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PHYSICS LETTERS B

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Physics Letters B 565 (2003) 229-236

www. elsevier. com/locate/npe

Extra gauge symmetry for extra supersymmetry

A.A. Zheltukhina b, D.V. Uvarovb

a Institute of Theoretical Physics, University of Stockholm, SCFAB, SE-106 91 Stockholm, Sweden b Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine

Received 17 April 2003; accepted 8 May 2003

Editor: P.V. Landshoff

Abstract

It is shown that the extra supersymmetry of tensionless superstring and super p-brane is accompanied by the presence of new bosonic gauge symmetries. It permits to use composed coordinates encoding all physical degrees of freedom of the model and invariant under these gauge symmetries and the enhanced k-symmetry. It is proved that the composed gauge invariant coordinates coincide with the components of symplectic supertwistor realizing a linear representation of the hidden OSp(1, 2M) symmetry of the super p-brane Lagrangian. A connection of the presented gauge symmetries with massless higher spin gauge theories and a symmetric phase of M/string-theory is discussed. © 2003 Elsevier B.V. All rights reserved.

1. Introduction

Tensionless (super)strings have recently been discussed in the frame of massless higher spin field theory and AdS/CFT correspondence [1-3]. The point is the relation gYMN = (R2/a')2 between the 't Hooft coupling constant g2MN and the rescaled string tension R2/a', where R is the radius of AdS5 x S5.1 This relation shows that the zero limit for the string tension T = 1/a' leads to a free gauge theory. The conjecture was advanced that conformal N = 4 SYM theory has to be dual to the theory of massless higher spin fields when N is large. Symplectic (super)symmetries OSp(1,2p) play an important role in the formulation of the massless higher spin field dynamics and may be linearly realized in the generalized spacetime described by the real symmetric matrix Yab (a,b = 1,..., 2p) and its superpartner [3,5]. If a, b are identified with the Majorana spinor indices the Yab components can be treated as spacetime coordinates added by tensor central charge (TCC) coordinates presented by antisymmetric spacetime tensors contracted with appropriate antisymmetrized products of y -matrices.

It was assumed in [3] that higher spin gauge theory may be treated as a symmetric phase of M-theory. This assumption is supported by the observation [6] on a nonlinear character of the OSp(1, 64) supersymmetry realization in D = 11 supergravity which is the low-energy limit of M/string-theory, where higher spin excitations are massive. The consideration of superstring theory as a spontaneously broken phase of higher spin gauge theory

E-mail address: aaz@physto.se (A.A. Zheltukhin). 1 The rescaled string tension as a perturbative parameter of string dynamics in curved spaces was previously considered in [4].

0370-2693/03/$ - see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0370-2693(03)00697-X

promises a new basis for understanding the microscopic structure of superbranes in terms of the spontaneous breaking parameters of higher spin gauge theory which correlates with the topological approach [7].

Thus, it is important to find the right kind of string/M-theory in terms of the new dynamical variables associated with orthosymplectic (super)symmetries and higher spins. This problem stimulates investigation of the superstring and superbrane dynamics in generalized spacetimes extended by the addition of TCC coordinates [8-10].

Using the twistor-like approach [10,11] the exactly solvable model of tensionless super p-brane preserving 3/4 of the D = 4 N = 1 supersymmetry was proposed in [12], which generalizes the superparticle model [13], where preserving 3/4 supersymmetry was earlier noted. This super p-brane saturates the exotic BPS state with the same symmetry whose existence has been proved in [14] applying the model independent analysis of the D = 4N = 1 superalgebra enlarged by tensor central charges. The model [12] preserves all its properties in the D-dimensional Minkowski space with D = 2, 3, 4 mod(8), where the supersymmetric Cartan form and an auxiliary twistor-like Majorana spinor, defining the superbrane Lagrangian, are the same modulo the spinor's dimension. As a result, the tensionless super p-brane will saturate the exotic BPS state spontaneously breaking only one from M global supersymmetries, where M is the Majorana spinor dimension. An interpretation of this tensionless super p-brane as the BPS state of M-theory was recently discussed in [15]. An important property of the tensionless super p-brane [12] and the appropriate BPS state is the linear character of the OSp(1, 2M) symmetry realization. The linear realization is a result of the transition to a new supertwistor variable previously considered in the superparticle dynamics [13,16,17]. The symplectic supertwistor YA [12] encodes all physical degrees of freedom contained in Yab and the Majorana spinor 0a which describe the original formulation of the model. This reduction of the initial variable number points out on the presence of hidden bosonic gauge symmetries strongly correlating with the extra k-symmetry. A fine tuning of these symmetries results in the minimal spontaneous breaking of the N = 1 global supersymmetry and manifests itself by the OSp(1,2M) symmetry appearance similarly to the superparticle case [13,18], where the first class constraints have signaled about the hidden bosonic symmetry presence.

