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Twelve-dimensional supersymmetric gauge theory as the large N limit1
22 April 1999
Physics Letters B 452 Ž1999. 265–273
Twelve-dimensional supersymmetric gauge theory as the large N limit 1 Hitoshi Nishino
2
Department of Physics, UniÕersity of Maryland at College Park, College Park, MD 20742-4111, USA Received 27 January 1999 Editor: M. Cveticˇ
Abstract Starting with the ordinary ten-dimensional supersymmetric Yang-Mills theory for the gauge group UŽ N ., we obtain a twelve-dimensional supersymmetric gauge theory as the large N limit. The two symplectic canonical coordinates parametrizing the unitary N = N matrices for UŽ N . are identified with the extra coordinates in twelve dimensions in the N ™ ` limit. Applying further a strongrweak duality, we get the ‘decompactified’ twelve-dimensional theory. The resulting twelve-dimensional theory has peculiar gauge symmetry which is compatible also with supersymmetry. We also establish a corresponding new superspace formulation with the extra coordinates. By performing a dimensional reduction from twelve dimensions directly into three dimensions, we see that the Poisson bracket terms which are needed for identification with supermembrane action arises naturally. This result indicates an universal duality mechanism that the ’t Hooft limit of an arbitrary supersymmetric theory promotes the original supersymmetric theory in Ž D y 1,1. dimensions into a theory in Ž D,2. dimensions with an additional pair of space-time coordinates. This also indicates interesting dualities between supermembrane theory, type IIA superstring with D0-branes, and the recently-discovered twelve-dimensional supersymmetric theories. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction There has been accumulating evidence that UŽ N . matrix theory w1x in the large N limit w2x corresponds to the strongly coupled type IIA superstring w3x, and therefore to the M-theory w4,5x. For example, we can explicitly compute the supergraviton effective potential that agrees with the that of eleven-dimensional Ž11D. 3 supergravity theory w6x. 1
This work is supported in part by NSF grant a PHY-93-41926. E-Mail: [email protected]. 3 We use 11D or Ds11 for eleven dimensions, when the signatures are not crucial. To specify the signatures, we use Ž s,t . for s positive space and t negative time signatures. 2
In the N ™ ` limit of an UŽ N . Yang-Mills ŽYM. theory, the commutators for N = N matrices Z and W are replaced by the Poisson bracket Z,W 4P ' Ž Ep Z .Ž EqW . y Ž Eq Z .Ž EpW . where p and q are commuting variables in the large N limit, which were originally non-commuting variables, satisfying w q, p x s 2p irN. It has been also conjectured w7x that the large N limit of UŽ N . YM theories with 16 supercharges are related to certain supergravity solutions. These recent developments indicate an universal mechanism relating a supersymmetric theory in Ž D y 1,1. dimensions with Ž D,2. dimensions involving two extra coordinates p and q, replacing all the non-Abelian commutators by the Poisson brackets in Ž D,2. dimensions. Our formulation is in a sense
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 2 5 9 - 2
H. Nishinor Physics Letters B 452 (1999) 265–273
266
similar to the formulation in w8x for supermembrane action, in the sense that the Poisson bracket terms are identified with the large N limit of UŽ N . supersymmetric YM theory. In this paper, we study if the mechanism as above works even in the case of 10D supersymmetric UŽ N . YM theory in the N ™ ` limit, with all the nonAbelian group commutators replaced by the Poisson brackets with respect to the extra coordinates in 12D. Interestingly, we will find that the resulting 12D theory has a peculiar ‘gauge’ symmetry, which is also consistent with 12D supersymmetry, similar to a previous formulation of 12D supersymmetric YM theory w9x. Our formulation here is similar to that in w10 x, in which Sp Ž2,R . symmetry for the positionrmomentum is treated as a local symmetry embedded in SO Ž d,2.. However, our formulation is also different in the sense that we do not deal with point particle action with bi-local fields, and SpŽ2,R. is not gauged, either. After establishing the N ™ ` action, we perform a dimensional reduction of our 12D theory into 3D, in order to compare the result with the supermembrane action. As desired, we see that the resulting action agrees with the action obtained by N ™ ` limit of the D0-brane action in 1D w1x, and therefore, it coincides with the supermembrane action in 3D. 2. Canonical variables p and q and large N limit We first review the parametrization of UŽ N . N = N matrices in terms of canonical variables p and q w11,12,1x, and its effect on the commutators in the large N limit w2x. Any N = N complex matrix Z can be expanded in terms of two unitary N = N matrices U and V, satisfying UV s e 2 p i r N VU , Ž 2.1a . 2 p i U s e i p , V s e i q , w q, p x s , Ž 2.1b . N with so-called canonical variables p and q, as U N sI ,
V N sI ,
Ny1
Zs
Ny1 m
z m nU V s
Ý m, ns0
zm n s
n
1 N
Ý
`
z Ž p,q . '
zm n e
e
,
Ž 2.2 .
zm n e im pe inq ,
Ý
Ž 2.3a .
m, ns0
dp
p
zm n s
dq
p
yi m pyi n q
Hyp 2p Hyp 2p z Ž p,q . e
Accordingly, we have p
trZ ™ N
dp
p
w Z,W x ™
2p i N
.
Ž 2.3b .
5
dq
Hyp 2p Hyp 2p z Ž p,q . , Ž Z , q W , p y Z , pW , q . .
Ž 2.4a . Ž 2.4b .
Eq. Ž2.4b. implies that an UŽ N . commutator can become a Poisson bracket in the large N limit. Note that due to the symplectic feature of these two coordinates, it is natural to have the indefinite signature Žq,y . in the Ž p,q . space, and it is convenient to use the coordinates Ž xq, xy . ' Ž Ž p q q . r'2 , Ž p y q . r'2 . . In principle, we can apply this aspect of the large N limit for UŽ N . to any YM theory in any dimensions. For example, after appropriate rescalings by the powers of N, the UŽ N . YM field strength Fmn ' An , m y Am , n q ig Am , An ,
Ž 2.5 .
in Ž D y 1,1. dimensions with the coordinates
4
im p inq
m , ns0
tr Ž Uym ZVyn . .
Eq. Ž2.1. implies that the eigenvalues of p and q can be chosen to be yp ,y p Ž N y 1.rN,y p Ž N y 2.r N, PPP ,y p r N, 0,p r N , 2p rN, PPP , Ž N y 1.pr N. 4 In the large N limit w2x, p and q become mutually commuting c-numbers, as Ž2.1b. shows. Moreover, the eigenvalues of p and q become continuous taking all the real values in yp F p - p ,y p F q p , behaving like a pair of coordinates for a phase space w11,12,1x. In such a limit, the Z in Ž2.2. becomes just an ordinary Fourier expansion in terms of p and q, which we call z Ž p,q .:
We use this convention instead of 0, p r N, 2p r N, PPP ,2 Ž N y1.p r N in w1,5x for a later purpose of decompactification. 5 In this paper we use the symbol like , p to denote the derivative E rE p. We avoid the usage of Em for E rE x m, because Am,q An ,y etc., are more compact than Ž Eq Am . Ž Ey An . , in equations like Ž2.6..
H. Nishinor Physics Letters B 452 (1999) 265–273
Ž x 0 , x 1 , PPP , x Dy1 . can be promoted into the field strength Fmn ' An , m y Am , n q g Ž Am ,q An ,yy Am ,y An ,q . ,
Ž 2.6 . 0
1
Dy1
q
in the Ž D,2. dimensions Ž x , x , PPP , x , x , xy .. The constant g in Ž2.5. is the usual YM coupling constant. The last term in Ž2.6. is nothing else than the Poisson bracket replacing the commutator when N is finite. Even though Ž2.6. seems rather unusual with the last term, we will see shortly how this makes sense as a field strength, transforming properly under our gauge transformation. The metric of the resulting Ž D,2. dimensions is Ž hmn . s diag. Žy,q,q, PPP ,q,y . with an additional pair of space-time coordinates, equivalent to the symplectic variables p and q. The geometrical meaning of this process is clear, from the viewpoint that the two variables p and q can be interpreted as coordinates of particles in quantum mechanics w1x. By adding two additional coordinates to the usual base manifold, the total space-time dimensions become now two dimensions higher than the original one. 6 As in Ž2.3., the range for the new variables p, q is to be wyp ,p .. This restriction of coordinates implies nothing else than ‘compactification’ on S 1 m S 1 of the extra dimensions. In order to get ‘decompactified’ system with the extra coordinates free of such a restriction, we need to adopt additional limiting procedure based on strongrweak duality. This can be done as follows. Consider the rescaling of the extra coordinates x Dq 1 and x Dq 2 by y Dq 1 s Rx Dq 1 , y Dq 2 s Rx Dq 2 Ž yp R F y Dq 1 - p R , yp R F y Dq 2 - p R . , Ž 2.7 . and take the limit R ™ ` , g ™ 0 with g˜ ' R 2 g fixed. Ž 2.8 . Now the field strength Ž2.6. stays formally the same, except that the derivatives ," are now with respect to the new coordinates y " with the ranges y` -
y Dq 1 - `, y` - y Dq 2 - `, and that the coupling constant g is now replaced by g. ˜ In other words, by taking this particular limit, we can realize the ‘decompactification’ from ŽMinkowski.10 m S 1 m S 1 into ŽMinkowski.12 . In the next section, we apply this prescription to the usual 10D supersymmetric YM theory for the gauge group UŽ N ., to get the decompactified 12D theory. Accordingly, all the field strengths used from now on are understood to be in terms of the new decompactified coordinates y` y Dq 1 - `,y ` - y Dq 2 - `, and the rescaled coupling constant g, ˜ even though we use their original symbols like x Dq 1, x Dq 2 and g, in order to simplify the notation.
