BPS PREONS AND M-THEORY

J. A. de AzcárragaDepartment of Theoretical Physics, Valencia University and IFIC (CSIC-UVEG)

46100-Burjassot (Valencia), Spain

RTN Network ‘Forces Universe’ 2nd Workshop and Midterm Meeting Napoli, October 9-13, 2006

[based on work with I. Bandos, J.M. Izquierdo, M. Picón and O. Varela]

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Summary

• The M-algebra

• (k/32)-supersymmetric BPS states and BPS preons (k=31) • Killing spinors, preonic spinors and the G-frame • Generalized curvature, holonomy and Killing spinors

• The G-frame method and k-supersymmetric solutions

• Implications of generalized holonomy for the existence of preonic solutions

• Conclusions

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(k/32-) Supersymmetric BPS states and preons[I.Bandos, J.A. de Azcárraga, J.M. Izquierdo and J. Lukierski, PRL 20, 4451 (2001), [hep-th/hep-th/0101113 ]]

[for D=11 for now, but all extends to arbitrary D]

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A preon state is a state preserving all supersymmetries but one. Hence it is labelled

Then, introducing a preonic spinor by the condition con , we see that

a preonic state may be equally labelled by

In general, when k ≤ 31 we may introduce n = 32-k linearly independent bosonic spinors orthogonal to the set of k spinors

A BPS state preserving k supersymmetries satisfies

where the k bosonic spinors , or Killing spinors (in sugra., see below) determinethe Grasmann parameters of the supersymmetries by .

Thus, a general BPS state preserving k supersymmetries may

characterized either by the k Killing spinors associated with thesupersymmetries preserved by the BPS state or by n=32-k preonic sp. orthogonal to the Killing spinors (the use of both will lead to the G-frame method).

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An equivalent definition of k/32-supersymmetric BPS states: BPS preons as constituents

A k/32-BPS state may also be defined as an eigenstate of the generalized momentum whose eigenvalue matrix is singular, of rank n-k, and positive (semi)definite,

Rank 1 is obtained when the matrix is given in terms of one spinor by the

generalized Penrose relation This will correspond to a preonic state (rank n=1, k=31); the case of a general k-BPS state will then correspond to the eigenvalues matrix

To see it, notice that this matrix can be diagonalized by the GL(32,R) automorphism symmetry. Further, all its eigenvalues are positive or zero, since in a suitable diagonal basis, , say, would imply

contradicting positivity. Hence, in that basis, becomes Now, since the 1’s in the diagonal may be obtained by multiplying elementary spinors in the original basis we may also write the eigenvalues matrix as a sum of products of spinors. Hence,

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Now, if the k preserved symmetries are generated by , implies for them that

i.e.

and k=32-n. The case k=1, n=31 corresponds to a BPS preon.

Thus, we may look at the equation at the top as a manifestation of the composite structure of a k/32-BPS state, since it is solved by

where the are elementary, BPS-preonic states characterized by the nlinearly independent preonic spinors , n=32-k.

Thus, for a k/32-BPS state, with , we find

n = number of preons composing the BPS state = # preonic spinors k = 32-n = number of preserved symmetries = # Killing spinors

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Killing spinors, preonic spinors and the G-frame[Bandos, de Azcárraga, Izquierdo, Picón and Varela, PRD 69, 105010 (2004) [hep-th/0312266]]

Since the set of bosonic k Killing and n=32-k preonic spinors are orthogonal,

they may be completed to obtain bases in the spaces of spinors with upper and lower indices by introducing n=32-k upper index spinors and k lower indices spinors

The G-frame is constructed from the k Killing spinors of the supersymmetriespreserved by a BPS state and the n=32-k bosonic, preonic spinors characterizing the BPS preons of which the BPS state is composed.

Each of these two dual bases defines a G-frame described by the non-singular matrices

where reproduces all products above and the second eqution is the unit decomposition or completeness condition.

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Generalized curvature and holonomy[See M.J. Duff and K.S. Stelle, PLB 253, 113 (1991); J. Figueroa-O'Farrill, G. Papadopoulos, JHEP 0303, 048 (2003) [hep-th/0211089]; M. Duff and J.T.

Liu,,NPB 674, 217 (2003); hep-th/0303140, C. Hull hep th/0305039, G. Papadopoulos and D. Tsimpsis hep-th/0306117; M. Duff, hep-th/0201062, 0403160] .