Therefore, the problem appears to find hidden gauge symmetries responsible for the reduction of the bosonic degrees of freedom described by the spacetime and TCC coordinates. Let us remind that the reduction of the fermionic degrees of freedom of the model is provided by its enhanced k-symmetry [12]. The required bosonic symmetries should match the enhanced k-symmetry and their description is important for understanding the structure of a symmetric phase of string/M-theory [3] and local symmetries compatible with the BPS states characterized by extra supersymmetry [19].

In this Letter we present a set of new bosonic gauge symmetries matched with the enhanced k-symmetry of the tensionless super p-brane. We show that these gauge symmetries result in the invariant reduction of the bosonic gauge degrees of freedom described by the original symplectic spin-tensor Yab. The invariant character of the reduction means that the symplectic supertwistor YA encoding all physical degrees of freedom and providing the OSp(1, 2M) symmetry of the superbrane Lagrangian is gauge invariant under the transformations of the bosonic and enhanced k-symmetries. It should be noted that the bosonic symmetries include the Weyl gauge symmetry which correlates with a global spacetime conformal symmetry of the tensionless string action as it was noted in [20].

2. Tensionless super p-brane with enhanced supersymmetry

The inclusion of TCC in the algebra of N = 1 supersymmetry generalizes anticommutator of the Majorana supercharges Qa to the form [21,22]

{Qa,Qb} = (Ym C-l)abPm + i (Ymn C~l)abZmn + (Ymnl C-l)abZmnl + •••. (1)

The r.h.s. of Eq. (1) contains antisymmetrized products of y -matrices multiplied by the TCC Zmn... which are antisymmetric Lorentz tensors in the Minkowski spacetime commuting with Qa. The real antisymmetric parameters zmn... corresponding to Zmn... are known as TCC coordinates which may be presented in the equivalent

spinor form

Zab = iZmn {y™" C-1)ab + Zmnl (ymnl C-1)ab + ■■■ (2)

by analogy with the spinor representation of the spacetime coordinates xm

Xab = Xm{ymC-1)ab. (3)

The coordinates zab and xab may be treated as the components of the real symmetric spin-tensor

Yab = Xab + Zab (4)

which can be identified with the symmetric matrix of generalized symplectic coordinates considered in [3,5]. The pairs (Yab,0a) unifying Yab and the Majorana spinor 0a form a generalized superspace invariant under the N = 1 global supersymmetry

SsOa = Sa, SsYab = 2i(0aSb + BbSa). (5)

The differential one-forms Wa and Wab of the generalized superspace

Wa = dB a, Wab = dYab - 2i(d0a Ob + dBb 0a) (6)

are the Cartan forms invariant under (5) which have been used to construct the exactly solvable model [12]

SP = \j drdPa p"{UaWfUb)

of tensionless super p-brane (p = 1, 2, 3,...) with extra k-symmetry. The action Sp includes an auxiliary Majorana spinor Ua [23] and the world-volume density p" [24] which are invariants of the N = 1 supersymmetry. The spinor Ua parametrizes the light-like density of the brane momentum. The action Sp is also invariant under the transformations of the enhanced k-symmetry