3. Supersymmetric gauge theory in 12D with peculiar gauge symmetry We first summarize our result on our 12D supersymmetric gauge theory after the large N limit and our decompactification limit, which is very similar to w9x, but with a peculiar gauge symmetry arising from the N ™ ` limit. As was mentioned, we need two time directions for promoting 10D supersymmetric YM theory to 12D, due to the two symplectic variables. Our notation is the same as the component formulation in w13x, i.e., our metric is Ž hmn . s diag. Žy,q, PPP ,q,q,y ., where we use the indices m , n , PPP s 0, 1, PPP , 9, 11, 12 for the 12D coordinates, Accordingly, our Clifford algebra is gm ,gn 4 s q2hmn , with e 012 PPP 9 11 12 s q1, and g 13 ' g 0 g 1 PPP g 9g 11 g 12 . We use null-vectors w9x defined by
ž ž ž ž
The geometrical significance and consistency with supersymmetry will be more elucidated, when we reformulate in superspace in a later section.
1
Ž n m . s 0,0, PPP ,0,q Ž nm . s 0,0, PPP ,0,q
'2 1
'2
Ž m m . s 0,0, PPP ,0,q Ž mm . s 0,0, PPP ,0,q
6
267
1
'2 1
'2
,y ,q
/ ' / ' / ' /
,q ,y
1
'2 1
2
1
2
1
2
,
, , .
Ž 3.1 .
We also use "-indices w14,15x, for the two extra dimensions in 12D: V "' 2y1 r2 Ž VŽ11. " VŽ12. ..
H. Nishinor Physics Letters B 452 (1999) 265–273
268
It then follows that nqs mqs q1, nys mys 0, and therefore n m nm s m m mm s 0 , m m nm s mqn qs my nys q1 .
Ž 3.2 .
The P ≠ , P x are our projection operators for the space of extra dimensions, satisfying the ortho-normality conditions w14x: P ≠ ' 12 nu mu s 12 gq gy,
P x ' 12 mu nu s 12 gy gq,
P≠ Px s Px P≠ s 0 ,
P ≠ P ≠ s qP ≠ ,
P x P x s qP x ,
coordinates between the 10D ones and the extra ones. The "-derivative terms in Ž3.6. are identified with the Poisson brackets with respect to the extra coordinates " as a reminiscent of the non-Abelian commutators for the adjoint representation of UŽ N . in 10D. Our supersymmetry transformation rule is similar to w16x with a slight difference:
dQ Am s Ž e P ≠ gm nu l . s Ž eg i nu l . dmi ,
P ≠ q P x s qI ,
Ž 3.3 .
dQ l s q 14 P x g m P x g n P x e Fmn s 14 P x g i je Fi j ,
with mu ' m mgm and nu ' n mgm . The symmetry of the g-matrices are such as
u . a b˙ s y Ž mu . ba ˙ , Žm ˙ , Ž nu . ab˙ s y Ž nu . ba
Ž P ≠ . a b s y Ž P x . ba ,
Ž 3.4 .
where undotted spinorial indices a , b , PPP s 1, 2 are for the negative chiral components, while ˙ 2˙ are for positive chirality. 7 a˙ , b˙ , PPP s 1, The field content for our 12D theory is Ž Am , l., where l is a Majorana–Weyl spinor satisfying g 13 l s yl. Our total action in 12D, obtained by the prescription in the last section, is now 2
I ' d 12 x y 14 Ž Fmn . q 12 Fmn F m r nn mr
H
2
q 12 Ž Fmn m m nn . q l P ≠ g m nu Dm l
ž
/
,
Ž 3.5 .
where our field strength and ‘covariant’ derivative are defined by Fmn ' An , m y Am , n q g Ž Am ,q An ,yy Am ,y An ,q . , Dm l ' l , m q g Ž Am ,q l ,yy Am ,y l ,q . .
Ž 3.6 .
Since these quantities are understood as the large N limit, there is no ‘hidden’ index like adjoint indices in 10D, and there is no need to take the trace in Ž3.5.. As has been mentioned at the end of Section 2, the ranges of the extra coordinates are y` - x 11 `, y` - x 12 - `, after our decompactification limit. Therefore there is no difference about the range of
Ž 3.7b . where the indices i, j, PPP s 0, 1, PPP , 9 are for the purely 10D coordinates. In Ž3.7a., we have dQ A "s 0 due to the property of nu and P ≠ . Our 12D system has a peculiar gauge symmetry which is understood as the N ™ ` reminiscent of the original 10D system. They are dictated with the infinitesimal parameter L by
dG Am s qL , m q g Ž Am ,q L ,yy Am ,y L ,q . , dG l s yg Ž L ,q l ,yy L ,y l ,q . .
We follow Refs. w9,14x for the dottedness of indices, which is opposite to the usual convention.
Ž 3.8 .
Clearly, the terms with " are the Poisson brackets with respect to our extra coordinates ", as the N ™ ` limit of the usual UŽ N . commutators. Accordingly, Fmn and Dm l transform
dG Fmn s yg Ž L ,q Fmn ,yy L ,y Fmn ,q . , dG Ž Dm l . s yg L ,q Ž Dm l . ,yy L ,y Ž Dm l . ,q .
Ž 3.9 . These are nothing but the N ™ ` limit of the the U Ž N . commutators in d G Fm n s yg w L , Fm n x, dG Ž Dm l . s yg w L, Dm lx for a finite N. Relevantly, we can confirm the closure of two gauge transformations Ž3.8. on Am :
dG1 , dG2 Am s gDm Ž L1,q L2,y y L1,y L2,q . ,
7
Ž 3.7a .
Ž 3.10 .
where Dm contains again the Poisson bracket terms, as in Ž3.6.. From these features, there seems to be no fundamental problem to interpret Fmn and Dm as
H. Nishinor Physics Letters B 452 (1999) 265–273
‘field strength’ and ‘covariant derivatives’ in our peculiar 12D space-time. Our field equations for Am and l are Dj F i j q 2 g l ,q P ≠ g i nu l ,y s 0 ,
Ž 3.11 .
mu g m nu Dm l s 0 .
Ž 3.12 .
ž
/
Note that the index m in Ž3.12. effectively takes only 10D values, due to the property of nu and mu . We can also easily confirm the closure of two supersymmetries in 12D, which is similar to w9x. Our result is
dQ Ž e 1 . , dQ Ž e 2 . s dj q dV q da ,
Ž 3.13 .
where dj is for the usual leading translation term, while dV and da are extra symmetries with respective parameters j m, V and a :
dj Am s j nAm , n , dV Am ' V nm , m
j ' Ž e 1 P≠ g
mn
dj l s j nl , n , da l ' P ≠ a , P x e 2 . nn s y Ž e 2 P ≠ g
Ž 3.14 . mn
P x e 1 . nn ,
V ' y 12 Ž e 1 P ≠ g i j P x e 2 . Fi j s q 12 Ž e 2 P ≠ g i j P x e 1 . Fi j ,
a ' j m Dm l .
Ž 3.15 .
Our action Ž3.5. is of course invariant under these extra symmetries. As usual, the closure on l in Ž3.13. holds only up to the l-field Eq. Ž3.12.. Despite of the peculiar property of our gauge symmetry, we can also confirm the Bianchi identity ŽBI.: Dr Fst q Ds Ftr q Dt Frs ' 0 .
Ž 3.16 .
Here the covariant derivative Dr contains the Poisson bracket term with "-derivatives. Relevantly, the following arbitrary variations for Fmn and Dm l are useful:
d Fmn s Dm Ž d An . y Dn Ž d Am . , d Ž Dm l . s Dm Ž dl . qg
Ž d Am . ,q l ,yy Ž d Am . ,y l ,q
.
