin,Pasti, Chekalov, Bergshoeff, Sezgin, …89-92] up to heterotic string without heterotic fermions [see Sorokin+Tonin, Ivanov+Sokatchev, P. Howe, 93 on the problem of heterotic fermions.]

Then it was used as a basis for superembedding approach to super-p-branes in [I. Bandos, P. Pasti, D. Sorokin, M. Tonin, D.V. Volkov 95] 10D superstrings and 11D M2-brane and [P. Howe and E. Sezgin 96] for 10D Dp-branes and 11D M5-brane.

Equivalent form of Superembedding equation:

where are (9-p) vector fields orthogonal to the worldvolume

superspace


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one finds one more equivalent form ofCombining this with

by adding (p+1) vectors tangential to the worldvolume superspace:

These are used to define on geometry induced by superembedding

We can complete their set of (D-p-1) vectors till moving frame

the superembedding equation

Pull-back of thebosonic vielbein of target superspace

bosonic vielbeinof the worldvolume

superspace

Moving framevariables

The pure bosonic limit of this equation makes sense and is known in Classical Surface Theory as well as in geometric or embedding approach to bosonic string [proposed by Omnis, Lund andRegge and developed further by Barbashov, Nesterenko, Zheltukhin, …]. However, in this pure bosonic case this is pure conventional constraint (off shell, just defining worldsheet zweibein).

In the SUSY case (embedding of superspaces) this is not conventional, nontrivial eq. which often (high SUSY, high D, high (D-p)) produces dynamical equations.


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Fermionic supervielbein and spin tensor h When Dp-branes are considered, it is convenient to define fermionic supervielbein

of by identifying it with the pull-back of, say, E¹,

Then the general decomposition of the pull-back of E² to reads

Integrability (selfconsistency) conditions for superembedding equations imply

The further investigations shows that the spin-tensor h encodes the info about the worldvolume gauge field strength Fab. It provides a ‘square root’ of k(F) SO(1,p)

For p<6 these also follow from the superembedding equations, which contain all the Info on the corresponding Dp-brane, including its dynamical equations of motion.


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Dp-branes in superembedding approach. II. Gauge field constraints on .

For p> 5 the superembedding equation is not sufficient to describe the dynamics of Dp-brane. This can be easily understood on the case of spacetime filling D9-brane, as far as for this case the superembedding equation is clearly pure conventional.

The equations of motion as well as the defining relations for the spin-tensor h

follow in these cases from the gauge field constraints on

where is the pull-back of the NS-NS gauge super-2-form with

or

The set of superembedding equations plus gauge field constraints always provides the complete description of the Dp-brane dynamics.

Notice that also in the p<6 cases it is convenient to use the gauge field constraints.


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D-instanton (D(-1)-brane) in superembedding approach. The further details of study the Dp-brane dynamics in superembedding approach are p-dependent, so it is convenient to do the study case by case.

For D-instanton all the directions are orthogonal to the ‘worldvolume’ (worldpoint, d=0).The superembedding equation can be written as

So that worldpoint superspace is purely fermionic

with the induced fermionic supervielbein

the spin tensor h should obey

This equation does not have any real solution, but the imaginary solution does exist:

It implies that

Hence the existence of D-instanton requires complex fermionic supervielbein.This is in agreement with the well known fact that the D-instanton exists in an Euclidean version, related with the standard type IIB by Wick rotation. This seems to be all the info which one can extract from the superembedding approach,which is not surprising as far as instanton is not dynamical ((-1)-brane or event).


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D0-brane in superembedding approach.

The worldvolume superspace has only one bosonic direction

Superembedding equation reads

and its (pure conventional) fermionic counterparts are

It is convenient to use the notation

keeping in mind that, due to presence of moving frame vectors, these matrices with the algebraic properties of 10D Pauli matrices are SO(1,9) invariant and not constant:

In this notation, the selfconsistency condition for the superembedding eq. implies

which is solved by

where

is the generalization of the mean curvatures of a line for the case of worldline superspace (of the second fundamental form of a surface in the case of p>0).

and


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D0-brane in superembedding approach II: Eq. of motion

Studying integrability conditions for the fermionic equations

one finds D0-brane equations of motion.

and

Generalizes 1d Dirac eq.

Generalizes 1d Klein Gordon eq.

