Complete nonlinear action for supersymmetric multiple D$0$-brane system

We present a complete nonlinear action for the dynamical system of nearly coincident multiple D$0$-branes (mD$0$) which possesses, besides manifest spacetime (target superspace) supersymmetry, also the worldline supersymmetry, a counterpart of the local fermionic $\kappa$-symmetry of single D$0$-brane (Dirichlet superparticle). The action contains an arbitrary non-vanishing function ${\cal M}({\cal H})$ of the relative motion Hamiltonian ${\cal H}$. The $D=10$ mD$0$ model with particular form of ${\cal M}({\cal H})$ can be obtained by dimensional reduction from the action of eleven-dimensional ($D=11$) multiple M-wave (mM$0$) system.


I. INTRODUCTION
Dirichlet p-branes or Dp-branes [42] are the supersymmetric extended objects on which the fundamental D = 10 superstring can have its ends attached [1,2]. Their especially important role in String Theory [3] was appreciated after the famous paper by J. Polchinski [4] where it was shown that they carry nontrivial charges with respect to Ramond-Ramond (RR) fields (see [5] for a comprehensive review).
The worldvolume action for single super-Dp-brane is known [6][7][8][9][10][11][12] to be given by the sum of supersymmetrised Dirac-Born-Infeld (DBI) term and a Wess-Zumino term describing the coupling to RR fields. Both terms contain the field strength of d = (p+ 1) dimensional worldvolume gauge field and in the weak field limit, after fixing the static gauge the first DBI term reduces to the action of the supersymmetric Abelian gauge field theory. Also the Wess-Zumino term in this gauge is expressed through the fields of Abelian super-Yang-Mills multiplet.
The quest for an effective action for the multiple Dpbrane system, i.e. the system of N nearly coincident Dp-branes and strings ending on these Dp-branes, can be followed back to the seminal paper by E. Witten [13] where he argued that the gauge fixed description of its weak field limit is given by the non-Abelian U(N ) super-Yang-Mills (SYM) action. Despite a number of very interesting results obtained during the passed 26 years [17][18][19][20][21][22][23][24][25][26][27][28] the complete nonlinear supersymmetric action for the dynamical system of multiple Dp-branes (mDp) is not known presently even for the simplest case of p = 0 [43].
In this paper we present a nonlinear action which possesses several properties expected from the action of mD0 system. Particularly, it is manifestly invariant under Poincaré symmetry, SU(N) gauge symmetry and spacetime (type IIA target superspace) supersymmetry, and also possesses local worldline supersymmetry generalizing the κ-symmetry of single D0-brane (massive type II D = 10 superparticle) action [44]. This latter fact is especially important because it guarantees that the ground state of this dynamical system is supersymmetric which is expected in the case of multiple D0-brane system.
The rest of the paper is organized as follows. In sec. II we present the complete supersymmetric and nonlinear candidate action for multiple D0-brane system. The rigid spacetime supersymmetry and local worldsheet supersymmetry transformations leaving this action invariant are described in sec. III. The technical details on the derivation of these results can be found in Appendix D which uses the approach and ingredients described in Appendices A-C. Sec. IV contains our conclusions and discussion of the results.

II. SUPERSYMMETRIC NONLINEAR ACTION
The nonlinear action which we have found is written in terms of center of energy variables of mD0 system, which are the same as in the case of single D0-brane, and matrix variables describing the relative motion of mD0 constituents. The set of center of energy variables contains coordinate functions describing the embedding of the center of energy worldline in flat type IIA superspace, bosonic 10-vector and two fermionic Majorana-Weyl spinors µ = 0, ..., 9, α = 1, ..., 16, as well as the spinor moving frame variables which we will describe below. The relative motion variables are matrix fields from the 1d extended (N = 16) SU(N ) SYM multiplet, the set of which can be split on matter fields, 9+9 bosonic and 16 fermionic Hermitean traceless N × N matrix fields i = 1, ..., 9, q = 1, ..., 16, and the bosonic anti-Hermitean traceless N × N matrix 1-form containing the su(N ) valued worldline gauge field A τ (τ ). Besides SU(N ) gauge transformations, the matrix fields are transformed by local SO(9) transformations according to their vector and spinor indices i = 1, ..., 9 and q = 1, ..., 16. These will also act on spinor frame variables and describe the gauge symmetry of the mD0 action. The action has the form where m and µ are constants of dimension of mass and has the meaning of the relative motion Hamiltonian. Actually the first line of (4) formally coincides with the action of single D0-brane, i.e. massive D = 10 type IIA superparticle in its moving frame formulation [27,34] (see below for the description of E 0 in it and Appendix B for some details). In this case m plays the role of the superparticle mass. In contrast, the constant µ characterizes the interaction of the center of energy and relative motion sector as well as the self-interaction of this latter. Notice that to simplify and to make more transparent the dependence of the action on this parameter we have chosen non-canonical dimensions for the matrix matter fields (2). In particular, with this choice of dimensions of matrix fields, the relative motion Hamiltonian H (5) is µ-independent. However its dimension becomes (mass 6 ) so that H/µ 6 is dimensionless. M in (4) is an arbitrary nonvanishing function of this dimensionless combination of the relative motion Hamiltonian and coupling constant, A particular case of the action (4) with can be obtained by dimensional reduction of the 11D multiple M-wave (multiple M0-branes or mM0) system action from [35,36] similar to dimensional reduction of its D = 4 counterpart described in [28]. Another representative of the family (4) with M = m was studied in [27] where it was noticed that it cannot be obtained by dimensional reduction from 11D mM0 action.
