On integrability of D0-brane equations on AdS_4 x CP^3 superbackground

Equations of motion for the D0-brane on AdS_4 x CP^3 superbackground are shown to be classically integrable by extending the argument previously elaborated for the massless superparticle model.


Introduction
Integrable structures have become nowadays the basic objects of study both in AdS 5 /CF T 4 [1] and Aharony-Bergman-Jafferis-Maldacena (ABJM) [2] dualities. In the former case classical integrability of the AdS 5 × S 5 superstring equations was proved in the seminal work [3] based on the AdS 5 × S 5 superstring description as a 2d sigma-model on the P SU(2, 2|4)/(SO(1, 4) × SO(5)) supercoset manifold [4], [5]. In the ABJM case theories conjectured to be dual share lower space-time (super)symmetry so that integrable structure is more difficult to unveil and explore. The gravity dual of ABJM gauge theory just in the special sublimit of the 't Hooft limit reduces to the IIA superstring theory on AdS 4 ×CP 3 superspace that is not isomorphic to a supercoset manifold. Only its subspace of dimension (10|24) can be described as the OSp(4|6)/(SO(1, 3) × U(3)) supermanifold. The OSp(4|6)/(SO(1, 3) × U(3)) sigma-model [6], [7] 2 corresponds to gauging away 8 fermionic coordinates for the supersymmetries broken by the AdS 4 × CP 3 superbackground in the complete superstring action constructed in [9]. In analogy with the AdS 5 × S 5 superstring case the OSp(4|6)/(SO(1, 3) × U(3)) sigma-model equations are classically integrable [6], [7] but integrability of the full set of AdS 4 × CP 3 superstring equations which depend nontrivially on those 8 fermions is by no means obvious. However, in [10] and [11] it was verified perturbatively in such fermionic coordinates up to the second order that this indeed the case.
In the present and companion papers [12], [13] we explore integrability of equations for the point-like dynamical objects in IIA superstring theory on AdS 4 × CP 3 superbackground. In [13] it was proved integrability of the equations of massless superparticle describing dynamics of the superstring zero modes. Here we extend the proof to the case of D0−brane, for which various formulations of the action and κ−symmetry gauge conditions were considered in [14].

Aspects of supergeometry of AdS 4 × CP 3 superspace
In this Section to make the presentation self-contained we sketch the construction of supervielbein and Ramond-Ramond (RR) 1-form on the OSp(4|6)/(SO(1, 3) × U(3)) supercoset manifold and the AdS 4 × CP 3 superspace that are 'building blocks' of the D0−brane action with the emphasis on the relation to D = 3 superconformal symmetry of the ABJM gauge theory [15]. Thorough treatment of the supergeometry of AdS 4 × CP 3 superspace is given in [9].
Using the isomorphism between osp(4|8) superalgebra and D = 3 N = 8 superconformal algebra left-invariant osp(4|8) Cartan forms admit decomposition over the D = 3 N = 8 superconformal generators [21], [22] whereĈ Eq. (11) contains so(2, 3) Cartan forms in conformal basis and Eq. (12) introduces Cartan forms for the so(8) generators in a basis corresponding to the Hopf fibration realization of the 7-sphere S 7 = CP 3 × S 1 . In such a basis one explicitly singles out generators of the su(4) ⊕ u(1) isometry algebra of CP 3 × S 1 . The generators ( V a b , T a , T a ) span the su(4) algebra and commute with the u(1) generator H. Remaining 12 generators ( T a , T a , V a 4 , V 4 a ) belong to the so(8)/(su(4) × u(1)) coset. Commutation relations and transformation to the conventional form of so (8) generators are discussed in [21], [22] and rely on a particular convenient realization [23] for the Kähler 2-form on CP 3 manifold. 3 The advantage of that choice of the Kähler 2-form consists in diagonalization of two projectors [17], [9] used to divide 32 fermionic generators of the osp(4|8) superalgebra (and associated coordinates) into 24 generators of the osp(4|6) superalgebra and 8 generators corresponding to the supersymmetries broken by the AdS 4 × CP 3 superbackground. Thus the first line in (13) contains fermionic generators of D = 3 N = 6 superconformal algebra while the second -the generators of the broken supersymmetries.
Then E m ′ (d) is identified with the tangent to AdS 4 components of the D = 10 supervielbein in KK frame, is the D = 10 dilaton superfield, and A L is the Ramond-Ramond (RR) 1-form potential. Explicit form of the entries of Lorentz rotation matrix is 4 D = 11 supervielbein bosonic components E a (d) and E a (d) in (15) do not depend on dy and thus can be directly identified with the tangent to the CP 3 components of AdS 4 × CP 3 supervielbein in the KK frame, hence they were not endowed with 'hats' in (15). This concludes characterization of the AdS 4 × CP 3 supervielbein bosonic components and RR 1-form potential -the constituents of the D0−brane action. For the analysis of D0−brane equations in analogy with those of the massless superparticle it turns helpful to expand G m ′ (d), E a (d), E a (d) and a(d) on the Z 4 −graded osp(4|6) Cartan forms (2) and dυ [13] and Coefficients at the differentials of the 'broken' fermions dυ represent corresponding AdS 4 ×S 7 supervielbein components while those at the osp(4|6) Cartan forms can be named 'previelbeins'. Due to the choice of the supercoset element (16) they are functions of υ only. To verify integrability of the D0−brane equations it is useful to have explicit expressions for them, as well as for G m ′ y and Φ that can be derived by specifying G br (see [13] for further details).