Consider a bosonic solution of CJS supergravity. Since ,the invariance under supersymmetry of a bosonic solution is guaranteed if

where we have writen constant and Grassmann odd, bosonic. A solution of the above Killing spinor equation is called Killing spinor; the J=1,…,k Killing spinors are in correspondence with the k preserved local supersymmetries.

Killing spinors have to satisfy an integrability or consistency equation in terms of the generalized curvature

The generalized curvature takes values in the generalized holonomy algebra . For the D=11 and type II sugras is known[C. Hull hep th/0305039 ; G. Papadopoulos and D. Tsimpsis hep-th/0306117]

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- If Killing spinors exist the inclusion is strict, , the 32 of sl(32,R) is reducible under , and the Killing spinors (supersymmetries preserved by

the bosonic supergravity solution) are invariant singlets in this decomposition.

- In particular, for a hypothetical preonic state, k=31, and thus

-Moreover, the following (see later) holds for the solution preserving of k supersymm.[C.Hull, hep-th/0305039 ; Papadopoulos and Tsimpis, CQG 20, L253 (2003) [hep-th/0307127], JHEP 0307, 018 (2003) [hep-th/0306117] ]

-In general, the generalized holonomy permits fractions k>16 of preserved supersymm. forbidden in the Riemannian (F=0) connection case, which limits k≤16 [see J. Figueroa O’Farrill, CQG 17, 2925 (2000) [hep.th/9904124], and refs. therein ]

Susy, generalized holonomy and Killing spinors-the connection and the curvature are the generalized ones (they include F) and generates , and

-The possible supersymmetric solutions of supergravity can be analized in terms of the generalized holonomies: the consistency cond’n for the existence of k Killing spinors, can be looked at as a restriction for the generalized holonomy group H:

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k/32-supersymmetric sugra solutions (present list) All fractions k/32 of preserved supersymmetries are allowed by the M-algebra.[ J.P. Gauntlett and C.M Hull, JHEP 0001, 001 (2000) [hep-th/9909098]; I. Bandos and J. Lukierski, XIIth Max Born, LNP (1999), hep-th/9812074 ]. Further, all Q-central charges are on the same footing and mixed by the GL(32,R) symmetry. Thus, a priori there is no reason for not finding backgrounds for all possibilities. But, can we actually find a configuration for any k in D=10, 11 supergravities?

k=n=16 : ‘standard’ BPS states, solutions of D=11 supergravity, as the M0 (M-wave), M2, M5, M-KK (D=11 Kaluza-Klein monopole), M9 branes [see M.J. Duff, R.R. Khuri and J.X. Lu, Phys. Rep. 259, 213 (1995) [hep-th/9412184]; K.S. Stelle, hep-th/9803116; C. Hull, NPB 509, 216 (1998) [hep th/9705162] ]

k ≤ 16, n ≥ 16: k/32 ≤ 1/2 states can be treated as supersposition of ½ states (intersecting branes) [P.K. Townsend, Cargèse lectures [hep-th/9712004]; see also J. Molins and J. Simon, PRD 62, 125019 (2000)

[hep- th/0007253]; A. Batrachenko, M.J. Duff, J.T. Liu and W.Y. Wen, hep-th/0312165 ; ……]

k > 16, n < 16 : ‘exotic’, k/32 > 1/2 states also have been found (from 2002) [see refs. above plus Cvetic+Lu+Pope [0203229]; Mickelson [0206204]; Lu+Vazquez-Poritz [0204001]; Bena+R. Roiban,[0206195]; Sakaguchi [0306009]…….]

At present the list of k-supersym. states is (bracketed numbers = missing (k)-solutions)

k = 0, 1, 2, 3, 4, 5, 6, (7), 8, (9), 10, (11), 12, (13), 14, (15), 16, (17),

18, (19), 20, (21), 22, (23), 24, (25), 26, (27), 28, (29), (30), (31), 32.

k=32, n=0 : vacuum, flat superspace; also the D=11 and the IIB-rel. supergravity solutions

[P.T. Chrusciel and J. Kowalski-Glikman, PLB 149, 107 (1984); J. Figueroa O’Farrill, G. Papadopoulos, JHEP 0108,036 (2002) [hep-th/0105308]; M. Blau, J. Figueroa O’Farrill, G. Papadopoulos, CQG 19, 4753 (2002) [hep-th/0202111]]