&K0a = Ka, 8 k Yab = -2i(0aKb + BbKa),

Ua = 0, 8K p" = 0, (8)

where the parameter k is restricted by the only one real condition

KaUa = 0. (9)

As a result, this model preserves (-^p) fraction of the /V = 1 supersymmetry, where M is the dimension of the correspondentMajorana spinor. The model yields a pure static general solution for the Goldstone fermion fj defined by the Lorentz invariant projection

n = -2i{Ua0a) (10)

encoding the spontaneously broken component of the supersymmetry. The exact solvability of the model implies the presence of hidden local symmetries which is revealed by the change of variables

iYa = YabUb -nBa (11)

introducing the Majorana spinor Ya as a new variable substituted for Yab. In terms of the new spinor variable the action (7) transforms to the form

SP=l-f dx dPa p^{[{Uad^Ya) - (d^UaYa)\- ijd^rj}. (12)

The action (12) is the component representation of the OSp(l, 2M) invariant action

SP = ^f drdPa pfldfI,YAGASYs (13)

in which the real OSp(l, 2M) supertwistor YA = (iUa, Ya, fj) and invariant supersymplectic metric Gas = (—1)as+1 Gsa have been used. The transition from the representation (7) to the supertwistor representation (12) (or (13)) is accompanied by the reduction of some original variables both in the fermionic and bosonic sectors. It means that the super p-brane Lagrangianis singular due to the presence of hidden gauge symmetries. The enhanced k-symmetry (8) is responsible for the reduction of the (M — 1) of the M components of the Majorana spinor 0a and one remaining fermionic variable fj (10) proved to be invariant under the k-symmetry transformations (8), (9). The invariance of the Goldstone fermion fj proves that (M — 1) fermionic gauge degrees of freedom have been reduced without gauge fixing. The question then appears about hidden gauge symmetries responsible for the reduction of the bosonic gauge degrees of freedom contained in Yab. We shall define these symmetries in the next section.

3. Gauge symmetries matched with extra k-symmetry

First of all we note that both of the representations (7) and (12) are invariant under the local Weyl symmetry including one real parameter A(t, a)

p' * = e—2Ap'\ U'a = eAUa, 0'a = 6a, Y'a b = Yab. (14)

The transformations (14) imply that x'ab = xab, z'ab = zab, but Ya and the supertwistor Y'A are not invariant under the Weyl transformations

Y'a= eAYa, Y's = eAYs. (15)

The invariant character of Yab (14) means that the Weyl symmetry does not participate in the discussed reduction of the bosonic coordinates and other gauge symmetries should be found. To connect the above-mentioned gauge symmetries with the OSp(1, 8) symmetry of the 4d higher spin theory [3] and the results [14] we present here a detailed analysis of the D = 4 N = 1 supersymmetry. The generalization of these results to the higher dimensions D = 2, 3,4 mod(8) will be clear from this analysis. In the 4d case the action Sp (7) acquires the form

i j drdpa p^(2uacoIÀaàiia + uaa)IÀapu13 + uacoIÂàpuP), (16)

where the supersymmetric one-forms and in the Weyl basis are

V^afi = —d^Zafi - 2i(dIÀ0a0p + d^0p0a),

v^à fi = -diz a fi - 2i (d!0a 0fi + di0fi 0a). (17)

For the search of hidden gauge symmetries it is efficient to introduce a basis in the spinor space of the model. To this end a linearly independent local Weyl spinor va may be added to ua. Then, without loss of generality, the Weyl spinors ua and va attached to the brane worldvolume may be identified with the local Neumann-Penrose dyad [25] defined by the well-known relations

uaua = 0, vava = 0, uava = uaeafivfi = 1 (18)

and their complex conjugate. The scalar products (18) will be invariants of the Weyl transformations (14) if the transformation V'a = e-AVa is taken into account. The Majorana bispinors Ua(t,a) = , Va(t,a) = {Vo), (y5U) a and (y5 V)a [12] will respectively form a basis in the Majorana bispinor space. The first of the desired gauge symmetries which transforms only xm is defined as

