D2 to M2 Procedure for D2-Brane DBI Effective Action

We apply the procedure that was suggested in [arXiv:0806.1639] to the case of abelian D2-brane Dirac-Born-Infeld effective action and discuss its limitation. Then we suggest an alternative form of this procedure that is based on an existence of interpolating action proposed in [hep-th/9707113,hep-th/9707011].


Introduction
It was proposed by Bagger and Lambert in collection of very nice papers [1,2,3] and independently by Gustavsson in [4] 1 following earlier works [5,6] that a certain class of N = 8 super-conformal theories in three dimensions are potential candidates for the worldvolume description of multiple M2-branes in M-theory. These constructions are based on introducing of an algebraic structure known as Lie 3-algebra that is needed for closure of supersymmetry algebra. The metric versions of the above theories fall into two classes that depend on whether the invariant bilinear form in 3-algebra space is positive definite or indefinite. The original theories proposed by Bagger-Lambert are Euclidean theories with positive definite bilinear form while more recent proposals [24,25] contain bilinear form that is indefinite and these Lie 3-algebras are known as Lorentzian 3-algebras.
It was claimed that the Lorentzian 3-algebra theories capture the low-energy worldvolume dynamics of multiple parallel M2-branes. This model has the required classical symmetries, but has several unresolved problems. In particular, the classical theory has ghosts, X ± . Moreover, the ghost-free formulation seems directly equivalent to the nonconformal D2-brane theory as was argued in [39,42,46]. Explicitly, it was shown very clearly in [46] how it is possible-starting from N = 8 SYM-systematically and uniquely recovery the theory [39,42].
when we take non-linear corrections into account. As the first step in this direction we try to apply EMP procedure to the case of single Dirac-Born-Infeld (DBI) action for D2-brane. We start our analysis with the remarkable form of 2 + 1 dimensional action that was proposed long ago in [92,93]. This action is an interpolating action that-after appropriate integration of some world-volume fields-either describes D2-brane DBI effective action in massive Type IIA supergravity or the directly dimensional reduced gauged M2-brane action. We show that in linearized level this action is equivalent to the abelian form of the action given in [46] and hence can be considered as the starting point for non-linear generalization of EMP procedure. On the other hand we argue that naive application of EMP procedure in this action leads to a puzzle. Explicitly, we argue that there is a unique ground state of this new action with infinite coupling constant. This is different from we would expect since M2 to D2-reduction is based on a presumption that the vacuum expectation value of X + can take arbitrary constant value. Equivalently, we would expect an infinite number of ground states that differ by vacuum expectation values of X + .
In order to resolve this problem we suggest that the natural object for the definition of non-linear EMP procedure is gauged M2-brane action. More precisely, it is well known that in the case of IIA supergravity it is possible to introduce non-zero cosmological constant proportional to m 2 with m a mass parameter [94]. Such backgrounds are essential for the existence of D8-branes whose charge is proportional to m [95]. The action for massive 11-dimensional theory has the same contain as the massless one (1.1) The action for these fields is manifestly 11-dimensional Lorentz covariant but it does not correspond to a proper 11-dimensional theory because, in order to write down the action, we need to introduce an auxiliary non-dynamic vector fieldk M such that the Lie derivatives of the metric and 3-form potential with respect to it are zero: An existence of this Killing vector is crucial for definition of massive M2-brane. In fact, the world-volume theory of massive branes 4 was extensively studied in the past, for example [96,97,98,99]. These actions have as a common characteristic that they are gauged sigma models. The gauged isometry is the same as an isometry that is needed in order to define the massive 11-dimensional supergravity theory. For example, the original un-gauged M2brane action is the same object as in the massless theory, i.e. the corresponding massless M2-brane. Let us again return to generalized EMP procedure. We argue that it can be naturally applied for gauged M2-brane action. As opposite to the original EMP procedure, where the Yang-Mills coupling constant vector g I Y M is replaced with a dynamical field X I + we replace the constant Killing vectork M with dynamical fieldX M + . Then we can easily find manifestly covariant form of the generalized action with infinite number of ground states that differ by vacuum expectation values ofX M + . It is remarkable that the gauged isometry that appears in massive M2-brane action is related to the gauge symmetry introduced in [46]. We hope that this observation will allow to find new geometrical interpretations of gauge symmetries that were introduced in [39,42,46].
The organization of this paper is as follows. In the next section (2) we introduce the interpolating D2-brane action and we argue that after appropriate redefinition of worldvolume fields it agrees with the abelian version of D2-brane action introduced in [46]. In section (3) we apply EMP prescription for gauged M2-brane action and we find covariant and non-linear version of M2-brane action that has all desired properties. In conclusion (4) we outline our results and suggest possible extension of our work. Finally, in Appendix A we explicitly show that the dimensional reduction in gauged M2-brane action leads to the interpolating action introduced in section (2).