D0−brane equations and their integrability
The D0−brane action is the sum of kinetic and Wess-Zumino (WZ) terms defined by the world-line pull-back of the RR 1-form potential A L with m measuring its flux. Introducing auxiliary 1d metric the action can be brought to the form without the square root Let us note at this point arbitrariness in the definition of the Lagrange multiplier e(τ ). The choice made in (23) gives conventional mass term em 2 independent of the superspace coordinates. It is also possible to 'absorb' Φ −2 L factor into the definition of e resulting in the action reducing in the m → 0 limit to that of the massless superparticle. The variation of (23) on e produces mass-shell constraint that is irrelevant for establishing integrability of other (dynamical) equations of motion and hence in the subsequent discussion we set e = 1/2, i.e. concentrate on the '1d sigma-model' case.
Considering the osp(4|6)/(so(1, 3) × u(3)) Cartan forms (2) as independent variation parameters allows to derive the set of 10 bosonic and 24 fermionic equations of motion. They can be written as a system of the first order ordinary differential equations with the coefficients given by the world-line pullbacks of the osp(4|6) Cartan forms to facilitate derivation of the Lax representation µc −y bω and In distinction to the superparticle case above introduced quantities take into account the WZ term contribution and the overall factor of Φ −2 L in the kinetic term. Finally there are 8 equations for the 'broken' fermions υ that can be brought to the following form where the shorthand notation was introduced ∂Gτ m ′ ∂υµ = (q m ′ µ ,q m ′ µ , s m ′ µ ,s m ′ µ ), 5 ∂aτ ∂υµ = (h µ ,h µ , p µ ,p µ ) and Analogously are defined ∂aτ ∂υµ and ∂Gτ · Gy ∂υµ . Equations of motion (24), (25) and (29) where each summand is given by the linear combination of the quantities introduced in (26)-(28)

Conclusion
In this note we centred in the proof of the classical integrability of equations of the D0−brane on AdS 4 × CP 3 superbackground. They can be written as the system of first order ordinary differential equations that admit Lax representation with the Lax pair components taking value in the osp(4|6) isometry superalgebra of AdS 4 ×CP 3 superbackground. This generalizes the proof of integrability of the massless superparticle equations given in [13]. However, the most important problem is to find a generalization to the superstring case, i.e. to find a zero curvature representation for the AdS 4 × CP 3 superstring equations. As follows from the discussion in [12] on the relation between the superparticle's Lax pair and the Lax connection of the superstring, the component L of the Lax pair captures the form of the linear in ℓ 2 contribution to the 2d Hodge dualized part of the Lax connection up to insertions of the functions ℓ 1 , ℓ 3 , ℓ 4 of the spectral parameter that is convenient to identify with ℓ 3 . It is thus interesting to find out whether the form of the remaining part of the Lax connection is also determined by the constituents of AdS 4 × S 7 supervielbein bosonic components and 'previelbeins' analogously to L .

Acknowledgements
The author is grateful to A.A. Zheltukhin for stimulating discussions.