D-branes, symplectomorphisms and noncommutative gauge theories

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D-branes, Symplectomorphisms

and Noncommutative Gauge Theories

I. Martin1, J. Ovalle 4 and A. Restuccia2 3

Departamento de Fısica, Universidad Simon Bolıvar, Venezuela

and1 Theoretical Physics Group, Imperial College , London University. e-mail: [email protected]; [email protected] Department of Mathematics, King’s College, London University. e-mail: [email protected] Invited talk at SSQFT, Kharkov 2000.4 e-mail: [email protected]

It is shown that the dual of the double compactified D=11 Supermembrane and a suitable compactified D=10Super 4D-brane with nontrivial wrapping on the target space may be formulated as noncommutative gaugetheories. The Poisson bracket over the world-volume is intrinsically defined in terms of the minima of thehamiltonian of the theory, which may be expressed in terms of a non degenerate 2-form. A deformation of thePoisson bracket in terms of the Moyal brackets is then performed. A noncommutative gauge theory in terms ofthe Moyal star bracket is obtained. It is shown that all these theories may be described in terms of symplecticconnections on symplectic fibrations. The world volume being its base manifold and the (sub)group of volumepreserving diffeomorphisms generate the symplectomorphisms which preserve the (infinite dimensional) Poissonbracket of the fibration.

1. Introduction

The formulation of D-brane theories in thepresence of constant antisymmetric backgroundfields and its relation to noncommutative gaugetheories has recently attracted a lot of interest[1]-[22]. It may well be that a noncommutativeformulation of the D=11 Supermembrane and theSuper M5-brane may indeed improve the under-standing of the quantum aspects of these theories.The spectrum of the D=11 Supermembrane ona Minkowski target space was shown to be con-tinuous from zero to infinite [23]. However notmuch it is known about the spectrum of the the-ory when the target space is compactified [24]-[26]. Even less it is known about the spectrumof the M5-brane. Nevertheless, one may extrap-olate some known aspects from the Supermem-brane case since both theories are U-dual. Thecovariant formulation of the M5-brane was foundin [27] and [28]. However the analysis of its phys-ical hamiltonian, the existence of singular config-urations, the topological instabilities and relatedproblems have not yet been discussed in a conclu-

sive way. It is possible that a formulation of thesetheories in terms of a noncommutative geometrymay allow an improvement in their analysis.

We describe in this talk a general approach toreach that formulation. The first step is to intro-duce a symplectic geometry intrinsic to the the-ory. This may be done when the target spaceis suitably compactified. The non degenerateclosed 2-form associated to the symplectic geom-etry may be obtained in a general way from theanalysis of the Born-Infield action. We discussthis problem in section 2. The second step in theconstruction is to obtain the hamiltonian of theD-brane with non trivial wrapping on the targetspace. We perform this construction for the dou-ble compactified D=11 Supermembrane [29], [30]and the compactified 4 D-brane in 10 dimensions.It turns out that the minima of these hamiltoni-ans are described by the dual configurations in-troduced in [31] and in section 2. The final stepis to introduce the geometrical objects describingthe noncommutative formulation. This is done interms of a symplectic fibration and a symplectic

2

connection over it. We also consider deformationsof the brackets introduced in these theories, al-lowing a construction of noncommutative gaugetheories in terms of the usual Moyal star prod-uct. There is a precise one to one correspondencein the sense of Kontsevich, between the originaltheories and their deformations.

2. The dual configurations

The Born-Infeld theory formulated over a Rie-mannian manifold M may be described by thefollowing D dimensional action

S (A) =

∫

M

(√

det (gab + bFab) −√

g)

dDx (1)

where gab is an external euclidean metric over thecompact closed manifold M . Fab are the com-ponents of the curvature of connection 1-form A

over a U(1) principle bundle on M .b is a constantparameter.

We may express the det (gab + bFab), using thegeneral formula obtained in [31], as

det (gab + bFab) = g

n∑

m=0

amb2m∗ [Pm ∧ ∗Pm]

≡ gW, (2)

where

Pm ≡ F ∧ . . . ∧ F︸ ︷︷ ︸

m

(3)

am are known constants, see [31].The first variation of (1) is given by

δS (A) =

∫

M

W−1

2

∑

m

mamb2mdδA∧Pm−1∧∗Pm(4)

which yields the following field equations

∑

m

mamb2mPm−1 ∧ d(

W−1

2 ∗Pm

)

= 0. (5)

We introduce now a set A of U(1) connection1-forms over M [31]. They are defined by thefollowing conditions,

∗Pm (A) = kmPn−m (A) , m = 0, . . . , n, (6)

where n = D2,i.e we assume the dimension D of M

to be an even natural number. (6) is the conditionthat the Hodge dual transformation maps the set{Pm , m = 0, . . . , n} into itself.

