Canonical approach to noncommutative gauge theory

Physics Letters B 683 (2010) 349–353

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Physics Letters B www.elsevier.com/locate/physletb

Canonical approach to noncommutative gauge theory ✩ ´ B. Sazdovic´ ∗ D.S. Popovic, Institute of Physics, University of Belgrade, 11001 Belgrade, PO Box 57, Serbia

a r t i c l e

i n f o

Article history: Received 16 March 2009 Received in revised form 7 December 2009 Accepted 10 December 2009 Available online 21 December 2009 Editor: M. Cvetiˇc

a b s t r a c t It is known that gauge transformation of the Kalb–Ramond field B μν with vector parameter Λμ is symmetry of the closed string world-sheet action. It fails at the end points of the open string and can be restored by introducing Maxwell field A μ . We show that the same conclusion valid for space–time equations of motion, because they are conditions for conformal invariance of the world-sheet action. This is example how the symmetries of space–time theory can be investigated using properties of σ -model energy–momentum tensor. We also show that the reducible part of the closed string symmetry transformation, with Λμ = ∂μ λ, turns to irreducible part of the open string one. As well as Maxwell field A μ , the parameter λ is nontrivial only on the string endpoints. We show that after quantization the symmetry transformations of the background fields B μν and A μ turn to the modified symmetries defined in terms of Moyal star product. The modified Λμ -transformation related different regularizations as well as Seiberg–Witten map. © 2009 Elsevier B.V. All rights reserved.

1. Introduction We will study physics of Dp-brane in the presence of constant antisymmetric background field B μν and coordinate dependent vector field A μ (x) living on the string endpoints. This problem has been investigated in detail in Refs. [1,2] using method of conformal field theory. We will use canonical approach and pay attention to the origin of Moyal structure of symmetry transformations. It is known that world-sheet action for closed string is invariant under gauge transformation of Kalb–Ramond antisymmetric field B μν with vector gauge parameter Λμ . For open strings, boundary conditions break this symmetry but it can be restored by introducing the vector field A μ on the string endpoints [1,3]. In our approach the essential role play symmetries of the space–time field equations. Let us stress that it is not necessary to know explicit form of these equations but it is enough to use their definition, as a condition for quantum conformal invariance of world-sheet theory. We use the method developed in Ref. [4] in order to describe the symmetries of closed string theory. Particularly, we will show that the space–time field equations are invariant under the same symmetry transformation as the worldsheet action and find the relevant symmetry generator. ✩ Work supported in part by the Serbian Ministry of Science, under contracts Nos. 141036 and 141037. Corresponding author. ´ [email protected] E-mail addresses: [email protected] (D.S. Popovic), ´ (B. Sazdovic).

*

0370-2693/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.12.027

We generalize the method of Ref. [4] to the case of the open string theory. As well as in the σ -model approach the symmetry which is broken by open string boundary conditions can be restored with the help of new vector field A μ on the string endpoints. In the case of the closed string, the symmetry transformation with parameter Λμ = ∂μ λ (the longitudinal part of Λμ ) is reducible. In the case of the open string, the appearance of vector field A μ turns this part of symmetry to irreducible one. Both background fields B μν and A μ depend on the transversal part of the gauge parameter Λμ . We investigate local gauge transformations of the background fields B μν and A μ on the quantum level. We find that as a consequence of space–time noncommutativity, transformation of both fields obtain new terms defined with the help of Moyal star product. With the help of these results, it is easy to find field strength and the action invariant under quantum λ-symmetry, which is just the noncommutative electrodynamics. We show that it is possible to chose gauge transformation which turns field A μ to zero and at the same time turns transformation of the Kalb–Ramond field to Seiberg–Witten map. 2. Symmetries of the space–time field equations We are going to consider the action of the bosonic string theory in the presence of the external fields: metric G μν (x) and antisymmetric tensor field B μν (x)

350

S=

D.S. Popovi´c, B. Sazdovi´c / Physics Letters B 683 (2010) 349–353

ηα β 4πα 



2.2. Improved open string theory invariant under Λ symmetry

d2 ξ G μν (x)∂α xμ ∂β xν Σ



+ εαβ

d ξ B μν (x)∂α xμ ∂β xν . 2

(2.1)

Σ

The corresponding space–time field equations can be derived from the requirement of Weyl invariance of the quantum worldsheet theory. This means that β -functions, relating to the backG B ground fields G μν and B μν vanish, βμν = 0 = βμν . On these conditions, the nonlinear σ -model (2.1) becomes conformal field theory represented by two independent copies of Virasoro algebra, satisfied by σ -model energy–momentum tensor T ±

  { T ± , T ± } = − T ± (σ ) + T ± (σ¯ ) δ  .

