Green-Schwarz action for Type IIA strings on $AdS_4\times CP^3$

We present the Green-Schwarz action for Type IIA strings on $AdS_4\times CP^3$. The action is based on a $\Zop_4$ automorphism of the coset $OSp(4|6)/(SO(1,3)\times SU(3)\times U(1))$. The equations of motion admit a representation in terms of a Lax connection, showing that the system is classically integrable.


Introduction
Following the papers of Bagger and Lambert [1] (see also [2]) there has been much interest in N = 8 d = 3 supersymmetric theories. 1 Recently it has been suggested that certain N = 6 D = 2+1 U(N)×U(N) Chern-Simons theories at level k with matter have a dual description as M-theory on AdS 4 × S 7 /Z Z k [4]. We may view the S 7 as a Hopf fibration with the radius of CP 3 proportional to Nk and the radius of the Hopf circle proportional to N 1/6 k −5/6 . The authors of [4] then identify a limit of the N-k parameter space in which the M-theory is described by a weakly coupled string theory on a AdS 4 × CP 3 background together with a four-form RR flux on AdS 4 and a two-form RR flux on CP 1 ⊂ CP 3 . In this note we propose a GS action for type IIA string theory (1.1). We follow the approach developed in a series of papers in the study of GS actions in Minkowski and AdS 5 ×S 5 spacetimes [5]. This approach has proven to be particularly useful in the studies of integrability of string theory; indeed we are able to construct a Lax connection for this action in analogy with the results of [6]. In section 2 we present an explicit realisation of the Z Z 4 automorphism and in section 3 we present the action, discuss κ-symmetry for the model and construct the Lax pair for it; conclusions are presented in section 4.

A Z Z 4 automorphism
The GS action presented in this paper will be based on the super-coset .
The Lie super-algebra can be represented by a super-matrix M written in block-matrix form where A is 4 × 4, B is 4 × 6 C is 6 × 4 and D is 6 × 6. In order to belong to the super-algebra OSp(4|6) we require that M satisfies where the supertranspose of M is given by while the fixed matrix Ω is and In terms of the block matrices this reduces to In particular A is a Sp(4) matrix, D is a SO(6) matrix. We also require the reality condition 8) with C defined in appendix A. The bosonic sub-algebra of M then generates Sp(2, R)×SO(6) ∼ SO(2, 3) × SU(4) (see appendix A for more detail).
Next we define the following automorphism and It is easy to see that G(M) ∈ OSp(4|6) since and so me may verify equation (2.7) explicitly It is also easy to show that which shows that G generates a Z Z 4 action on OSp(4|6). Since G(M) = g −1 Mg this is an automorphism of the Lie superalgebra, in other words Notice also that while iL ∈ SO(6), K ∈ Sp(4), and so g ∈ OSp(4|6); as a result G is an outer automorphism. As shown in appendix A the sub-algebra left invariant by G is 3 GS action for Type IIA string theory on AdS 4 × CP 3 In this section we construct the spacetime supersymmetric action for Type IIA string theory on AdS 4 × CP 3 . We first present the form of the action then discuss κ-symmetry and finally show that the action is supersymmetric.

The action
Let us briefly recall the construction of the GS action on a super-coset G/H. We require that: (i) H be bosonic and, (ii) G admit a Z Z 4 automorphism that leaves H invariant, acts by −1 on the remaining bosonic part of G/H, and by ±i on the fermionic part of G/H. The currents j α = g † ∂ α g can then be decomposed as where j (k) has eigenvalue i k under the Z Z 4 automorphism. The +1 eigenspace is where where A (0) satisfies the reality condition (2.8) and hence spans SO(1, 3) as shown in appendix A. Similarly we may describe the solutions of D = LDL in terms of linear combination of anti-symmetric matrices (M ij ) kl = δ ik δ jl − δ il δ jk . We find the condition D = LDL implies The explicit form of the −1 eigenspace is where with the γ µ4 and S µ defined in Appendix A In terms of the Z Z 4 -graded currents the GS action can be written as When restricted to bosonic fields (by setting j (1) and j (3) = 0) the above action reduces to the sigma model on Sp(4)/SO(1, 3) × SO(6)/SU(3) × U(1), in other words to AdS 4 × CP 3 . By construction this action has 24 supersymmetries, and has local κ symmetry. We take this as strong evidence that this action describes Type IIA string theory on the background (1.1).