We observe that these connections, if they existin a U(1) principle bundle over M , are solutionsof the field equations (5).

In fact, (6) implies

∗ [Pm ∧ ∗Pm] = km ∗ [Pm ∧Pn−m] = km ∗Pn(7)

but from (6), for m = n, we obtain

∗ Pn = kn (8)

which is constant. We thus have, for these con-nections,

W = constant. (9)

Finally, it results

d(

W−1

2 ∗Pm

)

= kmW−1

2 d (Pn−m) = 0, (10)

showing that (6), if they exits, define a set ofsolutions to the Born-Infeld field equations.

Let us analyse a particular case of (6). Let usconsider n = D

2= 1. We then have

∗P1 = ∗F = k1. (11)

This solution represents a monopole connectionover the D = 2 manifold M . When M is thesphere S2, (11) defines the U(1) connection de-scribing the Dirac monopole on the Hopf fibringS3 → S2. The constant k1 is determined from thecondition∫

M

F = 2π × integer (12)

which is a necessary condition to be satisfied fora U(1) connection, F being its curvature.

This solution was extended to U(1) connectionsover Riemann surfaces of any genus in [32] . In[29] it was shown that they describe the minima ofthe hamiltonian of the double compactified D =11 supermembrane dual.

3. Hamiltonian formulation

The hamiltonian formulation of the doublecompactified D=11 Supermembrane dual was ob-tained in [30]. Its hamiltonian density in the light

3

cone gauge is the following

H =1

2

1√W

(PMPM + det(∂aXM∂bXM )

+(Πar∂aXM )2 +

1

4(Πa

rΠbsǫabǫ

rs)2 +1

4W (∗F r)2

)

−Ar0∂cΠ

cr + Λǫab∂b

(∂aXMPM + Πc

rFrac√

W

)

(13)

where PM are the conjugate momentum to XM

while Πar are the corresponding momentum to Ar

a.The index r denote the 2 compactified directionson the target space. a is the world volume indexwhile M label the LCG transverse directions inthe target space.

Its supersymmetric extension may be obtainedin an straightforward way from the supermem-brane hamiltonian in the LCG by the proceduredescribed in [30].

We may solve explicitly the constraints on Πcr

obtaining

Πcr = ǫcb∂bΠr; r = 1, 2 (14)

Defining the 2-form ω in terms of Πr as

ω = ∂aΠr∂bΠsǫrsdξa ∧ dξb, (15)

the condition of non trivial membrane windingimposes a restriction on it, namely∮

Σ

ω = 2πn. (16)

With this condition on ω, Weil’s theorem en-sures that there always exist an associated U(1)principal bundle over Σ and a connection on itsuch that ω is its curvature. The minimal con-figurations for the hamiltonian (13) may be ex-pressed in terms of such connections.

In [29] the minimal configurations of the hamil-tonian of the double compactified supermem-brane were obtained. In spite of the fact that theexplicit expression (13) was not then obtained,all the minimal configurations were found. Theycorrespond to Πr = Πr satisfying

∗ ω = ǫab∂aΠr∂bΠsǫrs = n

√W n 6= 0 (17)

The explicit expressions for Πr were obtained inthat paper [29]. As mentioned before, they corre-spond to U(1) connections on non trivial principle

bundles over Σ. The principle bundle is charac-terized by the integer n corresponding to an irre-ducible winding of the supermembrane. Moreoverthe semiclassical approximation of the hamilto-nian density around the minimal configuration,was shown to agree with the hamiltonian den-sity of super Maxwell theory on the world sheet,minimally coupled to the seven scalar fields rep-resenting the coordinates transverse to the worldvolume of the super-brane.

As mention in the introduction these minimacorrespond to the dual solutions of section 2 cor-responding to 2D-brane. We now consider thehamiltonian of the D=10 4D-brane. It may beobtained by the following double dimensional re-duction procedure. We start from the PST ac-tion for the super M5-brane. We consider thegauge fixing condition which fixes the scalar fieldto be proportional to the world volume time. Wethen perform the usual double dimensional reduc-tion by taking one of the target space coordinatesX11 = σ5 where σ5 is one of the world volume lo-cal coordinates. After several calculations we endup with the following canonical lagrangian

L = PmXm + P ijBij −Hc (18)

Hc = λφ + λiφi + θi∂jPij (19)

where

φ =1

2P 2 + 2g + 2

(1

8P ijP klgikgjl + ∗Hi∗Hjgij

)

+1

32

(1

4ǫijklP

ijP kl

)2

(20)

φl = Pm∂lXm +

1

2ǫijklP

ij∗Hk (21)

∗Hi =1

6ǫijklHjkl (22)

We finally obtain the hamiltonian in the LCG

H =1√W

(1

2PMPM + 2g + 2

(1

8P ijP klgikgjl

+∗Hi∗Hjgij

)+

1

32

(1

4ǫijklP

ijP kl

)2

+Λlq∂q

(

PM∂lXM +

1

2ǫijklP

ij∗Hk

)

(23)

4

where Λlq are antisymmetric lagrange multipli-ers associated to the generator of volume preserv-ing diffeomorphisms.