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The G-frame method and k-supersym. solutions[I.A. Bandos,J.A. de Azcárraga, J.M. Izquierdo, M. Picón and O. Varela, PRD 69, 105010 (2004) [hep-th/0212347]

It may be checked that the above expression recovers that the allowed holonomies of the k-supersymmetric solution of CJS D=11 (and D=10) supergravity satisfy the mentioned result

General aim: to find solutions to , which has to be satisfied by a k-supersymmetric solution. Taking the generalized covariant derivative of the various orthogonality conditions of the spinors in the G-frame and in its inverse, one finds that the gen. curvature may be expressed as

where are the generalized Cartan forms in

involving the ‘complementary’ spinors u , w of the G- the inverse G-frame. Then,

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D=11 sugra and holonomy for preonic solutions[I.A. Bandos,J.A. de Azcárraga, J.M. Izquierdo, M. Picón and O. Varela, PRD 69, 105010 (2004) [hep-th/0212347] ]

Then, k=31. For G SL(32,R) , (which is always the case for D=11 and D=10 supergravities) . For k=31, this is simply A=0, and the the gen. curvature reduces that solves the consistency equation for a preonic solution satisfies

Oo

(1)+(2) (or their equivalent) are the eqns. to be satisfied by a preonic configuration.

Of course , as given by (1) still has to satisfy the bosonic eqs. of the motion, which may be written compactly as

An obvious sol. is trivial gen. holonomy group, H=1. Butthis k=31 dB=0 solution would actually be a k=32 one (fully supersymmetric vacuum); hence, if this were the only solution, (bosonic) preonic solutions would not exist in ‘free’ D=11 sugra. This would not yet exclude possible preonic solutions when considering ‘higher order’ (α')³-corrections, of third order in the Riemannian curvature [ see K. . Peeters,

P. Vanhove and A. Westerberg, CQG 18, 843 (2001) [hep-th/0010167, hep-th/0010182]; P.S. Howe and D. Tsimpis, JHEP 0309, 038 (2003) [hep-th/0305129]; B. Zwiebach,

PLB 156, 315 (1985)] and/or non-zero r.h.s’s for the equations of motion (D=11 sugra interacting with branes). As a result, for ‘free’ D=11sugra, no definite conclusion on preonic solutions may be reached yet.

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D=10: The case of IIA-supergravity

Type IIA fields may be obtained making the D=11 gravitino component fields independent of # by dimensional reduction. Then, =1,…32 denotes the components of a Majorana spinor composed of two Maj.-Weyl 16-spinors of different chiralities, and thus for the IIA type dilatino we have

In the IIA case the position of the index in can be changed by the 32×32 chargeconjugation matrix C that exchanges the 1 and 2 MW components.

a) Component fields

In D=11 CJS supergravity, the only fermionic field is the gravitino, which we maywrite

In contrast, in D=10, type II-sugra there are, besides two 16-component ‘spin 3/2’ (IIA or IIB) gravitini, two ‘spin ½’ 16-component fermionic dilatini fields .

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b) supersymmetry transformation properties

The IIA matrix M is given in terms of all the IIA fluxes (on-shell field strengths)

They can be written in the above compact notation as (a = 0,1,…,9)

where in which denotes the additional tensorial contributions to the IIA (similarly IIB) supercovariant derivatives. The supersym. transf. rules for the dilatini (derived for IIA dim. reduction for the # component) are algebraic and given by the matrix M [see F. Giani and M. Pernici, PRD 30, 325 (1984); I.C.G. Campbell and P.C. West, NPB 243, 112 (1984); M. Huq and

M. A. Namazie, CQG 2, 293 (1985) [E.-ibid. 2, 597 (1985)]; D. Marolf, L. Martucci and P.J. Silva, JHEP 0307, 019 (2003) (App. B) [hep-th/0306066] ] .

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A bosonic solution stable under k supersymmetric transformations satisfies a differential (Killing) equation (as before) plus an additional algebraic equation (coming from the dilatini transformation matrix M):

These two equations guarantee that the bosonic solution is supersymmetric.

For IIA preons, k=31, this implies that the algebraic equation, looked at as an equation for the matrix M, is solved if it can be written in terms of a preonic and of a certain spinor s as

Since M is given, this equation in the fluxes becomes a constraint that a preonic spinor must satisfy as well as a condition on the fluxes in M.