D2-Brane Action
We start with the action that was proposed in [92,93] and where where g mn , b mn are space-time metric and NS two form field respectively and where C where we used the fact that Then if we insert (2.4) into (2.1) we obtain an action in the form that is standard form of D2-brane in massive Type IIA background and that reduces to the massless Type IIA background when m = 0.
In order to see that the action (2.6) is related to abelian reduction of the action given in [46] we take following background: Further, let us impose static gauge Then in the quadratic approximation the action (2.6) takes the form (2.10) As the next step we introduce the gauge theory coupling constant through the standard relations so that after rescaling the action (2.10) takes the form (2.13) that has the same form as the abelian form of the action given in [46] 5 . Motivated by this result we perform the rescaling (2.12) in the action (2.6) and we obtain where we ignored term that contributes to the action as total derivative. Now we are ready to apply EMP procedure for (2.14). We introduce 8-dimensional vector g I Y M as g I Y M = ( 7 0, . . . , 0, g Y M ), I = 1, . . . , 8, and "covariant derivativeD" (2.15) Further, we rewrite g 2 Y M in manifest SO(8) covariant manner as g 2 Y M = g I Y M g J Y M δ IJ = |g Y M | 2 and then we replace vector g I Y M with dynamical field X I + so that the action (2.14) takes the form where we introduced auxiliary field C µ I that rendersX I + non-dynamical. 5 This is true up to total derivative term since Let us now analyze some properties of the action (2.16). We are mainly interested in the study of the ground state of this theory that has to solve the equations of motion that follow from the action (2.16). We presume that the ground state is characterized by following configuration of the world-volume fields Since for the ansatz (2.17) the matrix A µν is equal to In other words the ground state corresponds to the point v → ∞ that implies that there is unique ground state of the theory. As we argued in introduction this is not the same what we want since we would like to have a theory with infinite number of ground states that differ by vacuum expectation values ofX + . In order to find solution of this problem we suggest an alternative procedure how to introduce X I + as a new dynamical variable. In the next section we present such an alternative procedure that is based on the fact that the action (2.6) can be considered as dimensional reduction of massive M2-brane.

Gauged Theory for M2-brane
Let us again consider the action (2.6) and determine the equations of motion for V µ Inserting (3.1) back to (2.6) we obtain the action in the form

(3.2)
As was shown in [93] (and reviewed in appendix A) this action is very close to the action that one gets by direct dimensional reduction of the massive M2-brane that is also known as gauged M2-brane action. This action can be defined in the background with Killing vector isometryk M (X). Then the gauged M2-bane action takes the form [93] where the covariant derivative D µ is defined as where B µ is world-volume gauge field related to the Killing gauge isometry. To clarify meaning of this gauged form of the action let us consider following transformation where η(ξ) is a parameter of gauge transformations. This transformation immediately implies following transformation rules of background fields and transformation of covariant derivative where we postulate following transformation rule for gauge field B µ δB µ = −∂ µ η . In the same way we obtain that hence we see that the action is invariant under transformations (3.5) and (3.6).
Having clarified the fact that the D2-brane action (2.6) is related to the gauged M2-brane action we now introduce modified EMP procedure to the action (3.3). As the first step in our construction we will presume an existence of adapted system of coordinates wherek M = const. This is always possible to achieve in flat background g M N = η M N ,Ĉ M N K = 0. Further, in analogy with EMP prescription, we replace constant k M with dynamical fieldX M + and add to the action term 1 2 C µ M ∂ µX M + to render this field non-dynamical.
Further we rewrite the Wess-Zumino term in (3.3) as In fact it is easy to see that now the equation of motion for V µ that follow from (3.12) and hence when we insert it back to (3.12) we obtain the last term in (3.3). Note also that this expression is invariant under η transformations (up to total derivative) since In summary we derive the action in the form where we added term 1 2 C M µ G −1 µν η M N ∂ νX N + that rendersX M + constant on-shell and where we also introduced "generalized metric" G µν Following [46] we introduce fieldX M − and add to the action an expression in order the action will be invariant under additional shift symmetry Then the final form of the action takes the form Note also that in order to achieve thatX M + is constant on-shell and that the action possesses additional shift symmetry we can consider more general form of the action that can be finally written in a suggestive form as Let us now study properties of the action (3.20). Clearly it is invariant under shift symmetry (3.17). Further, the variation of this action with respect to C M µ implies that implies ∂ νX M + = 0. Let us again presume the ground state of the theory in the form It is easy to see that the equations of motion forX M , B µ , C M µ andX M − are obeyed for this ansatz. Finally, the problematic equation of motion forX M + takes the form that is clearly solved by (3.22) for any v M . Finally, let us impose the static gauge in the following formX µ = ξ µ , µ , ν = 0, 1, 2 so that the matrix A µν takes the form Then the action up to quadratic approximation can be written as where the Lagrangian density takes the form that is again very close to the abelian form of the action given in [46] and provides further support of our construction.

Conclusion
Let us summarize our results. We studied EMP procedure for Dirac-Born-Infeld action for D2-brane and we found its limitation. Then we suggested an alternative form of this procedure that is based on a formulation of gauged M2-brane action. This fact however implies that the theory should be defined in background with non-zero mass parameter m and this observation certainly deserves better understanding and more detailed study. Further, it would be also interesting to develop BRST Hamiltonian treatment of the action (3.20) and compare it with the similar analysis that was given in [24]. Finally, it will be extremely interesting to see whether there exists an non-abelian extension of the action (3.20). We hope to return to these problems in future.

Acknowledgement
This work was supported by the Czech Ministry of Education under Contract No. MSM 0021622409.

A. Direct Dimensional Reduction for Massive M2-brane
In this appendix we show that the gauged M2-brane action upon direct dimensional reduction in the direction X associated to the gauged isometry reduces to the action (3.2). To begin with we choose coordinates that are adapted to the isometry so thatk M = δ M x and we split eleven coordinatesX M into the ten 10-dimensional X m , m = 0, . . . , 9 and the extra scalarX x ≡ X. Using the relations between the 11-dimensional and 10-dimensional fieldsĝ xx = e  it is straightforward to see that Then if we insert (A.2) together with (A.1) into the action (3.3) we easily obtain that it reduces to the action (3.2).