κ-symmetry
Actions of the form (3.7) constructed over cosets with a Z Z 4 automorphism typically have κsymmetry. The fermions in this theory live in the 4 × 6 of the Sp(4) × SO (6) and so naively there are 24 real fermionic fields. In flat space and in AdS n × S n κ symmetry can be used to gauge away half of them. In other, less conventional, models it has been recently shown that κ symmetry is trivial on-shell and cannot be used to gauge away any fermionic degrees of freedom [8]. In the present model κ symmetry will allow us to remove 8 real fermionic degrees of freedom -leaving us with 16 physical fermions -the required number for Type IIA string theory on AdS 4 × CP 3 . In order to see that κ symmetry can indeed be used to gauge away 1/3 of the fermions we consider for simplicity the superparticle. 2 In this case the action is simply where e is the worldline metric. Picking the coset representatives as where g ferm (g bos ) is the exponential of a purely fermionic (bosonic) Lie-algebra element, it is a simple exercise (see Appendix B) to expand the above action to quadratic order in fermions. One finds that the two-fermion term is of the form L 2 ferm = dτ eK ai , bj θ ai ∂ τ θ bj = dτ e µ=0,...,3 p µ γ µ4 ac ω cb θ ai ∂ τ θ bi + µ=4,...9 p µ S µ ij ω ab θ ai ∂ τ θ bj , (3.10) where µ = 0, . . . , 3 and µ = 4, . . . , 9 is the spacetime AdS 4 and CP 3 index with p µ the spacetime momentum; a, b, c = 1, . . . , 4 and i, j = 1, . . . , 6 are Sp(2) and SO (6) indices; θ ai are the 24 real fermionic fields in the coset (2.1). The on-shell condition comes from the variation of e in the above action and is simply Str(j (2) τ j (2) τ ) = η µν p µ p ν = 0 . (3.11) We shown in appendix B that on-shell K ai , bj has rank 16. This indicates that κ-symmetry of the action can be used to gauge away 8 real fermions leaving us with 16 physical fermions.

Integrability
The equations of motion that follow from (3.7) are 3 (3.14) As is by now well known these equations follow from the zero curvature condition of a Lax connection [6] (we follow the notation of [7]) The parameters l i are constants which depend on the spectral parameter λ and are An infinite tower of local commuting charges may then be constructed using the monodromy matrix of the above Lax connection in the standard fashion.

Conclusions
In this paper we have constructed the Green-Schwarz action for Type IIA string theory on AdS 4 ×CP 3 . This construction relies on the presence of a Z Z 4 automorphism for the coset (2.1). We have argued that the action has an unconventional form of κ-symmetry which can be used to gauge-fix 1/3 of the fermionic fields. We have also shown that the model is classically integrable. Since the construction is purely algebraic it may also be used to construct the GS action for the background AdS 2 × CP 1 , in this case the coset being OSp(2|3)/(SO(2) × U(1)). We expect that it should be possible to construct the corresponding Berkovits string for this background as well; which presumably will also be integrable. It would be interesting to see the precise form of the ghost sector of that theory. Given the classical integrability of this string theory and the proposed [4] dual descriptions in terms of three-dimensional Chern-Simons theories coupled to matter it seems likely that these dualities may be investigated within the framework developed over the last few years in the study of AdS 5 × S 5 -N = 4 Super-Yang Mills duality [9]; indeed progress in this direction has already begun to appear [11].
Acknowledgements I am grateful to Arkady Tseytlin and Chris Hull for a critical reading of the manuscript. I would also like to thanks John McGreevy for many Bagger-Lambert sessions and Alessandro Torrielli for numerous discussions on integrability. This research is funded by EPSRC and MCOIF.