There is also a global constraint given by∮

Ci

(

PM∂lXM +

1

2ǫijklP

ij∗Hk

)

dσl = 0 (24)

where Ci is a basis of homology of dimension 1.We will not consider any further this global con-straint. We will work only with the diffeomor-phisms connected to the identity.

We now consider a target space with 4 com-pactified directions, M6xS1xS1xS1xS1. We con-struct the dual formulation associated to the com-pactified directions. We associate to each Xr,r = 1, ..., 4 a B3 3-form

dXr → dB3

r (25)

It is more convenient to work with the Hodge dualof the 3-form,

Brijk → Alr (26)

in the spatial world volume sector Brijo are La-grange multipliers. We denote Πrl the conjugatemomenta to Al

r.There is a constraint on Πrl which yields

Πrl = ∂lΠr (27)

We may then perform a canonical transformationto obtain

ΠrlAlr = Πr(∂lA

lr) (28)

that is, ∂lAlr is the conjugate momenta to Πr:

Πr ≡ Ar

∂lAlr ≡ Π r (29)

We then obtain the dual formulation to (23).For the determinant of the induced metric we

obtain

1

4!

(ǫi1...i4∂i1X

a1 ...∂i4Xa4

)2

→ 1

4!

(ǫi1...i4∂i1X

b1 ...∂i4Xb4

)2

+1

3!

(ǫi1...i4∂i1Ar∂i2X

b2 ...∂i4Xb4

)2

+1

2!

(ǫi1...i4∂i1Ar∂i2As∂i3X

b3∂i4Xb4

)2

+(ǫi1...i4∂i1Ar∂i2As∂i3At∂i4X

b4)2

+(ǫi1...i4∂i1Ar∂i2As∂i3At∂i4Au

)2

(30)

where the index b is used to denote the non com-pactified directions.

For the terms quadratic on the momenta to theantisymmetric field Bij we obtain similar terms,the connection 1-forms Ar replaces the corre-sponding terms where the compactified coordi-nates appear. In the same way the momenta Π r

replaces the conjugate momenta of the compacti-fied coordinates. That is, in the dual formulation(with the additional canonical transformation wementioned above) the compactified coordinatesare replaced by the connection 1-forms Ar. How-ever, it is clear from the dual formulation that Ar

may have non trivial transitions of a very specificform over a nontrivial bundle. This fact is diffi-cult to realize in terms of the original maps fromthe world volume to the compactified directionsof the target space.

4. The noncommutative formulation

We now introduce a symplectic 2-form in theprevious formulation. We take

Fijdσi ∧ dσj (31)

where Fij is the curvature of the connection 1-form which minimize the hamiltonian of the the-ory. For the D=11 Supermembrane we obtained

∗ F = n, i, j = 1, 2 (32)

as discussed before.For the D=10 Super 4D-brane we obtain

F ∝ ∗F (33)

∗ (F ∧ F ) ∝ n (34)

over the 4 dimensional spatial world volume.These are two of the dual solutions introducedin [31] and explained in section 2.

The procedure to obtain the symplectic non-commutative formulation for the double compact-ified D=11 Supermembrane was explicitly intro-duced in [30]. We will now obtain a similar for-mulation for the 4 D-brane in 10 dimensions.

5

The approach uses the symplectic 2-forms pre-viously introduced to obtain a description of thetheory in terms of symplectic connections over asymplectic fibration.

We consider the metric W ij defined by

W ij ≡ ΠirΠ

jr (35)

where

Πir ≡ F ij∂jAr (36)

The metric W ij is taken to be the metric over thespatial world volume for which F ij satisfies theduality conditions. Πi

r is a well defined vielbein.(36) defines Ar.

We then introduce the following Poissonbracket over the world volume

{B, C} ≡ F ij∂iB∂jC (37)

It satisfies the Jacobi identity

{{A, B}, C}+ {{C, A}, B} + {{B, C}, A}=

(F klF ij + F jlF ki + F ilF jk

)

.Dl (∂iA∂jB∂kC)

= kǫklij

√W

Dl(∂iA∂jB∂kC) = 0 (38)

Where Dl denotes the covariant derivative withrespect to the metric W ij . A, B and C are scalarfields.

We now introduce the rotated covariant deriva-tive

Dr ≡ ΠirDi (39)

Dr = Dr + {Ar, } (40)

and curvature

Frs = DrAs − DsAr + {Ar,As} (41)

The hamiltonian density of the double com-pactified Supermembrane was expressed in termsof these geometrical objects in [30]. We now per-form the analogous formulation for the 4 D-brane.