Coming back to the first line of (4), in it E 0 is the projection of (the pull-back of) 10D Volkov-Akulov 1form to one of the vector fields, u 0 µ (τ ), of moving frame attached to the worldline. That is described by Lorentz group valued 10×10 matrix composed of the moving frame vectors which obey The spinor moving frame described by Spin (1,9) valued matrix v α q ∈ Spin(1, 9) provides a kind of square root of the above described moving frame in the sense of Cartan-Penrose-like relations (see Appendix A for more details) In distinction to their D = 4 counterparts (described in [37] and e.g. [28]) Eqs. (12) impose strong constraints on the spinor moving frame field v α q = v α q (τ ) reducing the number of its components from the original 16×16=256 to 45 = dim(SO (1,9)).
This spinor frame matrix field v α q (τ ) and its inverse v α q (τ ) are used to construct the fermionic forms E 1q and E q 2 which enter the last term of the action (4), The covariant derivatives in the second line of (4) contain, beside the SU(N ) gauge field (3), also the composite SO(9) connection (Cartan form)

III. LOCAL WORLDLINE SUPERSYMMETRY
The action (4) is manifestly invariant under the rigid super-Poincaré supergroup transformations, including spacetime (target 10D IIA superspace) supersymmetry with constant fermionic parameters ǫ α1 and ǫ α 2 acting nontrivially only on the center of energy variables, It is also invariant under the SU(N ) gauge symmetry acting on the matrix matter fields by its adjoint representation, provided the su(N ) valued 1-form A transforms as SU(N ) connection, as well as under the SO(9) symmetry acting by vector representation on index i of u i µ , X i , P i and by its spinor representation on index q of Ψ q and v q α . Furthermore the action is invariant under local fermionic worldline supersymmetry parametrized by fermionic function κ q (τ ) carrying spinor index of SO (9). It acts on the center of energy variables exactly in the same manner as irreducible κ-symmetry of single D0brane in its spinor moving frame formulation [27,34] (hence notation κ q (τ )), The action of worldline SUSY on the matrix fields includes essentially nonlinear terms some of which are proportional to the derivative of the function M with respect to its argument and, hence to additional power of 1 The worldline supersymmetry transformations of the matrix matter fields are (see Appendix D for their derivation by method described in Appendix C) This latter is related to the worldline supersymmetry where In terms of the above blocks the worldline supersymmetry variation of the SU(N ) connection 1-form (gauge field) can be written as (see Appendix D for its derivation) IV. CONCLUSION AND DISCUSSION Thus, we have found that the action (4) is invariant, besides the manifest spacetime (target superspace type IIA) supersymmetry (18), also under 16-parametric local worldline supersymmetry transformations (19), (21)-(23) and (29). Its counterpart in the case of single p-branes, local fermionic κ-symmetry, is considered as an exclusive property of the supersymmetric extended objects of String/M-theory. It guarantees that the ground state of the dynamical system preserves a part (one-half) of the spacetime supersymmetry.
The form of this worldline supersymmetry depends strongly on the choice of the function M(H/µ 6 ) in the action (4). This is restricted by the requirement of nonsingularity M = 0 but otherwise is arbitrary [45].
The simplest model obtained by setting M = m = const was studied earlier in [27]. In this case M ′ = 0 and worldline supersymmetry transformations of the matrix fields (21)-(23), (29) simplify drastically and provides the local supersymmetry generalization of the rigid d = 1 N = 16 supersymmetry of 10D SU(N ) SYM model reduced to d = 1. The local supersymmetry of the action is provided by coupling of this 1d SYM to the composed worldline supergravity on the worldline induced by the center of energy motion. This is described by 1d graviton 1-form (einbein) E 0 and 16 1d gravitini 1-forms E 1q − E 2 q constructed from the center of energy variables according to (8) and (14).