Ž 3.17 . Needless to say, all of these covariant derivatives contain the "-derivatives. Note also that Dm satisfies the Leibnitz rule: Dm Ž AB . s Ž Dm A . B q
269
A Ž Dm B . , enabling us to perform partial integrations under Hd 12 x. Using this with Ž3.17., it is now straightforward to obtain our field Eqs. Ž3.11. and Ž3.12., and also to confirm the invariance of our total action dQ I s 0 under our supersymmetry Ž3.7.. The transformation rule dQ A "s 0 in Ž3.7. poses no problem, due to the effective absence of A " in our action Ž3.5.. Even though the extra components F" i and Fqy are effectively absent from our action Ž3.5., the system is not reduced to just an infinite identical copies of supersymmetric YM theory in 10D, or a rewriting of the latter ‘in disguise’. This is due to the non-trivial Poisson bracket terms in Fi j which are the non-trivial reminiscent of the non-Abelian terms in the original 10D theory. One crucial question is whether or not our 12D theory can be Lorentz covariant. Even though we still lack a Lorentz invariant lagrangian yet, we emphasize as in w16,13x that all of our field equations can be made entirely Lorentz covariant, by expressing the null-vectors in terms of two scalars: nm ' 2 w, m , mm ' w˜ , m , satisfying w, mn s 0, w˜ , mn s 0, Ž w , m . 2 m s 0, Ž w˜ , m . s 0, w, m w˜ , s q1 w16,13x. From this viewpoint, our system has another non-trivial feature, even for Lorentz covariance. The possibility of a Lorentz invariant lagrangian is now under study.
4. Superspace formulation in 12D Once we have understood the component formulation of our 12D system, the next natural task is to reformulate this system in superspace. This is done mainly by studying the geometrical significance of supercovariant derivatives, and satisfaction of BIs for superfield strength in superspace. Our superspace coordinates are Ž Z A . s Ž x a , u a . , where we use the superspace index convention: A s Ž a, a ., B s Ž b, b ., PPP , with the bosonic coordinates a, b, PPP s 0, 1, PPP , 9,q,y and the fermionic coordinates a , b , PPP s 1, 2, PPP , 32. As usual, our starting point is the super-gauge covariant derivative, defined in our case by
=A ' DA q g Ž A A ,q Dyy A A ,y Dq . , DA ' EAME M
Ž 4.1 .
where is the usual superspace covariant derivative, while A A,"' D " A A ' E " A A . We regard
H. Nishinor Physics Letters B 452 (1999) 265–273
270
the last two terms in Ž4.1. as a ‘gauge connection’ term, generating the Poisson bracket terms in the superfield strength, as will be seen. Note that only D " instead of = " are needed for these terms. Now our superfield strength FA B is defined by the commutator as
w =A ,=B 4 s g Ž FA B ,q Dyy FA B ,y Dq . ,
Ž 4.2a .
FA B ' D w A A B . y TACB A C q g Ž A A ,q A B ,yy A A ,y A B ,q . ,
Ž g i nu . Ž a b < Ž g i nu l. < g . ' 0 , Ž 4.2b .
where the last Poisson bracket terms are similar to the component case Ž2.6., as the reminiscent of the commutator of UŽ N . generators in the large N limit. From the commutator defining FA B , we see the first signal of the geometric significance of our formulation. Even though it may be expected in a certain sense, it is remarkable that our new superfield strength FA B satisfies the BIs
=w A FB C . y Tw DA B < FD
Ž 4.3 .
=w A TBDC . y Tw EA B < TED
Ž 4.4 . TACB ,
where Ž4.4. is the BI for the torsion superfield and =A acts on FB C with the Poisson bracket terms as in Ž4.1.. These BIs are confirmed by the Jacobi identity ww=w A ,=B 4 ,=C .4 ' 0. To specify the components in Ž4.3., we call it Ž ABC .-type BI. Our next task is to satisfy all the components of the BI Ž4.3. and Ž4.4.. As usual in superspace formulations, we postulate a set of constraints: Tacb s Ž g c d . ab n d q Ž P ≠ x . a b n c s Ž P≠ g c dPx . a b n c ,
final superfield Eqs. Ž4.7. and Ž4.8. below. The difference is that the fields l or FA B are not subject to any extra constraints, such as Fa b n b s 0 w9,17x. The satisfaction of BI Ž4.4. is rather trivial, due to c the non-vanishing component of Tab in Ž4.5a.. The confirmation of the BI Ž4.3. is as easy as the other 12D cases w9,17x, up to some points peculiar to this system. First, the Ž abg .-type BI is proportional to
Ž 4.5a .
Fa b s y Ž P ≠ g b nu l . a q n b xa ,
Ž 4.5b .
=a lb s y 14 Ž P ≠ g c P ≠ g d P ≠ . ab Fc d ,
Ž 4.5c .
=a xb s q 12 Ž P ≠ g a nu . ab Fa b m b ,
Ž 4.5d .
=a Fb c s q Ž P ≠ g w b < nu= < c x l . a y nw b=cx xa .
Ž 4.5e .
There is similarity as well as difference between this superspace formulation and that in w9,17x or that in w14x. First, Ž4.5a. is exactly the same as that in w14x, in particular with the P ≠ x -term. Another similarly is that the x-field is a kind of auxiliary field, needed for the Ž ab c .-type BI, but it disappears from the
Ž 4.6 .
confirmed by 12D g-matrix algebra as in w9,13x. The Ž ab c .-type BI is easily shown to be satisfied due to Ž4.5c., while Ž a bc .-type BI gives Ž4.5e.. At this stage, we can get the l-superfield equation as =Ž a Ž=b . l b . y =a ,=b 4l b ' 0. This l-superfield equation in turn gives the F-superfield equation ag by taking its spinorial derivative like Ž g a . =g Ž la field equation. s 0, both in agreement with Ž3.11. and Ž3.12.. In the present notation they are b
=j F i j q 2 g l a,q Ž g i nu . a lb ,ys 0 , b
Ž mu g a nu . a=a lb s 0 .
Ž 4.7 . Ž 4.8 .
As has been already mentioned, even though the extra components F" i and Fqy are absent in Ž4.7., the Poisson bracket terms with extra derivatives ," in these superfield equations differentiate our 12D system from merely a rewriting of 10 supersymmetric theory, or the latter just in disguise. It is interesting that our newly-defined superfield strength FA B reveals so much geometrical significance and consistency with supersymmetry, quite parallel to conventional superspace formulations. This already suggests much deeper physical and geometrical significance of the incorporation of the symplectic canonical variables p, q as a part of the space-time coordinates, forming the total space-time with two time coordinates.
5. Dimensional reduction into 3D As an important test of our 12D theory, we perform a dimensional reduction into 3D, and see if the resulting action coincides with the N ™ ` limit of D0-brane action in 1D w1x. This is because the the two canonical symplectic variables p, q parametriz-
H. Nishinor Physics Letters B 452 (1999) 265–273
ing the unitary N = N matrices in 1D form additional two extra coordinates, promoting it to a 3D theory, compatible with the action Žhamiltonian. w11x of supermembrane w18x. Our dimensional reduction prescription is the usual one, namely we require all the fields to be independent of the internal 9D coordinates x 1 , x 2 , PPP , x 9 , so that the space-integrals over these coordinates in the action become an over-all trivial factor. Only in this section we use hats on fields and on indices in 12D in order to distinguish them from those in 3D. In this notation, our 12D lagrangian Ž3.5. is rewritten as 2
ˆ
I s d 12 xˆ y 14 Fˆiˆˆj q lˆ P ≠ˆ gˆ i nˆu Dˆ iˆlˆ
ž /
H
ž
/
,
Ž 5.1 .
where i,ˆ jˆ, PPP s 0, 1, PPP , 9 are 10D coordinate ˆˆ are indices. Other components in Fˆmn ˆ ˆ such as Fiq effectively absent from Ž5.1., due to the second and third terms in Ž3.5.. All the 12D fields are dimensionally reduced to 3D with the coordinates Ž x 0 , xq, xy . by the rules, such as
°Fˆ s g Ž X X y X X . , Fˆ s~ ¢Fˆ s X q g Ž A X y A i ,q
ij
iˆˆj
j ,y
i ,0
0i
i ,y
0,q
i ,y
0
žl/
,
~gˆ
i
i
0
0
i 0,y X ,q
. ' D0 X i ,
mt 3 ,
H
q'2 g lG
i
q'2 Ž lG 0 D 0 l . .
ž
I32 0
nˆu s I32 m tq,
0 , 0
/
Ž 5.3 .