We obtain

1

4!

[ǫi1...i4∂i1X

b1 ...∂i4Xb4

]2

→ 1

4!

[{Xb1, Xb2}{Xb3, Xb4}

]2(42)

1

3!

[ǫi1...i4∂i1Ar∂i2X

b2 ...∂i4Xb4

]2

→ 1

3!

[DrX

b2 .{Xb3 , Xb4}]2

(43)

1

2!

[ǫi1...i4∂i1Ar∂i2As∂i3X

b3∂i4Xb4

]2

→ 1

2!

[Frs.{Xb3 , Xb4}

]2(44)

[ǫi1...i4∂i1Ar∂i2As∂i3At∂i4X

b4]2

→[FrsDtX

b}]2

(45)

[ǫi1...i4∂i1Ar...∂i4Au

]2 → [FrsFtu]2

(46)

where it is understood that the antisymmetricpart on the b indexes and the target space r, s,...indexes is taken.

Similar formulae may be written for all otherterms in the hamiltonian density for the 4 D-brane. They may be rewritten in terms of thebracket (37), the covariant derivative and the cur-vature Frs.

The expression for the double compactifiedD=11 Supermembrane was [30]:

H =

∫

Σ

H =

∫

Σ

1

2√

W

[(PM )2 + (Π r )2

+1

2W{XM , XN}2 + W (DrX

M )2

+1

2W (Frs)

2

]

+

∫

Σ

[1

8

√Wn2

−Λ(DrΠ

r + {XM , PM})]

(47)

The geometrical interpretation of the above for-mulation (47) was given in [30] in terms of sym-plectic fibrations and connection 1-form over it.We have shown here that the same geometricaldescription may be given for the 4 D-brane. Wewill discuss it in more detail shortly. Before then,we would like to remark that there is a naturaldeformation of the above formulations in termsof the Moyal bracket. Let us see how this defor-mation may be realized preserving the symplecticstructure on the fibration, for the case of the dou-ble compactified Supermembrane.

We replace in (47) the Poisson bracket

{B, C} = F ij∂iB∂jC (48)

6

by the Moyal bracket {, }M . (48) is the first termin the expansion of the Moyal bracket. We noticethat under the gauge transformation generatedby the first class constraint

δXM = {ξ, XM} (49)

δAr = −Drξ = − (Drξ + {Ar, ξ}M ) (50)

δDrXM = {ξ,DrX

M}M (51)

These properties ensure that the hamiltoniandensity transform as

δH = {ξ,H}M (52)

The integral over a compact world volume ren-ders the canonical lagrangian invariant under thegauge transformations.

We notice that the physical degrees of freedomof both theories, the double compactified D=11Supermembrane (47) and its Moyal deformation,are exactly the same. Moreover it can be shownthat there is a one to one correspondence, in thesense of Kontsevich, between both theories.

In [30] a geometrical description of the sym-plectic non commutative gauge theory was intro-duced. The same geometrical interpretation maybe used to describe its deformation in terms of theMoyal bracket and the compactified 4 D-brane wehave discussed.

We consider a symplectic fibration with basemanifold the spatial world volume, which is aclosed (without boundary) manifold. Over thefibration we consider the (infinite dimensional)Poisson bracket of the sections XM (σ), PM (σ):

[XM (σ), PM (σ′)] = δ(σ, σ′) (53)

This Poisson structure is preserved under thetransition maps of the fibration. These maps aredefined by

δXM = {ξ, XM} (54)

δPM = {ξ, PM} (55)

over Uα

⋂Uβ , Uα is a covering of the base mani-

fold. We notice that the transformation maps aredefined in terms of the (finite dimensional) Pois-son bracket or Moyal bracket over the world vol-ume, and they preserve the (infinite dimensional)

Poisson bracket on the fibration. Ar define a sym-plectic connection over the fibration. That is, thePoisson bracket on the fibration is preserved un-der the holonomy generated by Ar.

The three theories we have discussed, thedouble compactified D=11 Supermembrane, itsMoyal deformation and the compactified 4 D-brane all admit the same geometrical interpreta-tion. They describe the dynamics of a symplecticconnection over a symplectic fibration.

5. Conclusions

We formulated the double compactified D=11Supermembrane and the compactified Super 4D-brane in terms of a symplectic noncommuta-tive gauge theory. We constructed a deforma-tion of the compactified D=11 Supermembranein terms of the Moyal brackets. There exist a oneto one correspondence between both theories inthe sense of Kontsevich. A unified geometricaldescription of these theories was given in termsof a symplectic fibration over the world volumeand the dynamics of symplectic connections overit. We hope the analysis for the D=10 4D-branemay be extended to describe the M5-brane in 11dimension. Once that formulation is available wemay start analysing the corresponding quantumfield theory. We hope to report on that shortly.

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