Thus the nonlinearity of the previously proposed candidate action with M = m =const [27] does not go beyond that of the non-Abelian Yang-Mills. In contrast the action (4) with a generic function M(H/µ 6 ), particularly the one with (7) which can be obtained by dimensional reduction from 11D mM0 action of [35], shows essential nonlinearity beyond the level of SYM one, as it has been expected for the multiple D0-system. It is impressive that such a nonlinearity can be reached with preserving the local worldline supersymmetry characteristic for mD0 system, and that this can be done for essentially arbitrary function M(H/µ 6 ). Also the above mentioned connection with 11D mM0 system, the details of which will be published in a forthcoming paper [38], is another important advantage of the functional (4) as a candidate mD0 action.
The problem of what choice of the function M(H/µ 6 ) leads to the true mD0-brane action requires additional study. A natural way to make this choice through using T-duality (which was the main argument for construction of bosonic actions in [15]) requires as a first step to construct the candidate action for type IIB multiple D1branes (mD1), the problem we are planning to address in the future. A more detailed study of the properties of the model (4) with arbitrary function M(H/µ 6 ), including the solution of its equations of motion and describing its BPS states, can be also useful to single out the true mD0-brane action or to clarify why so big set of models possesses the expected properties.
For a moment, an especially interesting in String/Mtheoretic perspective looks the model (4) with function M(H/µ 6 ) given in (7) because, as we will show in the forthcoming paper [38], this can be obtained by dimensional reduction of the action for multiple M0-brane (mul-tiple M-wave or mM0) constructed in [35]. However, this argument implies the uniqueness of the action [35] as the one having the properties expected for mM0 system. On the other hand, in the light of the found multiplicity of the 10D actions with the properties expected for mD0 system, it is tempting to search for possible essentially nonlinear generalizations of the 11D mM0 action of [35].
Also the generalization of the action (4) for the case of multiple Dp-brane system with 1 < p ≤ 9 and for the case of curved target IIA supergravity superspace are intriguing and important problems. The multiple D0-brane action, presented in the main text, is presently known only in its spinor moving frame formulation involving the auxiliary variables which we are going to describe in some details.
The Spin(1, 9)/Spin(9) spinor moving frame variables and their moving frame vector companions appropriate to the description of D0 brane and multiple D0 (mD0) systems are elements of, respectively, 16×16 and 10×10 matrices (11) and (9) (see [34] and [27]) The condition that moving frame variables form the SO(1, 9) valued matrix implies (10) and The spinor moving frame variables obey the constraints (1,9) Lorentz invariance of the 10D generalization of the relativistic Pauli matrices σ µ αβ = σ µ βα andσ αβ µ =σ βα µ , and also makes the spinor frame matrix to describe double covering of the Lorentz group element represented by the moving frame matrix (see [37,39,40]). Roughly speaking this statement can be formulated by saying that spinor frame variables (also called Lorentz harmonics [37,39,40]) are square roots of the moving frame variables (also called vector harmonics [41]).
Choosing the SO(9) invariant representation where γ i qp = γ i pq are d = 9 gamma matrices, we find that Eqs. (A3) acquire the form of (12) and Similarly, we find is the inverse spinor moving frame matrix v α q ∈ Spin (1,9): The derivatives of the moving frame and of the spinor moving frame variables are expressed in terms of Cartan forms Taking exterior derivatives of Eqs. (A12) (see Appendix C for definitions) we can find the Maurer-Cartan equations Appendix B: Single D0-brane in spinor moving frame formulation and its κ-symmetry The action of the moving frame formulation of the 10D D0-brane in flat type IIA superspace, which also appears as a part of the multiple D0-brane action (4) describing the center of mass dynamics of this system, reads [34] Here d = dτ ∂/∂τ =: dτ ∂ τ , τ is proper time variable parametrizing the D0-brane worldline W 1 defined as a line in target D = 10 type IIA superspace Σ (10|32) with 10 bosonic and 16 + 16 = 32 fermionic coordinates by corresponding coordinate functions The constant m entering both terms of (B1) is the mass of D0-brane and E 0 is the contraction of the pull-back to the worldline of the 10D Volkov-Akulov 1-form with the vector field u 0 µ = u 0 µ (τ ). The pull-back of a differential form on target superspace is obtained by substituting the coordinate functions for coordinates; so that Eq. (B5) actually includes (B7) Notice that, to simplify notation, below and below, as well as in the main text, we use the same symbols for the differential forms on the target superspace and their pull-backs to the worldline W 1 . The same applies to the superspace coordinates (B2) and the coordinate functions (B3). Particularly, in the second term of (B1) θ 1α and θ 2 α denote θ 1α (τ ) and θ 2 α (τ ). A very important property of the action (B1) is that, besides manifest D = 10 N = 2 spacetime supersymmetry, it is also invariant under the following local fermionic κ-symmetry transformations where κ q = κ q (τ ) with q = 1, ..., 16 are arbitrary fermionic functions.