The gauge invariance under transformations (2.3) has failed for the open string, because of the boundary contributions. It can be restored by introducing a vector field A μ (x) living on string endpoints, if we vary it with respect to the same parameter Λμ , δ A μ (x) = −Λμ , [1,3]. Again, we are going to derive above result for space–time field equations. In the open string theory the generator is defined with different integration interval ◦

π ΓΛ = 2

dσ Λμ (x)x μ ,

(2.10)

0

(2.2)

so that instead of (2.8) we have

2.1. Symmetries of the closed string field equations

◦

It is well known that this σ -model action (2.1), with closed string boundary conditions, is symmetric under the transformation (see for example [3])

δ(σ − π ) − δ(σ + π ) and it vanishes as a consequence of periodic-

δΛ B μν = ∂μ Λν − ∂ν Λμ .

(2.3)

We are interested in the corresponding symmetry of the space– time equations of motion. In papers [4], very interesting method has been developed for such purpose. The authors proved general statement that if space–time equations of motion are symmetric under some transformations δΛ ϕ , of the background fields ϕ ∈ {G μν , B μν } with parameter Λ, then there exist an expression ΓΛ such that

  δΛ T ± (σ ) = ΓΛ , T ± (σ ) .

(2.4)

It is easy to see that Eq. (2.4) is condition that energy– momentum tensor T ± (σ ) + δΛ T ± (σ ) also satisfies the Virasoro algebra, and consequently does not change the physics. But, such transformation change the background fields and so it generate space–time symmetry transformation. The components of the energy–momentum tensor, for the action (2.1), can be express in terms of currents j ±μ [5]

T± = ∓

πα  2

G μν j ±μ j ±ν ,



j ±μ = πμ + 2 B μν ±



1 4πα 

ν

G μν x .

(2.5)

In the case of the closed string, the last term is of the form ity condition. In the case of the open string, boundary contribution breaks the invariance because (2.11) is not of the form (2.6), proμ portional to j ± j ν∓ . We want to construct open string theory, symmetric under transformation (2.9). We are going to add a new term to the current in order to cancel boundary part. For our purpose it is enough to use momentum and chirality independent expression  j μ (x), so that improved open string currents and corresponding energy– momentum tensor components are ◦

(2.6)

2

Following [4], we chose the generator ΓΛ proportional to tive of the coordinate xμ ,

π ΓΛ = 2

dσ Λμ xμ = 2πα 

−π

π

μ

μ

d σ Λμ j + − j − ,

σ deriva-

(2.7)

−π



2  μ ΓΛ , T ± (σ ) = 2 πα  (∂μ Λν − ∂ν Λμ ) j ± j ν∓ .

(2.8)

With help of (2.4), (2.6) and (2.8) we can find expressions of the symmetry transformations generating by ΓΛ

δΛ B μν = ∂μ Λν − ∂ν Λμ ,

δΛ G μν = 0.

T± = ∓

πα  2

G μν ◦ j ±μ ◦ j ±ν .

δΛ ◦ T ± (σ ) =

◦

 ΓΛ , ◦ T ± (σ ) .

(2.9)

So, we can conclude that for closed string theory the transformation (2.3) is symmetry of the space–time field equations.

(2.12)

(2.13)

Now, instead of (2.6) we have



2 μ μ δΛ ◦ T ± = 2 πα  δΛ B μν ◦ j ± ◦ j ν∓ ∓ πα  ◦ j ± δΛ  j μ ,

(2.14)

while the right-hand side of (2.11) remains unchanged because  j μ does not depend on momenta

◦



2 μ μ ΓΛ , ◦ T ± = 2 πα  (∂μ Λν − ∂ν Λμ ) ◦ j ± ◦ j ν∓ ∓ 2πα  ◦ j ± Λμ   × δ(σ − π ) − δ(σ ) . (2.15)

From last three equations we can conclude that

δΛ B μν = ∂μ Λν − ∂ν Λμ ,   δΛ  j μ = 2Λμ δ(σ − π ) − δ(σ ) .