8xxaa = exuaua (19)

and it is a local symmetry of the action Sp (16), due to the relation uaua = 0. We shall call this one-parametric real transformation as X-shift. It shifts xm by the light-like 4-vector (uamu)

Sxxm = --cx{u<Jmu). (20)

The change (uamu) ^ (UymU) in (20) lifts the X-shift to the bispinor representation and shows that it is also the symmetry of the high dimensional action (7) due to the relation (UaUa) = 0 satisfiable for the Majorana bispinors. The next gauge symmetry of Sp (16) transforms only the TCC coordinates zap

8TZap = ^Tuaup, 8tZa j}=<=Tua up. (21)

It includes one complex parameter eT and we shall call it as T-shift. Note that the local spin-tensor uaup in (21) has zero norm, i.e., (ua up)(uauP) = 0, and defines a set of local null planes attached to the super p-brane world-volume. Therefore, the T-shifts are a generalization of the X-shifts (19), described by the field of null vectors, to the shifts defined by the field of null bivectors, as it can be seen from the representation (21) in the tensor form

$TZmn= ^ [^r {uGmnM U(7mnll^ (ll(7mnU U(7mnU^J, (22)

where and are real parameters

4*> = I(er+e-r), 4') = J_(er_?r). (23)

In terms of the Majorana bispinor Ua the transformation (22) is presented as

&TZmn = (Uymnu) + (Uymny5U)], (24)

where the bivectors (UymnU) and (Uymny5U) belong to the field of null or isotropic bivectors [11] defined by the conditions

[UymnU )2 = 0, {UymnYsU )2 = 0, (25)

and are interpreted as the egenvalues of the generalized TCC Zmn (1). We see that the local translations of the TCC coordinates zmn by the null bivectors (25) produce a new type of gauge symmetry due to which the TCC coordinates zmn proved to be defined modulo the shift by the null bivectors. In the higher-dimensional Minkowski spacetimes additional multivector translations presented by the bilinear covariants similar to (UYmn...iU) will appear as admissible gauge symmetries of the action (7).

To continue the description of the next gauge symmetries we note that the dyad space (18) is symmetric under the u ^ v spinor permutation which transforms the null spin-tensors uaup (21) to the null spin-tensors vavp. But, it is not a symmetry of the action (16). However, the local shift of zap by the null tensor vavp similar to the null shift (21) may be compensated if simultaneous shift of xaa by the correspondent null vector vava similar to (19) will be added. As a result, we find a new one-parametric gauge symmetry of Sp (16)

= eTRVa va, $TRZap = eTRVaVp, sTRza p= e TRva Vp, (26)

because the correspondent variation of Sp (16)

SYRSp = j drdpa p*d^YR [{uava )2 — 1] = 0 (27)

equals to zero due to (18). Let us call the transformation (26) as 7R -shift. The local space-like vectors m(+land m(— ^

m(c+a) = UaVa + vaUa, = i(uava — vaua), n(+ = uaua, n{~a) = vava, (28)

orthogonal to the real local null vectors uaUa (19) and va va (26), form the local tetrade attached to the superbrane worldvolume.

The local shifts of the x -coordinates in the transverse directions m ^

8$(+)Xa a = e0(+)m<a+l!, 8$(—)Xaa = , (29)

which we shall call 0(±)-shifts, change Sp (16)

&&(+)S = j dr dpa p*e@(+) (uad*ua + uad*ua),

8#(—)S = i j drdpap*€0(—) (uad*ua — uad*ua). (30)

However, just as in the previous case the variations (30) are exactly compensated by the correspondent shifts of the TCC coordinates zap

&®(+)zaP = 2^0(+)u{a vp}, 8$(+)za a = 2e0(+)U{aVp},

8$(—)Zap = 2ie$(—)u{a vp}, &&(—)Za p = —2ie$(—)u{aVp}, (31)

where the symmetrized production u{avp) = \{uavp + upva) was introduced.