This lagrangian is still in terms of 10D spinor l, which is to be further reduced to 16 component spinors to fit the SO Ž9. symmetry we need to compare with the result in w1x. This can be done by the SO Ž9. g-matrix representations w1,5x:
ls
žu/
G 0s
0
ž
,
0 I16
l s Ž 0, u T . , yI16 , 0
/
G is
ž
0
gi
gi
0
/
,
Ž 5.4 .
yielding the lagrangian 2
2
i i q'2 g u Tg i Ž X ,q u ,yy X ,y u ,q .
q'2 Ž u T D 0 u . .
Ž 5.5 .
Accordingly, we can perform the similar dimensional reduction to our supersymmetry transformation Ž3.7., to get
dQ A 0 s '2 Ž e Tu . ,
dQ Aqs dQ Ays 0 ,
¢ Pˆ≠ s
2
Ž X ,qi l ,yy X ,yi l ,q .
dQ X i s y'2 Ž e Tg iu . ,
3
sG mt , gˆqs I32 m tq , gˆys I32 m ty ,
gˆ iˆs
2
i j i j I s d 3 x q 12 Ž D 0 X i . y 14 g 2 Ž X ,q X ,y y X ,y X ,q .
H
lˆ s Ž 0, l . Ž I32 m t 1 . s Ž l ,0 . , Ž 5.2b .
°gˆ s G
Ž5.2., l is a 10D Majorana–Weyl spinor with maximally 32 components, and t 1 , t 2 , t 3 are the usual 2 = 2 Pauli matrices, and t "' Ž t 1 " it 2 . r2. Applying the useful relations Ž5.2. to the action Ž5.1., and integrating over the internal 9D coordinates x 1 , PPP , x 9 , we get
i j i j I s d 3 x q 12 Ž D 0 X i . y 14 g 2 Ž X ,q X ,y y X ,y X ,q .
j ,q
Ž 5.2a . lˆ s
271
dQ u s y 12 g ie D 0 X i q
0 Pˆx s 0
ž
m ˆu s I32 m ty,
0 I32 ,
/
Ž 5.2c . Ž 5.2d .
where i, j, PPP s 1, PPP , 9 in this section are for the spatial 9D, as those used in w1x. Therefore Ž G 0 , G i . in Ž5.2c. realize the Clifford algebra for 10D. In
1 4
i j i j gg i je Ž X ,q X ,y y X ,y X ,q ..
Ž 5.6 . Our results Ž5.5. and Ž5.6. agree with the 3D result in w11x after the large N limit, up to non-essential scaling factors such as '2 . We have thus seen that our 12D supersymmetric YM theory directly gives rise to the 3D theory w11x corresponding to the supermembrane theory w18x after taking the large N limit in the D0-brane action.
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6. Concluding remarks In this paper we have presented a 12D supersymmetric gauge theory, with a very peculiar gauge symmetry associated with Poisson bracket as a reminiscent of the commutators in non-Abelian generators in the original 10D theory. The Poisson bracket arises in the ’t Hooft N ™ ` limit for UŽ N ., whose p, q variables are now regarded as two extra coordinates in the total 12D. The extra coordinates have the ranges yp F p - p ,y p F q - p , implying a compactification on S 1 m S 1 within the total 12D. These extra dimensions are further decompactified by the strongrweak duality by rescaling the coordinates by x 11 ' Rp, x 12 ' Rq, and taking the limit R ™ `, g ™ 0 with R 2 g fixed. Subsequently, we have also studied the geometrical significance in superspace of our superfield strength with the Poisson brackets, and found that a superspace formulation is equally possible like the conventional supersymmetric YM theory. This superspace formulation elucidates the geometrical significance of our theory, in terms of field strength superfield defined by commutators between super-gauge covariant derivatives, with our peculiar ‘gauge’ symmetry. We have also studied the dimensional reduction of our 12D theory into 3D, that yields the desirable Poisson brackets in 3D, corresponding to the action for supermembrane theory w18,11x. In a recent development in the duality between Anti-de Sitter ŽAdS. and conformal field theory w19x, it is conjectured that the N s 4, UŽ N . supersymmetric YM theory in 4D is dual to type IIB theory on ŽAdS.5 m S 5 in the large N limit. It was pointed out w19x that the group SO Ž2,4. = SO Ž6. in the N s 4 supersymmetric YM theory suggests a 12D realization with two time coordinates. In our present paper, we have given a first explicit example, in which the direct connection between the supermembrane theory and 10D supersymmetric YM is much more natural than before. Since the N s 4 supersymmetric YM in 4D has the 10D origin which is promoted to be 12D in the large N limit, we have seen another duality link between superstring theory w3x, N s 4 supersymmetric YM in 4D, and supermembrane theory w18x Õia 12D supersymmetric YM theory w9x. We have observed similarity as well as difference between our new superspace formulation and that in
w9,17x. The most conspicuous difference is the nonvanishing Poisson bracket terms in FA B making the system non-trivial. We have also clarified the geometrical significance of our peculiar field strength superfield in terms of superspace language. We also stress that our present paper gives a clearer link between supersymmetric YM theory in 12D w9,17x and M-theory w4,5x. In a usual non-supersymmetric theory, the loss of Lorentz covariance makes the system completely ambiguous, because we can always put null-vectors anywhere by hand in any equation in the system. However, as was also stressed in w13,17x, this is no longer the case with supersymmetric theories, in which the coefficients in field equations are tightly fixed by supercovariance. The incorporation of symplectic variables as target coordinates makes stronger sense in supersymmetric theories. Our formulation has been also strongly motivated by the recent development in the matrix theory approach w1x to M-theory w4,5x. In our formulation, due to the absence of supergravity, the loss of local Lorentz covariance is not crucial. In this connection, we mention that only global Lorentz covariance plays an important role in the study of non-perturbative aspects of M-theory. In fact, in the conjecture by Maldacena w19x about the duality between the large N limit of D s 4, N s 4 supersymmetric Yang-Mills and type IIB superstring compactified on Ž AdS .5 m S 5, the global isometry group is SO Ž4,2. = SO Ž6., being further promoted to SO Ž10,2. w19,20x, indicates the existence of 12D supersymmetric theory. Note that the SO Ž10,2. symmetry is global, and therefore the loss of local Lorentz covariance in 12D supergravityrsupersymmetry formulations as in w13x is not crucial in such a formulation of M-theory w4,5x. Our result also provides a new link between the F-theory w21x in 12D and the M-theory w4x in 11D, that has not been explicitly presented before. 8 Our result also clarifies general dualities connecting a supersymmetric YM theory in Ž D y 1,1. dimensions
8
In Ref. w22x, an idea of deleting even the world-line for the D0-brane was presented, but with no reference to the 12D YM field strength defined with Poisson bracket in terms of extra coordinates.
H. Nishinor Physics Letters B 452 (1999) 265–273
with another supersymmetric theory formulated in Ž D,2. dimensions under the large N limit. It also gives a new connection between the conventional supersymmetryrsupergravity theories in D F 11 with higher-dimensional theories, such as the 12D supergravity w14,15,13x, or more supersymmetric YM theories in D G 12 w17x. In particular, the evidence for the importance of the results in w9,17x is now rapidly mounting.
Acknowledgements We are indebted to I. Bars, S.J. Gates, Jr., J.H. Schwarz, and W. Siegel for valuable suggestions.
References w1x T. Banks, W. Fischler, S.H. Shenker, L. Susskind, Phys. Rev. D 55 Ž1997. 5112. w2x G. ’t Hooft, Nucl. Phys. B 72 Ž1974. 461. w3x M. Green, J.H. Schwarz, E. Witten, Superstring Theory, vols. IrII, Cambridge University Press, 1987. w4x P.K. Townsend, M-Theory from its Superalgebra, hepthr9712004, and references therein.