To prove the κ-invariance of the single D0-brane action and also the invariance of multiple D0-brane action under its generalization, the worldline supersymmetry, we have used the formalism of generalized Lie derivatives based on formal exterior derivatives of differential forms which we are going to describe in the next Appendix C.

Appendix C: Differential forms and variations
Let Ξ q be differential q-form in a superspace with coordinates Z M , where ∧ is the exterior product of the differential forms.
In the simplest case of basic 1-forms given by differentials of the superspace coordinates, where ǫ(M ) ≡ ǫ(Z M ) is the so-called Grassmann parity of Z M defined by in the case of D = 10 type IIA superspace with coordinates Z M = (x µ , θ α1 , θ 2 α ). For any bosonic p-and q-forms in particular, In the case of the forms which can be also fermionic In particular, (C2) implies that all products of the supercoordinate differentials are antisymmetric but The exterior derivative of the differential forms, which maps q-forms into (q + 1)-forms, is defined by where ∂ N = ∂ ∂Z N and [...} denotes graded antisymmetrization over the enclosed indices, in particular The exterior derivative operator d obeys the nilpotency condition and the (generalized) Leibniz rule (C8) The variation of differential forms under generic transformations of coordinates can be calculated using the socalled Lie derivative formula, where i δ is the contraction with variation symbol defined by Notice that this implies The contraction i δ maps differential q-forms into (q − 1)-forms and obeys its own counterpart of the Leibnitz rule: The variation of the Lagrangian D-form L of a Ddimensional field theory can be calculated using the Lie derivative formula with formal exterior derivative [46] The total derivative term d(i δ L) is not essential when we derive the equations of motion and can be conventionally omitted if one does not study effects of boundary contributions.
In the models with manifest gauge symmetry it is more convenient to define the variations of differential forms given by covariant Lie derivative where D is covariant derivative including the connection of the gauge symmetry group and A is an index (or multiindex including the index) of a representation of the gauge group carried by the differential q-form. Clearly for the Lagrangian D-form, which is invariant under the gauge symmetry, DL = dL and the covariant Lie derivative prescription coincides with the standard Lie derivative one (C13). As a warm-up exercise let us apply this method to vary the Lagrangian 1-form of the action (B1) of single D0-brane in flat 10D type IIA superspace [34]: The formal exterior derivative of E 0 = Π µ u 0 µ in the first term of the Lagrangian form is given by where (C16) To find that we have used as well as Eqs. (12) and (A12). The derivative of the second, Wess-Zumino term of the D0-brane action is Now after an elementary algebra we find that the formal exterior derivative of the Lagrangian form of single D0-brane can be written as where Ω i is the covariant Cartan form defined in (A11). Then, using the Lie derivative formula (C13), we find where i δ Ω i defines essential variation of the spinor frame variable by δv α q = i δ Dv α q = 1 2 γ i qp v α p i δ Ω i . This equation can be obtained from the i δ contraction of (A14) by setting i δ Ω ij = 0.
To conclude, let us note that in this formalism the local fermionic κ-symmetry transformations δ κ (B8) leaving invariant the D0-brane action (B1) can be described by (i κ d := δ κ ) Indeed substituting the above i κ for i δ in (C20), we find δ κ L D0 = 0.
where K := tr(X i P i ), ν q is defined in (28) and H is the relative motion Hamiltonian (5). The derivatives of these 'blocks', which also enter (D3), read dH = tr P i DP i + 1 16 2. Worldline supersymmetry (κ-symmetry) transformations of the center of energy variables The previous experience with lower-dimensional counterparts of the mD0 system [28] suggests to assume that the worldline supersymmetry acts on the center of energy variables of the mD0 system (i.e. on the superspace coordinate functions and spinor frame variables) as the κ-symmetry of the single D0-brane action (see sec. B) acts on their single-brane counterparts. Namely, we set [47] and