(2.16)

The variation of the additional part  j μ is nontrivial only on the string endpoints. Therefore, it is useful to express  j μ in terms of new field A μ (x), living on the string endpoints

   j μ = −2 A μ δ(σ − π ) − δ(σ ) .

and obtain



◦

j ±μ = j ±μ +  j μ ,

The form of  j μ (x) we can obtain from the requirement that improved open string theory is symmetric under transformations (2.9), which means that

The variation with respect to background fields is of the form

  1 μ δΛ T ± = πα  2πα  δΛ B μν ± δΛ G μν j ± j ν∓ .



2 μ μ ΓΛ , T ± = 2 πα  (∂μ Λν − ∂ν Λμ ) j ± j ν∓ ∓ 2πα  j ± Λμ   × δ(σ − π ) − δ(σ ) . (2.11)

(2.17)

Then, the complete expressions for currents and symmetry transformations take a form ◦



j ±μ = πμ + 2 B μν ±



1 4πα



G x ν  μν

 − 2 A μ δ(σ − π ) − δ(σ ) , δΛ B μν = ∂μ Λν − ∂ν Λμ ,

δΛ A μ = −Λμ .

(2.18) (2.19)

D.S. Popovi´c, B. Sazdovi´c / Physics Letters B 683 (2010) 349–353

So, as it is well known, the open string theory can depend only on gauge invariant combination

Fμν = B μν + ∂μ A ν − ∂ν A μ .

(2.20)

It is easy to find the corresponding σ -model Lagrangian ◦ L = πμ x˙ μ − ◦ H, with standard definition of the canonical Hamiltonian, ◦ H = ◦ T − ◦ T . After elimination of the momenta π , on theirs − + μ equations of motions we obtain ◦

S=

ηα β 4πα 

+ εαβ

d2 ξ B μν (x)∂α xμ ∂β xν Σ











dτ A μ x˙ μ σ =π − A μ x˙ μ σ =0 ,

+2

(2.21)

4πα 

d2 ξ G μν (x)∂α xμ ∂β xν

Σ



+ εαβ

d2 ξ Fμν (x)∂α xμ ∂β xν .

(2.22)

2.3. Reducible symmetry of the closed string turns to irreducible part of the open string The symmetry of closed string theory is reducible, because the gauge parameters are not mutually independent. For Λμ = ∂μ λ π from (2.7) we have ΓΛ=∂λ = 2 −π dσ λ = 0 which means that this part of gauge parameter does not remove any degree of freedom. The same conclusion we can obtain from the relation (2.9), δΛ=∂λ B μν = 0. The open string theory is not reducible, because for Λμ = ∂μ λ we have

ΓΛ=∂λ ≡ ◦ Γλ = 2

π

  dσ λ = 2 λ(x)|σ =π − λ(x)|σ =0 .

(2.23)

0

The corresponding gauge symmetry is Abelian and has the form

δλ B μν = 0,

(3.1)

δλ A μ = −∂μ λ.

(2.24)

So, the Kalb–Ramond field B μν is invariant under reducible part of transformation, while the vector field A μ (x) transforms as in standard Maxwell theory. T ν − If we separate parameter Λμ to the transversal Λμ = (δμ ∂μ ∂ ν ∂2

∂μ ∂ ν ∂2

)Λν and longitudinal part Λμ = Λν ≡ ∂μ λ then transformation of B μν depend only on the transversal one while the transformation of A μ depend on both parts T δΛ B μν = ∂μ ΛνT − ∂ν Λμ ,

L

2 1 ik θ i j = − 2πα  G − Fkq G qj , eff

2 eff G i j = G i j − 4πα  Fik G kq Fqj .

T δΛ A μ = −Λμ − ∂μ λ.