273
w5x A. Bilal, MŽatrix. Theory: A Pedagogical Introduction, hepthr9710136; J.H. Schwarz, Introduction to M Theory and ADSrCFT Duality, hep-thr9812037; and references in them. w6x E. Cremmer, B. Julia, N. Scherk, Phys. Lett. B 76 Ž1978. 409; E. Cremmer, B. Julia, Phys. Lett. B 80 Ž1978. 48; Nucl. Phys. B 159 Ž1979. 141. w7x N. Itzhaki, J. Sonnenschein, S. Yankielowicz, Phys. Rev. D 58 Ž1998. 046004. w8x I. Bars, Phys. Lett. B 245 Ž1990. 35. w9x H. Nishino, E. Sezgin, Phys. Lett. B 388 Ž1996. 569. w10x I. Bars, C. Deliduman, O. Andreev, Phys. Rev. D 58 Ž1998. 066004. w11x B. de Wit, M. Luscher, H. Nicolai, Nucl. Phys. B 305 Ž1988. ¨ 545. w12x D. Fairlie, P. Fletcher, C. Zachos, Jour. Math. Phys. 31 Ž1990. 1088; J. Hoppe, Int. Jour. Mod. Phys. A 4 Ž1989. 5235. w13x H. Nishino, Supergravity Theories in stD G 12, hepthr9807199, to appear in Nucl. Phys. B 542, March, 1999. w14x H. Nishino, Phys. Lett. B 428 Ž1998. 85. w15x H. Nishino, Phys. Lett. B 437 Ž1998. 303. w16x H. Nishino, Phys. Lett. B 426 Ž1998. 64. w17x H. Nishino, Nucl. Phys. B 523 Ž1998. 450. w18x E. Bergshoeff, E. Sezgin, P.K. Townsend, Phys. Lett. B 189 Ž1987. 75; Ann. of Phys. 185 Ž1988. 330. w19x J.M. Maldacena, Adv. Theor. Math. Phys. 2 Ž1998. 231. w20x I. Bars, Hidden Symmetries, stAdSDmS n , and the Lifting of One-Time Physics to Two-Time Physics, hep-thr9810025. w21x C. Vafa, Nucl. Phys. B 469 Ž1996. 403. w22x V. Periwal, Phys. Rev. D 55 Ž1997. 1711.
Physics Letters B 452 Ž1999. 265–273
Twelve-dimensional supersymmetric gauge theory as the large N limit 1 Hitoshi Nishino
2
Department of Physics, UniÕersity of Maryland at College Park, College Park, MD 20742-4111, USA Received 27 January 1999 Editor: M. Cveticˇ
Abstract Starting with the ordinary ten-dimensional supersymmetric Yang-Mills theory for the gauge group UŽ N ., we obtain a twelve-dimensional supersymmetric gauge theory as the large N limit. The two symplectic canonical coordinates parametrizing the unitary N = N matrices for UŽ N . are identified with the extra coordinates in twelve dimensions in the N ™ ` limit. Applying further a strongrweak duality, we get the ‘decompactified’ twelve-dimensional theory. The resulting twelve-dimensional theory has peculiar gauge symmetry which is compatible also with supersymmetry. We also establish a corresponding new superspace formulation with the extra coordinates. By performing a dimensional reduction from twelve dimensions directly into three dimensions, we see that the Poisson bracket terms which are needed for identification with supermembrane action arises naturally. This result indicates an universal duality mechanism that the ’t Hooft limit of an arbitrary supersymmetric theory promotes the original supersymmetric theory in Ž D y 1,1. dimensions into a theory in Ž D,2. dimensions with an additional pair of space-time coordinates. This also indicates interesting dualities between supermembrane theory, type IIA superstring with D0-branes, and the recently-discovered twelve-dimensional supersymmetric theories. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction There has been accumulating evidence that UŽ N . matrix theory w1x in the large N limit w2x corresponds to the strongly coupled type IIA superstring w3x, and therefore to the M-theory w4,5x. For example, we can explicitly compute the supergraviton effective potential that agrees with the that of eleven-dimensional Ž11D. 3 supergravity theory w6x. 1
This work is supported in part by NSF grant a PHY-93-41926. E-Mail: [email protected]. 3 We use 11D or Ds11 for eleven dimensions, when the signatures are not crucial. To specify the signatures, we use Ž s,t . for s positive space and t negative time signatures. 2
In the N ™ ` limit of an UŽ N . Yang-Mills ŽYM. theory, the commutators for N = N matrices Z and W are replaced by the Poisson bracket Z,W 4P ' Ž Ep Z .Ž EqW . y Ž Eq Z .Ž EpW . where p and q are commuting variables in the large N limit, which were originally non-commuting variables, satisfying w q, p x s 2p irN. It has been also conjectured w7x that the large N limit of UŽ N . YM theories with 16 supercharges are related to certain supergravity solutions. These recent developments indicate an universal mechanism relating a supersymmetric theory in Ž D y 1,1. dimensions with Ž D,2. dimensions involving two extra coordinates p and q, replacing all the non-Abelian commutators by the Poisson brackets in Ž D,2. dimensions. Our formulation is in a sense
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 0 2 5 9 - 2
H. Nishinor Physics Letters B 452 (1999) 265–273
266
similar to the formulation in w8x for supermembrane action, in the sense that the Poisson bracket terms are identified with the large N limit of UŽ N . supersymmetric YM theory. In this paper, we study if the mechanism as above works even in the case of 10D supersymmetric UŽ N . YM theory in the N ™ ` limit, with all the nonAbelian group commutators replaced by the Poisson brackets with respect to the extra coordinates in 12D. Interestingly, we will find that the resulting 12D theory has a peculiar ‘gauge’ symmetry, which is also consistent with 12D supersymmetry, similar to a previous formulation of 12D supersymmetric YM theory w9x. Our formulation here is similar to that in w10 x, in which Sp Ž2,R . symmetry for the positionrmomentum is treated as a local symmetry embedded in SO Ž d,2.. However, our formulation is also different in the sense that we do not deal with point particle action with bi-local fields, and SpŽ2,R. is not gauged, either. After establishing the N ™ ` action, we perform a dimensional reduction of our 12D theory into 3D, in order to compare the result with the supermembrane action. As desired, we see that the resulting action agrees with the action obtained by N ™ ` limit of the D0-brane action in 1D w1x, and therefore, it coincides with the supermembrane action in 3D. 2. Canonical variables p and q and large N limit We first review the parametrization of UŽ N . N = N matrices in terms of canonical variables p and q w11,12,1x, and its effect on the commutators in the large N limit w2x. Any N = N complex matrix Z can be expanded in terms of two unitary N = N matrices U and V, satisfying UV s e 2 p i r N VU , Ž 2.1a . 2 p i U s e i p , V s e i q , w q, p x s , Ž 2.1b . N with so-called canonical variables p and q, as U N sI ,
V N sI ,
Ny1
Zs
Ny1 m
z m nU V s
Ý m, ns0
zm n s
n
1 N
Ý
`
z Ž p,q . '
zm n e
e
,
Ž 2.2 .
zm n e im pe inq ,
Ý
Ž 2.3a .
m, ns0
dp
p
zm n s
dq
p
yi m pyi n q
Hyp 2p Hyp 2p z Ž p,q . e
Accordingly, we have p
trZ ™ N
dp
p
w Z,W x ™
2p i N
.
Ž 2.3b .
5
dq
Hyp 2p Hyp 2p z Ž p,q . , Ž Z , q W , p y Z , pW , q . .
Ž 2.4a . Ž 2.4b .
Eq. Ž2.4b. implies that an UŽ N . commutator can become a Poisson bracket in the large N limit. Note that due to the symplectic feature of these two coordinates, it is natural to have the indefinite signature Žq,y . in the Ž p,q . space, and it is convenient to use the coordinates Ž xq, xy . ' Ž Ž p q q . r'2 , Ž p y q . r'2 . . In principle, we can apply this aspect of the large N limit for UŽ N . to any YM theory in any dimensions. For example, after appropriate rescalings by the powers of N, the UŽ N . YM field strength Fmn ' An , m y Am , n q ig Am , An ,
Ž 2.5 .
in Ž D y 1,1. dimensions with the coordinates
4
im p inq
m , ns0
tr Ž Uym ZVyn . .
Eq. Ž2.1. implies that the eigenvalues of p and q can be chosen to be yp ,y p Ž N y 1.rN,y p Ž N y 2.r N, PPP ,y p r N, 0,p r N , 2p rN, PPP , Ž N y 1.pr N. 4 In the large N limit w2x, p and q become mutually commuting c-numbers, as Ž2.1b. shows. Moreover, the eigenvalues of p and q become continuous taking all the real values in yp F p - p ,y p F q p , behaving like a pair of coordinates for a phase space w11,12,1x. In such a limit, the Z in Ž2.2. becomes just an ordinary Fourier expansion in terms of p and q, which we call z Ž p,q .:
We use this convention instead of 0, p r N, 2p r N, PPP ,2 Ž N y1.p r N in w1,5x for a later purpose of decompactification. 5 In this paper we use the symbol like , p to denote the derivative E rE p. We avoid the usage of Em for E rE x m, because Am,q An ,y etc., are more compact than Ž Eq Am . Ž Ey An . , in equations like Ž2.6..