In the case of flat target superspace these are

In general type IIA supergravity background the fermionic eq. acquires the r.h.s

from ‘’fermionic fluxes’’ = pull-backs of D1Φ, D2Φ,

The bosonic equation for D0-brane in general type IIA supergravity background

has the r.h.s contribution

from the 2-form RR flux R2=dC1: and from dilaton.


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On gauge field constraints on the worldline superspace.

p=0

Bianchi identities p=0

This is satisfied by

much simpler

Lesson: even in the cases when this is not necessary (p<6, where all Dp-brane equations follow from the superembedding equation) it is convenient to use also gauge field constraints on the worldvolume superspace.


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Multiple D0-branes in superembedding approach • As far as single D0-brane can be described by superembedding equation, and this has a clear geometrical meaning,

it is natural to expect that a center of mass motion of an interacting system of N D0-branes could be described in 1d =16 SSP by

• This is of course not a rigorous proof. However, the universality of super-embedding eq. ( ½ BPS p-branes!) and difficulties which one meets in attempt to modify it makes reasonable to accept the above framework: it should give at least a reasonable approximation (beyond the U(N) SYM one).

• As far as superembedding equation for the embedding of 1d =16 worldvolume superspace into 10D type IIA superspace has the D0-brane equations of motion among its consequences, the center of mass motion of N D0-bane system would be described by D0-brane equations but with tension (mass) x N. (This looks natural for the classical description of multiple brane system, although for Myers action this is not the case).

• In this framework the only possibility to obtain multiple D0-brane description is to put the SU(N) gauge superform on the D0-brane worldvolume super-space, and to search for a proper set of 1d =16 superspace constraints


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Non-Abelian SYM constraints on the D0-brane worldvolume superspace.

Introduce su(N) valued gauge potential with the field strength

This should be considered on the worldvolume superspace of D0-brane and subject to the constraint. For d=1 =16 case the natural proposition is

A clear candidate for the description of relative motion of D0-branes comes from dim 1 field strength!

Bianchi identities are satisfied if

and

It is natural to call this superembedding like equationIt gives su(N) generalization of the gauge fixed version of the linearized superembedding equation

It has the form of the superfield equation for

matter, and indeed, determines the relative

motion of D0 constituents.


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Multiple D0-branes in flat type IIA superspace

Studying the selfconsistency of the superembedding-like equation

Algebra of covariant derivatives is simpler

one finds

and, then, the Dirac-like equation for su(N) valued fermion Ψ

and the bosonic equations for su(N) valued X

and

Bosonic equations for relative motion of N D0-branes Gauss constraint characteristic

for (d=1 dim reduction) of gauge theories Why?


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Multiple D0-branes in flat superspace, 10D SYM and Matrix model

Our equations, in the part corresponding to relative motion of D0-branes, appear to be 10D SYM equations dim. reduced to d=1.

Thus our description of relative motion of D0-brane constituents is close to the very first approximate U(N) SYM model for N Dp: it is given by max. SU(N) SYM model, but with center of mass motion and U(1) multiplet described a single D0-brane equations. This provides the Lorentz covariance of our formalism.

Why?

This is because in flat d=1 N=16 worldvolume superspace our SYM constraints can be obtained by dimensional reduction of the D=10 SYM constraints

When fields are independent on spatial coordinates, are just scalars in adjoint representation of su(N) and the minimal d=1, N=16 field strength is

our constraints

This is important, in particular, because it indicates the relation with the Matrix model, which is described by 1d dimensional reduction of the 10D U(N) SYM action. This was also the first model used to describe D0-brane dynamics (before DBI+WZ actions)


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Furthermore, our superembedding approach allows to obtain the equations for multiple D0-branes in curved type IIB supergravity background, and it is unclear how to reach this by dimensional reduction from 10D SYM.

It is tempting to state that our superembedding description of multiple D0-system, based on superembedding equation for the center of mass motion and simplest 1d =16 SYM Constraints is `complete’ (like DBI+WZ action for a single D-brane).

However, of SUSY deformations of the 10D SYM equations and of its reduction to d=1 (Cederwall, Nilsson et al 2001; A.Schwarz + Moschev 0910.0620 )

suggests that our superembedding description might be approximate.