Thus, we found six real bosonic gauge symmetries of the action (7) in the 4d Minkowski spacetime generated by the X, T, T, Tr,@(+) and which form six parametric Abelian group of translations in the 10d symplectic subspace of the 11d symplectic superspace. In the next section we shall show that these gauge symmetries are responsible for the invariant reduction of six bosonic gauge degrees of freedom contained in Yab (4).

4. The gauge invariance of the supertwistor YA

Here we show that the supertwistor YA is invariant under the above presented bosonic gauge symmetries and the enhanced k-symmetry. The invariance of YA will prove the gauge invariant character of the considered reduction of bosonic and fermionic degrees of freedom. To this end let us consider the transformation properties of the supertwistor Ya = (iUa, Ya, rj ) under the fermionic k-symmetry (8), (9) and bosonic symmetries described by the X-shifts (19), T-shifts (21), Y -shifts (26) and the i>(±)-shifts (29), (31).

Then we find that invariant character of the Ua and rj follows from the definitions of the transformation rules of the enhanced k-symmetry and the bosonic symmetries.

Using this observation one can present the gauge transformations of the remaining supertwistor component i Ya

i&Ya = 8YabUb -rj80a. (32)

The substitution of the k-symmetry transformations (8) in (32) together with using the relations (9) and (10) yields the required result

&Ja = 0. (33)

Taking into account the invariance of 0a under the bosonic gauge symmetries one can simplify the variation (32) to the form

iSTa = SYabUb = (~yup +^XaàU a \ (34)

V Sxaaua - 8zapup z The substitution of the X-shifts (19) and T-shifts (21) into (34) results in the relations

Sxxaaua = exua(u auua) = 0, &rzapup = eju^upu^ = 0 (35)

and their complex conjugate which prove the invariance of Ya under these shifts.

The invariance Ya under the TR -shifts (26) follows from the cancellation of x and z contributions given by

&TRXaaua - STRZapuP = eTRVa - eTRVa = 0. (36)

The analogous cancellations take place between the x and z contributions

&0(+)Xaàua - &$(+)Zapup = e0(+) [m^ua - 2u{aVp}up] = 0,

S0(-)Xaaûa - &$(-)ZapuP = e$(-) [m(~^uua - 2iu{aVp}up] = 0 (37)

and their complex conjugate generated by the 0(±) -shifts (29), (31).

It completes the proof of the invariance of YA under the discussed six bosonic and three fermionic gauge symmetries.

5. Conclusion

Symmetries of tensionless superstring and superbrane with extra supersymmetry were studied. It was shown that the k-symmetry enhancement is accompanied by the appearance of bosonic gauge symmetries including the Weyl transformation and the local Abelian translations of the spacetime and TCC coordinates. In the case of D = 4N = 1 supersymmetry the translations are presented by the local vectors and bivectors constructed from the components of an auxiliary spinor field parametrizing the momentum density of the brane. In the high dimensional spaces gauge multivector translations of TCC coordinates will appear. Due to these gauge symmetries the original brane's coordinates are defined modulo these gauge translations resulting in the appearance of new composed coordinates encoding the physical degrees of freedom contained in the original coordinates. The new variables are unified in the components of a symplectic supertwistor realizing linear representation of the OSp(1, 2M) symmetry. We proved that this supertwistor is an invariant of the gauge translations and the enhanced k -symmetry. So, the linearly realizable symplectic supersymmetries describing massless higher spin gauge theories appear as a result of a fine tuning of the set of local and global symmetries of the Lagrangians of tensionless superstring and superbrane. The tensionless objects are connected with the high energy limit E > Mpianck of the string theory, where masses of all particles are negligible and a hidden large symmetry takes shape [26]. It hints that symplectic supertwistor describing tensionless strings and branes may be found relevant variable for the description of a symmetric phase of M-theory and quantum field string theory at the high energy scale.

Acknowledgements

A.Z. thanks Fysikum at the Stockholm University for the kind hospitality and Ingemar Bengtsson for the useful discussion. The work was partially supported by the grant of the Royal Swedish Academy of Sciences and Ukrainian SFFR project 02.07/276.

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