H. Nishinor Physics Letters B 452 (1999) 265–273
Ž x 0 , x 1 , PPP , x Dy1 . can be promoted into the field strength Fmn ' An , m y Am , n q g Ž Am ,q An ,yy Am ,y An ,q . ,
Ž 2.6 . 0
1
Dy1
q
in the Ž D,2. dimensions Ž x , x , PPP , x , x , xy .. The constant g in Ž2.5. is the usual YM coupling constant. The last term in Ž2.6. is nothing else than the Poisson bracket replacing the commutator when N is finite. Even though Ž2.6. seems rather unusual with the last term, we will see shortly how this makes sense as a field strength, transforming properly under our gauge transformation. The metric of the resulting Ž D,2. dimensions is Ž hmn . s diag. Žy,q,q, PPP ,q,y . with an additional pair of space-time coordinates, equivalent to the symplectic variables p and q. The geometrical meaning of this process is clear, from the viewpoint that the two variables p and q can be interpreted as coordinates of particles in quantum mechanics w1x. By adding two additional coordinates to the usual base manifold, the total space-time dimensions become now two dimensions higher than the original one. 6 As in Ž2.3., the range for the new variables p, q is to be wyp ,p .. This restriction of coordinates implies nothing else than ‘compactification’ on S 1 m S 1 of the extra dimensions. In order to get ‘decompactified’ system with the extra coordinates free of such a restriction, we need to adopt additional limiting procedure based on strongrweak duality. This can be done as follows. Consider the rescaling of the extra coordinates x Dq 1 and x Dq 2 by y Dq 1 s Rx Dq 1 , y Dq 2 s Rx Dq 2 Ž yp R F y Dq 1 - p R , yp R F y Dq 2 - p R . , Ž 2.7 . and take the limit R ™ ` , g ™ 0 with g˜ ' R 2 g fixed. Ž 2.8 . Now the field strength Ž2.6. stays formally the same, except that the derivatives ," are now with respect to the new coordinates y " with the ranges y` -
y Dq 1 - `, y` - y Dq 2 - `, and that the coupling constant g is now replaced by g. ˜ In other words, by taking this particular limit, we can realize the ‘decompactification’ from ŽMinkowski.10 m S 1 m S 1 into ŽMinkowski.12 . In the next section, we apply this prescription to the usual 10D supersymmetric YM theory for the gauge group UŽ N ., to get the decompactified 12D theory. Accordingly, all the field strengths used from now on are understood to be in terms of the new decompactified coordinates y` y Dq 1 - `,y ` - y Dq 2 - `, and the rescaled coupling constant g, ˜ even though we use their original symbols like x Dq 1, x Dq 2 and g, in order to simplify the notation.
3. Supersymmetric gauge theory in 12D with peculiar gauge symmetry We first summarize our result on our 12D supersymmetric gauge theory after the large N limit and our decompactification limit, which is very similar to w9x, but with a peculiar gauge symmetry arising from the N ™ ` limit. As was mentioned, we need two time directions for promoting 10D supersymmetric YM theory to 12D, due to the two symplectic variables. Our notation is the same as the component formulation in w13x, i.e., our metric is Ž hmn . s diag. Žy,q, PPP ,q,q,y ., where we use the indices m , n , PPP s 0, 1, PPP , 9, 11, 12 for the 12D coordinates, Accordingly, our Clifford algebra is gm ,gn 4 s q2hmn , with e 012 PPP 9 11 12 s q1, and g 13 ' g 0 g 1 PPP g 9g 11 g 12 . We use null-vectors w9x defined by
ž ž ž ž
The geometrical significance and consistency with supersymmetry will be more elucidated, when we reformulate in superspace in a later section.
1
Ž n m . s 0,0, PPP ,0,q Ž nm . s 0,0, PPP ,0,q
'2 1
'2
Ž m m . s 0,0, PPP ,0,q Ž mm . s 0,0, PPP ,0,q
6
267
1
'2 1
'2
,y ,q
/ ' / ' / ' /
,q ,y
1
'2 1
2
1
2
1
2
,
, , .
Ž 3.1 .
We also use "-indices w14,15x, for the two extra dimensions in 12D: V "' 2y1 r2 Ž VŽ11. " VŽ12. ..
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It then follows that nqs mqs q1, nys mys 0, and therefore n m nm s m m mm s 0 , m m nm s mqn qs my nys q1 .
Ž 3.2 .
The P ≠ , P x are our projection operators for the space of extra dimensions, satisfying the ortho-normality conditions w14x: P ≠ ' 12 nu mu s 12 gq gy,
P x ' 12 mu nu s 12 gy gq,
P≠ Px s Px P≠ s 0 ,
P ≠ P ≠ s qP ≠ ,
P x P x s qP x ,
coordinates between the 10D ones and the extra ones. The "-derivative terms in Ž3.6. are identified with the Poisson brackets with respect to the extra coordinates " as a reminiscent of the non-Abelian commutators for the adjoint representation of UŽ N . in 10D. Our supersymmetry transformation rule is similar to w16x with a slight difference:
dQ Am s Ž e P ≠ gm nu l . s Ž eg i nu l . dmi ,
P ≠ q P x s qI ,
Ž 3.3 .
dQ l s q 14 P x g m P x g n P x e Fmn s 14 P x g i je Fi j ,
with mu ' m mgm and nu ' n mgm . The symmetry of the g-matrices are such as
u . a b˙ s y Ž mu . ba ˙ , Žm ˙ , Ž nu . ab˙ s y Ž nu . ba
Ž P ≠ . a b s y Ž P x . ba ,
Ž 3.4 .
where undotted spinorial indices a , b , PPP s 1, 2 are for the negative chiral components, while ˙ 2˙ are for positive chirality. 7 a˙ , b˙ , PPP s 1, The field content for our 12D theory is Ž Am , l., where l is a Majorana–Weyl spinor satisfying g 13 l s yl. Our total action in 12D, obtained by the prescription in the last section, is now 2
I ' d 12 x y 14 Ž Fmn . q 12 Fmn F m r nn mr
H
2
q 12 Ž Fmn m m nn . q l P ≠ g m nu Dm l
ž
/
,
Ž 3.5 .
where our field strength and ‘covariant’ derivative are defined by Fmn ' An , m y Am , n q g Ž Am ,q An ,yy Am ,y An ,q . , Dm l ' l , m q g Ž Am ,q l ,yy Am ,y l ,q . .
Ž 3.6 .
Since these quantities are understood as the large N limit, there is no ‘hidden’ index like adjoint indices in 10D, and there is no need to take the trace in Ž3.5.. As has been mentioned at the end of Section 2, the ranges of the extra coordinates are y` - x 11 `, y` - x 12 - `, after our decompactification limit. Therefore there is no difference about the range of
Ž 3.7b . where the indices i, j, PPP s 0, 1, PPP , 9 are for the purely 10D coordinates. In Ž3.7a., we have dQ A "s 0 due to the property of nu and P ≠ . Our 12D system has a peculiar gauge symmetry which is understood as the N ™ ` reminiscent of the original 10D system. They are dictated with the infinitesimal parameter L by
dG Am s qL , m q g Ž Am ,q L ,yy Am ,y L ,q . , dG l s yg Ž L ,q l ,yy L ,y l ,q . .
We follow Refs. w9,14x for the dottedness of indices, which is opposite to the usual convention.
Ž 3.8 .
Clearly, the terms with " are the Poisson brackets with respect to our extra coordinates ", as the N ™ ` limit of the usual UŽ N . commutators. Accordingly, Fmn and Dm l transform
dG Fmn s yg Ž L ,q Fmn ,yy L ,y Fmn ,q . , dG Ž Dm l . s yg L ,q Ž Dm l . ,yy L ,y Ž Dm l . ,q .
Ž 3.9 . These are nothing but the N ™ ` limit of the the U Ž N . commutators in d G Fm n s yg w L , Fm n x, dG Ž Dm l . s yg w L, Dm lx for a finite N. Relevantly, we can confirm the closure of two gauge transformations Ž3.8. on Am :
dG1 , dG2 Am s gDm Ž L1,q L2,y y L1,y L2,q . ,
7
Ž 3.7a .
Ž 3.10 .
where Dm contains again the Poisson bracket terms, as in Ž3.6.. From these features, there seems to be no fundamental problem to interpret Fmn and Dm as
H. Nishinor Physics Letters B 452 (1999) 265–273
‘field strength’ and ‘covariant derivatives’ in our peculiar 12D space-time. Our field equations for Am and l are Dj F i j q 2 g l ,q P ≠ g i nu l ,y s 0 ,
Ž 3.11 .
mu g m nu Dm l s 0 .
Ž 3.12 .