The corrections for the case of SYM in curved 1d N=16 superspace -curved worldline superspace of a D0-brane in curved type IIA superspace – have not been studied yet. But even if exist, they have to be complicated and their use is hardly practical (without computer). This makes our approach potentially useful (even if approximate)


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Multiple D0-branes in arbitrary IIA SG background

fermionic bilinears of the type

Equations acquire r.h.s.-s with background flux contributions.

Comparing with the single D0-brane equations (with center of mass motion)

We see that the multiple D0-brane system possesses coupling to the higher form NS-NS and RR fluxes, which do not interact with a single D0-brane.

This is the famous ‘dielectric brane effect’ [Emparan 98, Myers 99] = polarization of the D0-brane by external fluxes which simulates higher Dp-brane interactions.


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Conclusions and discussion• We have developed the superembedding description of multiple D0-brane system

and show that in the case of general curved back-ground it results in equations of motion describing the ‘dielectric brane effect’ by Emparan and Myers, i.e. coupling of multiple D0-brane system to higher form gauge fields thus simulating higher Dp-brane interactions.

• In the case of flat target superspace the superembedding approach produces a covariant generalization of the Matrix model equations; eqs. for SU(N) SYM plus D0-brane equations for the coordinate functions and DBI dynamics of related U(1) gauge field. In generic case these latter should be the nonlinear equations describing DBI+WZ dynamics and making the worldvolume superspace curved.

• Thus the bosonic limit of our equations is simpler than the (non-supersymmetric and not Lorentz covariant) equations following from the Myers action. However, their advantage is that they are SUSY and Lorentz covariant. The question of whether it gives an approximation to a more complete action is to be clarified.

• Our approach based on superembedding equation describing the motion of the center of mass of the multiple D0-brane system and the simplest 1d N=16 SU(N) SYM constraints might allow for modification. However, the form of the known susy deformations of SYM in flat 1d N=16 superspace, which suggests this, looks very complicated to be practical. This allows us to hope that our approach gives, at least, a potentially useful covariant approximation (generically, beyond the non-covariant U(N) SYM one).


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Directions for future study• If a modification of the superembedding equation resulting in a more

complicated interaction of D0-constituents did exist, our present study might provide a basis in search for such a hypothetical modification. To try to move in the direction it is important to understand the relation of our construction with the model of [S. Panda and D. Sorokin, JHEP 2003]; this might help in searching for a modification of the superembedding equation (if any existed).

• Our study has shown the existence of the superembedding description of multiple D0-brane (D-particle) system. It will be interesting to understand whether such a description exists for the case of type IIB D1-brane (D-string) and type IIA D2-branes (D-membranes) and higher Dp-branes.

• In particular, it will be interesting to apply the superfield (superform) generalization of the Bucher T-duality rules [B.Kulik + R. Roiban JHEP 02, I.Bandos+B.Julia JHEP 03] to our type IIA multiple D0- superembedding to obtain a candidate type IIB multiple D1- equations in such a way.


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Thank you for your

attention!


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Appendix A: 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 quantum 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 failed].

• Even if the separate quantization of boundary fermions were found to fail to provide a covariant multiple Dp-brane action of the ‘usual’ type, the approach by Howe, Lindstrom & Wulff would give a correct,- but pre-quantized or true classical description of the system.


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Appendix B: on [Sorokin & Panda 2003]

• To give an idea of the problems one meets on the way of searching for generalization of our approach which might incorporate nonlinear interactions suggested in [Sorokin & Panda 2003], let us notice that,

• although the consideration of [Sorokin & Panda 2003] uses a purely bosonic worldline, the identification of the κ-symmetry with worldline supersymmetry [STV 89] can be used to identify the corresponding superembedding equation

• This appears to be where is some worldvolume function ( in the notation of [Sorokin + Panda 03]) and is the pull-back of the RR 1-superform of the type IIA supergravity ( in the case of flat superspace considered in [Sorokin + Panda 03]).

• The problem with such a generalization of superembedding equation is that it is not invariant under the gauge symmetry of the RR 1-form.


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Appendix C: Structure of the fermionic equation

Structure of the equation for the su(N) valued fermion Ψ, defined in

i.e. of the fermionic equations of multiple D0-branes, in the presence of fluxes:

Documents

Miami 09, 15/12/2009I. Bandos, Superembedding N Dp1 Superembedding approach in search for multiple D-brane equations. D0 story. Igor A. Bandos Based on