(3.2)

In order to consider quantization procedure, instead of classical variables, the coordinates xi and corresponding momenta πi we introduce the corresponding operators xˆ i and πˆ i , and replace Poisson brackets by commutators. Because commutation relations on different string endpoints differ only in sign, for definiteness we will consider the σ = 0 case



Σ

◦

(σ = 0, π )

3.2. Quantization

in agreement with Lagrangian approach, [1,3]. We can rewrite the last Lagrangian in terms of gauge invariant variable Fμν , as

 ηα β



xi , x j = ±θ i j

where



S=

It is well known that in the presence of the antisymmetric tensor field, Dp-brane world-volume becomes noncommutative manifold, if open string ends on Dp-brane, [6,7]. For Dp-brane coordinates xi (i = 0, 1, . . . , p ), satisfying Neumann boundary conditions, the noncommutativity appears only on the string endpoints. For constant background fields we have

d2 ξ G μν (x)∂α xμ ∂β xν Σ

◦

3.1. Dp-brane as noncommutative manifold





351

(2.25)

The transversal and longitudinal parts are generated with ◦ ΓΛT and ◦ Γλ , defined in (2.10) and (2.23) respectively. 3. Quantum space–time symmetry transformations In this section we will repeat the procedure of Section 2 taking into account noncommutativity of space–time coordinates (3.1). We will show that after quantization gauge transformations of the background fields B μν and A μ obtain noncommutative forms.



xˆ i , xˆ j = i θ i j .

(3.3)

In order to investigate the algebra of functions depending on the operators xˆ i , we must introduce prescription which uniquely assign the operator ˆf (ˆx) to any function f (x). Denoting by ˜f (k) the Fourier transformation of the given function f (x)

˜f (k) =

 d p x f (x)e ikx ,

(3.4)

we define the corresponding operator ˆf (ˆx) as

ˆf (ˆx) =

1



(2π ) p

d p k ˜f (k)e −ikxˆ .

(3.5)

This prescription is known as Weyl or symmetric ordering prescription. We define the star product of the functions to be isomorphic to the operator multiplication

ˆf (ˆx) · gˆ (ˆx) = hˆ (ˆx) ∼ f (x)  g (x) = h(x).

(3.6)

Using Baker–Hausdorff formula it is easy to show that i

f (x)  g (x) = e 2

θ i j ∂∂a

∂ i ∂b j



f (x + a) g (x + b)a=b=0 ,

(3.7)

which is known as the Moyal star product. This product is noncommutative but it is associative. If the functions f and g vanish rapidly enough at infinity, one can integrate by parts



 d2 x f  g =

 d2 x g  f =

d2 x f g .

(3.8)

Consequently, some trivial Poisson brackets { f (x), g (x)} = 0, after quantization yields nontrivial commutators, which in representation with Moyal star product can be written as





f (x), g (x)  ≡ f (x)  g (x) − g (x)  f (x).

(3.9)

In our case transformations of both fields in (2.25) obtain additional parts, originated from coordinate noncommutativity.

352

D.S. Popovi´c, B. Sazdovi´c / Physics Letters B 683 (2010) 349–353

3.3. Gauge transformations of noncommutative theory Let us consider quantization of open string theory in order to find modified gauge transformations of the background fields B i j and A i . We will use the representation with Moyal star product. The quantum form of fundamental open string symmetry condition (2.13), turns to the relation

δΛ ◦ T ± (σ ) = −i

◦

 ΓΛ , ◦ T ± (σ ) .

(3.10)

Using expressions (2.12) for the energy–momentum tensor components, the last condition imply

 ΓΛ , G i j ,   δΛ ◦ j ±i (σ ) = −i ◦ ΓΛ , ◦ j ±i (σ ) . δΛ G i j = − i

◦

(3.11) (3.12)

For constant metric tensor, G i j = const, the first commutator turns to zero and we are going to solve the second one. With the help of the relations

  δΛ ◦ j ±i = 2δΛ B i j x j − 2δΛ A i δ(σ − π ) − δ(σ ) ,   −i ◦ ΓΛ , ◦ j ±i (σ )      j = 2 ∂i ΛTj − ∂ j ΛiT + 2 ΛTj , A i  δ(σ ) + δ(σ − π ) x

  + 2 ΛiT + D i λ δ(σ − π ) − δ(σ ) ,

(3.13)

δΛ A i (σ ) =

−ΛiT

− D i λ,

(3.14)

(3.15) (3.16)

(3.17)

3.4. Lagrangian of noncommutative electrodynamics The noncommutative electrodynamics is the theory invariant under quantum λ symmetry. As a consequence of the relation (3.16) for Λ T = 0 the noncommutative field strength

(3.18)

has following gauge transformation

δλ F i j = 2(λ  F i j − F i j  λ).