ž
/
Note that the index m in Ž3.12. effectively takes only 10D values, due to the property of nu and mu . We can also easily confirm the closure of two supersymmetries in 12D, which is similar to w9x. Our result is
dQ Ž e 1 . , dQ Ž e 2 . s dj q dV q da ,
Ž 3.13 .
where dj is for the usual leading translation term, while dV and da are extra symmetries with respective parameters j m, V and a :
dj Am s j nAm , n , dV Am ' V nm , m
j ' Ž e 1 P≠ g
mn
dj l s j nl , n , da l ' P ≠ a , P x e 2 . nn s y Ž e 2 P ≠ g
Ž 3.14 . mn
P x e 1 . nn ,
V ' y 12 Ž e 1 P ≠ g i j P x e 2 . Fi j s q 12 Ž e 2 P ≠ g i j P x e 1 . Fi j ,
a ' j m Dm l .
Ž 3.15 .
Our action Ž3.5. is of course invariant under these extra symmetries. As usual, the closure on l in Ž3.13. holds only up to the l-field Eq. Ž3.12.. Despite of the peculiar property of our gauge symmetry, we can also confirm the Bianchi identity ŽBI.: Dr Fst q Ds Ftr q Dt Frs ' 0 .
Ž 3.16 .
Here the covariant derivative Dr contains the Poisson bracket term with "-derivatives. Relevantly, the following arbitrary variations for Fmn and Dm l are useful:
d Fmn s Dm Ž d An . y Dn Ž d Am . , d Ž Dm l . s Dm Ž dl . qg
Ž d Am . ,q l ,yy Ž d Am . ,y l ,q
.
Ž 3.17 . Needless to say, all of these covariant derivatives contain the "-derivatives. Note also that Dm satisfies the Leibnitz rule: Dm Ž AB . s Ž Dm A . B q
269
A Ž Dm B . , enabling us to perform partial integrations under Hd 12 x. Using this with Ž3.17., it is now straightforward to obtain our field Eqs. Ž3.11. and Ž3.12., and also to confirm the invariance of our total action dQ I s 0 under our supersymmetry Ž3.7.. The transformation rule dQ A "s 0 in Ž3.7. poses no problem, due to the effective absence of A " in our action Ž3.5.. Even though the extra components F" i and Fqy are effectively absent from our action Ž3.5., the system is not reduced to just an infinite identical copies of supersymmetric YM theory in 10D, or a rewriting of the latter ‘in disguise’. This is due to the non-trivial Poisson bracket terms in Fi j which are the non-trivial reminiscent of the non-Abelian terms in the original 10D theory. One crucial question is whether or not our 12D theory can be Lorentz covariant. Even though we still lack a Lorentz invariant lagrangian yet, we emphasize as in w16,13x that all of our field equations can be made entirely Lorentz covariant, by expressing the null-vectors in terms of two scalars: nm ' 2 w, m , mm ' w˜ , m , satisfying w, mn s 0, w˜ , mn s 0, Ž w , m . 2 m s 0, Ž w˜ , m . s 0, w, m w˜ , s q1 w16,13x. From this viewpoint, our system has another non-trivial feature, even for Lorentz covariance. The possibility of a Lorentz invariant lagrangian is now under study.
4. Superspace formulation in 12D Once we have understood the component formulation of our 12D system, the next natural task is to reformulate this system in superspace. This is done mainly by studying the geometrical significance of supercovariant derivatives, and satisfaction of BIs for superfield strength in superspace. Our superspace coordinates are Ž Z A . s Ž x a , u a . , where we use the superspace index convention: A s Ž a, a ., B s Ž b, b ., PPP , with the bosonic coordinates a, b, PPP s 0, 1, PPP , 9,q,y and the fermionic coordinates a , b , PPP s 1, 2, PPP , 32. As usual, our starting point is the super-gauge covariant derivative, defined in our case by
=A ' DA q g Ž A A ,q Dyy A A ,y Dq . , DA ' EAME M
Ž 4.1 .
where is the usual superspace covariant derivative, while A A,"' D " A A ' E " A A . We regard
H. Nishinor Physics Letters B 452 (1999) 265–273
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the last two terms in Ž4.1. as a ‘gauge connection’ term, generating the Poisson bracket terms in the superfield strength, as will be seen. Note that only D " instead of = " are needed for these terms. Now our superfield strength FA B is defined by the commutator as
w =A ,=B 4 s g Ž FA B ,q Dyy FA B ,y Dq . ,
Ž 4.2a .
FA B ' D w A A B . y TACB A C q g Ž A A ,q A B ,yy A A ,y A B ,q . ,
Ž g i nu . Ž a b < Ž g i nu l. < g . ' 0 , Ž 4.2b .
where the last Poisson bracket terms are similar to the component case Ž2.6., as the reminiscent of the commutator of UŽ N . generators in the large N limit. From the commutator defining FA B , we see the first signal of the geometric significance of our formulation. Even though it may be expected in a certain sense, it is remarkable that our new superfield strength FA B satisfies the BIs
=w A FB C . y Tw DA B < FD
Ž 4.3 .
=w A TBDC . y Tw EA B < TED
Ž 4.4 . TACB ,
where Ž4.4. is the BI for the torsion superfield and =A acts on FB C with the Poisson bracket terms as in Ž4.1.. These BIs are confirmed by the Jacobi identity ww=w A ,=B 4 ,=C .4 ' 0. To specify the components in Ž4.3., we call it Ž ABC .-type BI. Our next task is to satisfy all the components of the BI Ž4.3. and Ž4.4.. As usual in superspace formulations, we postulate a set of constraints: Tacb s Ž g c d . ab n d q Ž P ≠ x . a b n c s Ž P≠ g c dPx . a b n c ,
final superfield Eqs. Ž4.7. and Ž4.8. below. The difference is that the fields l or FA B are not subject to any extra constraints, such as Fa b n b s 0 w9,17x. The satisfaction of BI Ž4.4. is rather trivial, due to c the non-vanishing component of Tab in Ž4.5a.. The confirmation of the BI Ž4.3. is as easy as the other 12D cases w9,17x, up to some points peculiar to this system. First, the Ž abg .-type BI is proportional to
Ž 4.5a .
Fa b s y Ž P ≠ g b nu l . a q n b xa ,
Ž 4.5b .
=a lb s y 14 Ž P ≠ g c P ≠ g d P ≠ . ab Fc d ,
Ž 4.5c .
=a xb s q 12 Ž P ≠ g a nu . ab Fa b m b ,
Ž 4.5d .
=a Fb c s q Ž P ≠ g w b < nu= < c x l . a y nw b=cx xa .
Ž 4.5e .
There is similarity as well as difference between this superspace formulation and that in w9,17x or that in w14x. First, Ž4.5a. is exactly the same as that in w14x, in particular with the P ≠ x -term. Another similarly is that the x-field is a kind of auxiliary field, needed for the Ž ab c .-type BI, but it disappears from the
Ž 4.6 .
confirmed by 12D g-matrix algebra as in w9,13x. The Ž ab c .-type BI is easily shown to be satisfied due to Ž4.5c., while Ž a bc .-type BI gives Ž4.5e.. At this stage, we can get the l-superfield equation as =Ž a Ž=b . l b . y =a ,=b 4l b ' 0. This l-superfield equation in turn gives the F-superfield equation ag by taking its spinorial derivative like Ž g a . =g Ž la field equation. s 0, both in agreement with Ž3.11. and Ž3.12.. In the present notation they are b
=j F i j q 2 g l a,q Ž g i nu . a lb ,ys 0 , b
Ž mu g a nu . a=a lb s 0 .
Ž 4.7 . Ž 4.8 .
As has been already mentioned, even though the extra components F" i and Fqy are absent in Ž4.7., the Poisson bracket terms with extra derivatives ," in these superfield equations differentiate our 12D system from merely a rewriting of 10 supersymmetric theory, or the latter just in disguise. It is interesting that our newly-defined superfield strength FA B reveals so much geometrical significance and consistency with supersymmetry, quite parallel to conventional superspace formulations. This already suggests much deeper physical and geometrical significance of the incorporation of the symplectic canonical variables p, q as a part of the space-time coordinates, forming the total space-time with two time coordinates.