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No preonic solutions in IIA supergravities [I.A. Bandos, J.A. de Azcárraga and O. Varela, JHEP 09, 009 (2006) hep-th/0607060]

A similar analysis for IIB produces the same negative answer, previously found by [U.Gran, J.Gutowski, G.Papadopoulos and D.Roest, hep-th/0606049] using different methods. Hence, there are no preonic, k=31 solutions among the bosonic solutions of type II, D=10 supergravities i.e., in the low energy approximations to IIA and IIB string theories.

In the IIA case, the preonic and the auxiliary spinor are Majorana ones,

Then, the equation splits into

and, since , the l.h.s of (a) and (d) are equal and thus One now uses this result and the other blocks to find that which, now in (b) and (c), allows us to conclude that

either or The first case excludes preonic solutions; the second implies the vanishing of M and hence all IIA supergravity fluxes,(spin connection). Then, the Killing spinor equation involves the ordinary connnection and may have supersymmetries, not 31.

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Could D=10 preonic solutions still exist?The previous negative results for IIA supergravities refer to the classical, low energyapproximations of type II string theories.

Stringy corrections are difficult [see K. Peeters, P.Vanhove and A. Westerberg, CQG 18, 843 (2001) [hep-th/0010167] ;

D.J. Gross and E. Witten, NPB 277, 1 (1986) ]. But it looks promising that the D=11 generalized connection discussed in [ H. Lu, C.N. Pope, K.S. Stelle and P.K. Townsend, JHEP 0507, 075 (2005) [hep-th/0410176]; H. Lu, C.N. Pope and K.S. Stelle,

NPB 741, 17 (2006) [hep-th/0509057] ] includes terms that under dimensional reduction lead to the above type modification of the IIA M-matrix.

On account of the fundamental role played by preons in the classification of BPS states,it is tempting to say that the negative results for type II point out towards the need ofincorporating ‘stringy’[ -(and perhaps higher)] corrections to the supergravity equations and transformation rules. In the IIA case the previous negative results would not follow if the M-matrix acquired new terms invalidating the previous analysis. If these were be of the form

where one could not longer conclude all fluxes equal zero.

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Conclusions

For classical IIA and IIB supergravities, no preonic solutions exist, but since the hypothetical preons are M-theory objects, a conclusive analysis of supergravity with ‘quantum stringy corrections’ remains to be done.

In contrast, and even for ‘free’ D=11 CJS supergravity, no definite conclusion on the presence of D=11 preonic solutions can be made yet.

All the supersymmetric BPS states of M-theory preserving k supersymmetries can be treated as composites of n=32-k primary constituents or BPS preons [the statement is true for any spacetime D with 32 replaced by the corresponding spinor dimension]. Usually, BPS states are described by solutions of D=10, D=11 supergravities. Are there any preonic solutions?

(In contrast, there are always preonic solutions in Chern-Simons type supergravities[I..A. Bandos,J.A. de Azcárraga, J.M. Izquierdo, M. Picón and O. Varela, PRD 69, 105010 (2004) [hep-th/0212347] )

If preonic solutions were found only when corrections are considered (‘stringy’ corrections in type II, ‘radiative’ higher order corrections in Riemann curvature to the Einstein equations in D=11), this could indicate that preons are intrinsically quantum objects that cannot be seen in the low energy aproximation to M-theory.

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Note, finally, that all searches for preonic solutions have been concerned with purely bosonic solutions, a restriction not implied by the preon conjecture. [I.A. Bandos, J.A. de Azcárraga, J.M. Izquierdo and J. Lukierski, PRL 86, 4451 (2001) [hep-th/0101113] ]

If the preonic solutions were finally shown to be forbidden in supergravity, a natural question would be to know the degree of the ‘preon conspiracy’ determining the minimal number of preons of a supergravity solution.

And, since preonic p-brane actions are easily constructed in maximally extended, tensorial superspaces [see I. Bandos and J. Lukerski, Mod. Phys. Lett. 14, 1257 (1999) [hep-th/9811022], id.+D. Sorokin, PRD 61, 045002 (2000)

[hep-th/9904109] ] , perhaps one should take enlarged superspaces seriously [in the spirit of [C. Chryssomalakos, J.A. de Azcárraga, J.M. Izquierdo andJ.C. Pérez Bueno, NP B567, 293 (2000) [hep-th/9904137];

;J.A. de Azcárraga and J.M. Izquierdo, AIP Conf. Proc. 589, 3 (2001) [hep-th/0105125] ]