(3.19)

It produces



δλ F i j  F i j = F i j  F i j  λ − λ  F i j  F i j ,

(3.20)

and with the help of (3.8) we can show that the action



S=

dx p +1 F i j  F i j ,

Bˆ i j = B i j + ∂i A j − ∂ j A i − 2[ A i , A j ] .

(4.1)

[ A i , A j ] = i θ kq ∂k A i ∂q A j .

(4.2)

i

[ A i , A j ] = − F ik θ kq F qj , 4

(4.3)

and with help of (4.1)

Consequently, on the quantum level not only λ symmetry (acting on the string endpoints), but also Λ symmetry (acting in the bulk) are deformed to Moyal symmetry. The last term in expression (3.15) is in agreement with Ref. [2]. The corresponding δ functions appears because the vector field A i is boundary valued. Note that according to (3.2) noncommutative parameter θ i j depend on background field B i j and consequently it is changed under Λ transformation. As has been shown in [2] it is not possible to remove noncommutativity by gauge choice.

F i j = ∂i A j − ∂ j A i − 2[ A i , A j ] ,

It is shown in Ref. [1] that Pauli–Villars and point-splitting regularizations lead to theories with ordinary and noncommutative gauge symmetries, respectively. The explanation is in the fact, that theories with different regularizations are related with coupling constant redefinition. In particular case it means redefinition of space–time fields, because they are the coupling constant of the world-sheet Lagrangian. The relation between ordinary and noncommutative descriptions of gauge symmetries is known as Seiberg–Witten map [1]. Let us discus relation between Λ-transformation, defined in relations (3.15) and (3.16), with Seiberg–Witten map. In fact, on the string end points we can chose such gauge transformation Λi = A i , that turns field A i to zero and (3.15) to the expression

For the constant field strength, F i j = ∂i A j − ∂ j A i = const, the λtransformation can be chosen in such a way that A i = − 12 F i j x j . Consequently,

where we introduce Moyal covariant derivative

D i λ ≡ ∂i λ + 2[λ, A i ] .

4. Seiberg–Witten map

On the other hand, definition of the star product up to first order produces

we obtain

 

  δΛ B i j (σ ) = ∂i ΛTj − ∂ j ΛiT + ΛTj , A i  − ΛiT , A j 

× δ(σ ) + δ(σ − π ) ,

is gauge invariant under λ symmetry. It is in agreement with Ref. [1], where the last expression has been obtained in different way.

(3.21)

Bˆ i j = B i j + F i j +

i 2

F ik θ kq F qj ,

(4.4)

which in fact describes Seiberg–Witten map. 5. Conclusion In this Letter using canonical method we derived noncommutative symmetry transformations of the background fields B i j and A i . This approach help us to discussed origin of noncommutative transformations. We used the nice feature of paper [4], which shows how to obtain symmetries of the space–time equations of motion, using conformal invariance of the world-sheet action. In Section 2, we generalize this approach to the open string case. Note that space– time fields are background fields of the σ -model. Classically, the closed string field equations are invariant under symmetry transformation of the Kalb–Ramond field B i j with vector gauge parameter Λi . In order to preserve this symmetry for the open string field equations the vector field A i must be introduced on the string endpoints. The reducible part of the closed string symmetry transformation with parameter Λi = ∂i λ, turns to the irreducible one of the open string symmetry transformation. Both vector field A i and parameter λ live on the string endpoints. On the quantum level, as a consequence of coordinates noncommutativity, new terms appear in the symmetry transformations. They are described in terms of Moyal star product so that both Λ T and λ symmetries are deformed to a Moyal symmetries. The basic relations (3.15) and (3.16) show that, as well as in the classical case, the transformation of B i j depend only on Λ T while the transformation of A i depend on both Λ T and λ.

D.S. Popovi´c, B. Sazdovi´c / Physics Letters B 683 (2010) 349–353

We present the known fact, that the theory invariant under quantum λ-symmetry is a noncommutative electrodynamics. In fact the λ gauge transformation of the field A i , which has the standard form in the classical theory turns to noncommutative form in the quantum theory. Let us comment deformed Λ-symmetry. First, because noncommutative parameter θ i j depend on antisymmetric field B i j , it is not invariant under Λ-transformation. Second, in Section 4 we discuss connection between this modified Λ-symmetry and Seiberg– Witten map. Both of them relate commutative and noncommutative descriptions of the theory obtained by different regularization methods. References [1] N. Seiberg, E. Witten, JHEP 9909 (1999) 032.

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