5. Dimensional reduction into 3D As an important test of our 12D theory, we perform a dimensional reduction into 3D, and see if the resulting action coincides with the N ™ ` limit of D0-brane action in 1D w1x. This is because the the two canonical symplectic variables p, q parametriz-
H. Nishinor Physics Letters B 452 (1999) 265–273
ing the unitary N = N matrices in 1D form additional two extra coordinates, promoting it to a 3D theory, compatible with the action Žhamiltonian. w11x of supermembrane w18x. Our dimensional reduction prescription is the usual one, namely we require all the fields to be independent of the internal 9D coordinates x 1 , x 2 , PPP , x 9 , so that the space-integrals over these coordinates in the action become an over-all trivial factor. Only in this section we use hats on fields and on indices in 12D in order to distinguish them from those in 3D. In this notation, our 12D lagrangian Ž3.5. is rewritten as 2
ˆ
I s d 12 xˆ y 14 Fˆiˆˆj q lˆ P ≠ˆ gˆ i nˆu Dˆ iˆlˆ
ž /
H
ž
/
,
Ž 5.1 .
where i,ˆ jˆ, PPP s 0, 1, PPP , 9 are 10D coordinate ˆˆ are indices. Other components in Fˆmn ˆ ˆ such as Fiq effectively absent from Ž5.1., due to the second and third terms in Ž3.5.. All the 12D fields are dimensionally reduced to 3D with the coordinates Ž x 0 , xq, xy . by the rules, such as
°Fˆ s g Ž X X y X X . , Fˆ s~ ¢Fˆ s X q g Ž A X y A i ,q
ij
iˆˆj
j ,y
i ,0
0i
i ,y
0,q
i ,y
0
žl/
,
~gˆ
i
i
0
0
i 0,y X ,q
. ' D0 X i ,
mt 3 ,
H
q'2 g lG
i
q'2 Ž lG 0 D 0 l . .
ž
I32 0
nˆu s I32 m tq,
0 , 0
/
Ž 5.3 .
This lagrangian is still in terms of 10D spinor l, which is to be further reduced to 16 component spinors to fit the SO Ž9. symmetry we need to compare with the result in w1x. This can be done by the SO Ž9. g-matrix representations w1,5x:
ls
žu/
G 0s
0
ž
,
0 I16
l s Ž 0, u T . , yI16 , 0
/
G is
ž
0
gi
gi
0
/
,
Ž 5.4 .
yielding the lagrangian 2
2
i i q'2 g u Tg i Ž X ,q u ,yy X ,y u ,q .
q'2 Ž u T D 0 u . .
Ž 5.5 .
Accordingly, we can perform the similar dimensional reduction to our supersymmetry transformation Ž3.7., to get
dQ A 0 s '2 Ž e Tu . ,
dQ Aqs dQ Ays 0 ,
¢ Pˆ≠ s
2
Ž X ,qi l ,yy X ,yi l ,q .
dQ X i s y'2 Ž e Tg iu . ,
3
sG mt , gˆqs I32 m tq , gˆys I32 m ty ,
gˆ iˆs
2
i j i j I s d 3 x q 12 Ž D 0 X i . y 14 g 2 Ž X ,q X ,y y X ,y X ,q .
H
lˆ s Ž 0, l . Ž I32 m t 1 . s Ž l ,0 . , Ž 5.2b .
°gˆ s G
Ž5.2., l is a 10D Majorana–Weyl spinor with maximally 32 components, and t 1 , t 2 , t 3 are the usual 2 = 2 Pauli matrices, and t "' Ž t 1 " it 2 . r2. Applying the useful relations Ž5.2. to the action Ž5.1., and integrating over the internal 9D coordinates x 1 , PPP , x 9 , we get
i j i j I s d 3 x q 12 Ž D 0 X i . y 14 g 2 Ž X ,q X ,y y X ,y X ,q .
j ,q
Ž 5.2a . lˆ s
271
dQ u s y 12 g ie D 0 X i q
0 Pˆx s 0
ž
m ˆu s I32 m ty,
0 I32 ,
/
Ž 5.2c . Ž 5.2d .
where i, j, PPP s 1, PPP , 9 in this section are for the spatial 9D, as those used in w1x. Therefore Ž G 0 , G i . in Ž5.2c. realize the Clifford algebra for 10D. In
1 4
i j i j gg i je Ž X ,q X ,y y X ,y X ,q ..
Ž 5.6 . Our results Ž5.5. and Ž5.6. agree with the 3D result in w11x after the large N limit, up to non-essential scaling factors such as '2 . We have thus seen that our 12D supersymmetric YM theory directly gives rise to the 3D theory w11x corresponding to the supermembrane theory w18x after taking the large N limit in the D0-brane action.
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6. Concluding remarks In this paper we have presented a 12D supersymmetric gauge theory, with a very peculiar gauge symmetry associated with Poisson bracket as a reminiscent of the commutators in non-Abelian generators in the original 10D theory. The Poisson bracket arises in the ’t Hooft N ™ ` limit for UŽ N ., whose p, q variables are now regarded as two extra coordinates in the total 12D. The extra coordinates have the ranges yp F p - p ,y p F q - p , implying a compactification on S 1 m S 1 within the total 12D. These extra dimensions are further decompactified by the strongrweak duality by rescaling the coordinates by x 11 ' Rp, x 12 ' Rq, and taking the limit R ™ `, g ™ 0 with R 2 g fixed. Subsequently, we have also studied the geometrical significance in superspace of our superfield strength with the Poisson brackets, and found that a superspace formulation is equally possible like the conventional supersymmetric YM theory. This superspace formulation elucidates the geometrical significance of our theory, in terms of field strength superfield defined by commutators between super-gauge covariant derivatives, with our peculiar ‘gauge’ symmetry. We have also studied the dimensional reduction of our 12D theory into 3D, that yields the desirable Poisson brackets in 3D, corresponding to the action for supermembrane theory w18,11x. In a recent development in the duality between Anti-de Sitter ŽAdS. and conformal field theory w19x, it is conjectured that the N s 4, UŽ N . supersymmetric YM theory in 4D is dual to type IIB theory on ŽAdS.5 m S 5 in the large N limit. It was pointed out w19x that the group SO Ž2,4. = SO Ž6. in the N s 4 supersymmetric YM theory suggests a 12D realization with two time coordinates. In our present paper, we have given a first explicit example, in which the direct connection between the supermembrane theory and 10D supersymmetric YM is much more natural than before. Since the N s 4 supersymmetric YM in 4D has the 10D origin which is promoted to be 12D in the large N limit, we have seen another duality link between superstring theory w3x, N s 4 supersymmetric YM in 4D, and supermembrane theory w18x Õia 12D supersymmetric YM theory w9x. We have observed similarity as well as difference between our new superspace formulation and that in
w9,17x. The most conspicuous difference is the nonvanishing Poisson bracket terms in FA B making the system non-trivial. We have also clarified the geometrical significance of our peculiar field strength superfield in terms of superspace language. We also stress that our present paper gives a clearer link between supersymmetric YM theory in 12D w9,17x and M-theory w4,5x. In a usual non-supersymmetric theory, the loss of Lorentz covariance makes the system completely ambiguous, because we can always put null-vectors anywhere by hand in any equation in the system. However, as was also stressed in w13,17x, this is no longer the case with supersymmetric theories, in which the coefficients in field equations are tightly fixed by supercovariance. The incorporation of symplectic variables as target coordinates makes stronger sense in supersymmetric theories. Our formulation has been also strongly motivated by the recent development in the matrix theory approach w1x to M-theory w4,5x. In our formulation, due to the absence of supergravity, the loss of local Lorentz covariance is not crucial. In this connection, we mention that only global Lorentz covariance plays an important role in the study of non-perturbative aspects of M-theory. In fact, in the conjecture by Maldacena w19x about the duality between the large N limit of D s 4, N s 4 supersymmetric Yang-Mills and type IIB superstring compactified on Ž AdS .5 m S 5, the global isometry group is SO Ž4,2. = SO Ž6., being further promoted to SO Ž10,2. w19,20x, indicates the existence of 12D supersymmetric theory. Note that the SO Ž10,2. symmetry is global, and therefore the loss of local Lorentz covariance in 12D supergravityrsupersymmetry formulations as in w13x is not crucial in such a formulation of M-theory w4,5x. Our result also provides a new link between the F-theory w21x in 12D and the M-theory w4x in 11D, that has not been explicitly presented before. 8 Our result also clarifies general dualities connecting a supersymmetric YM theory in Ž D y 1,1. dimensions
8
In Ref. w22x, an idea of deleting even the world-line for the D0-brane was presented, but with no reference to the 12D YM field strength defined with Poisson bracket in terms of extra coordinates.
H. Nishinor Physics Letters B 452 (1999) 265–273
with another supersymmetric theory formulated in Ž D,2. dimensions under the large N limit. It also gives a new connection between the conventional supersymmetryrsupergravity theories in D F 11 with higher-dimensional theories, such as the 12D supergravity w14,15,13x, or more supersymmetric YM theories in D G 12 w17x. In particular, the evidence for the importance of the results in w9,17x is now rapidly mounting.
Acknowledgements We are indebted to I. Bars, S.J. Gates, Jr., J.H. Schwarz, and W. Siegel for valuable suggestions.
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