Abstract
We construct the type IIA Green-Schwarz superstring action on a ten-dimensional IIA plane wave background with 24 supersymmetries keeping the full 32 fermionic coordinates. Starting from the symmetry superalgebra for the maximally supersymmetric eleven dimensional plane wave background, we obtain the eleven dimensional superfields. The Kaluza-Klein reduction leads to the ten dimensional superfields for the IIA plane wave background, from which the type IIA superstring action is constructed. We show that the superstring action reduces correctly to the previously known light-cone gauge fixed action upon imposing the light-cone κ-symmetry fixing condition.
Introduction
Recent years have seen the tremendous development in our understanding of AdS 4 /CFT 3 correspondence. Important inflection point is proposal by J. Schwarz [1] that underlying CFT 3 can be written as Chern-Simons matter (CSM) theories without the usual kinetic term for the gauge fields, especially for higher supersymmetric theories with N ≥ 4. Once that proposal is realized in specific examples, the understanding of AdS 4 /CFT 3 correspondence has grown by leaps and bounds. After the realization that the Bagger-Lambert-Gustavsson theory [2] [3] [4] [5] [6] can be written as the usual SU(2) × SU(2) Chern-Simons matter theory [7] , there appeared a paper by Gaiotto and Witten [8] where the attempt is made to write down N = 4 Chern-Simons matter theory with matter hypermultiplets. The attempt was generalized in [9] which includes twisted hypermultiplets as well, thereby writing down the general classes of N = 4 Chern-Simons matter theories. The special case of such construction is the famous N = 6 theory, known as ABJM theory [10, 11] , describing coincident M2 branes on C 4 /Z k where (k, −k) is the Chern-Simons level for two gauge groups of ABJM theory.
Introduction
If the level k is taken to be infinite, the correspondence becomes that between the ABJM theory and type IIA superstring theory on AdS 4 × CP 3 .
Introduction
For the further study of the AdS 4 /CFT 3 correspondence, the first step in the bulk side is to have the IIA superstring action on AdS 4 × CP 3 , which has been constructed based on the super coset formulation in [12] . However, as pointed out already in the coset formulation and emphasized in a subsequent work [13] , the constructed superstring action contains only 24 fermionic components although type IIA superstring has 32 fermionic components, and thus is not complete. This problem has been cured by incorporating the missing 8 fermionic components through a suitable extension of the coset superspace, and the complete type IIA superstring action on AdS 4 × CP 3 has been constructed in [13] .
Introduction
The complete type IIA superstring action opens up the way of investigating various possible string configurations. However, it has rather complicated structure, which may make the full understanding of string or the quantization of string on the AdS 4 × CP 3 background not so easy. In this case, some simple background would be helpful. One typical and successful example of such simplification may be the Penrose limit of the AdS 5 × S 5 background which leads to the IIB plane wave background [14, 15] . The type IIB superstring action constructed on this IIB plane wave background has turned out to be exactly solvable [16] , and triggered the significant progress in the study of AdS 5 /CFT 4 correspondence [17] .
Introduction
One may expect that a similar simplified theory can be constructed also in the present case. Indeed, it has been shown in [18] [19] [20] that a suitable Penrose limit of the AdS 4 × CP 3 background leads to the previously known IIA plane wave background [21] [22] [23] . Similar to the IIB case, the type IIA superstring theory on this background is also solvable and can be quantized [24] . This manageable situation through the simplification has been exploited in various works on the study of AdS 4 /CFT 3 correspondence [25] [26] [27] [28] [29] [30] . We expect this type of investigation would lead to better understanding of the stringy aspects of AdS 4 /CFT 3 correspondence.
Introduction
In this paper, we revisit the type IIA Green-Schwarz superstring action on the IIA plane wave background and try to construct the complete action containing all the 32 fermionic components. Basically, the motivation comes from the fact that the IIA plane wave background is not maximally supersymmetric. As pointed out in [13] and explored in more detail in [31] , when the superstring action on a less supersymmetric background is considered, the κ-symmetry fixing condition should be chosen carefully in studying a particular motion of string. For example, the type IIA superstring action on AdS 4 × CP 3 constructed based on the super coset does not give the correct description of string moving only in AdS 4 space, because it fixes from the beginning 8 fermionic components corresponding to the broken supersymmetries though those components play important role in describing such string motion. In this sense, the complete action containing all the 32 fermionic components is required for studying various possible string configurations or motions.
Introduction
In the previous construction of type IIA superstring action on IIA plane wave background [21, 22] , the Penrose limit was taken for the membrane action on AdS 4 × S 7 [32] , and the double dimensional reduction was performed after taking the light-cone gauge fixing condition in eleven dimensions for simplicity. We could take the same steps for obtaining the complete action without any gauge fixing. In this paper, however, we will take a different route. Starting from the symmetry superalgebra of eleven dimensional plane wave background, the eleven dimensional superfields will be obtained by following the recipe conceived in [33] . Then, through the Kaluza-Klein (KK) reduction, we will derive the ten dimensional superfields which are necessary in constructing the superstring action. In this way, we obtain another example of the Green-Schwarz superstring action with κ-symmetry for a background with non-maximal supersymmetry, whose supergravity constraints can be completely solved.
Introduction
The organization of this paper is as follows. In the next section, we review the approach of [33] , where one can read off the 10d/11d superfields for a background described by superalgebra and the associated coset structure. In section 3, the superfields for the eleven dimensional plane wave background are obtained starting from the symmetry superalgebra of the background. The KK reduction is performed in Sec. 4, which leads to the ten dimen-sional superfields. In Sec. 5, the complete type IIA superstring action on IIA plane wave background is written down. As a check, we show that the action reduces correctly to the light-cone gauge fixed action constructed previously. Sec. 6 is devoted to conclusion. Our notation is summarized in an appendix.
Preliminary
If underlying geometry can be described by superalgebra and associated supercoset structure, the supergravity constraints can be completely solved. We review this approach by closely following [33] and apply it to 11-dimensional plane wave background in Sec. 3.
Preliminary
The general superalgebra is of the form
Preliminary
where Br and Fâ are the bosonic and fermionic generators, respectively and f are the structure constants. The differential operator is defined to be
Preliminary
where Lr, Lâ are the left-invariant Cartan one-forms. The equation D 2 leads to the usual Mauer-Cartan equation
Preliminary
If the underlying space has the supercoset structrure, the Mauer-Cartan equation can be solved completely. Writing
Preliminary
we find that By differentiating both sides and using the superalgebra, we obtain
Preliminary
With the initial conditionLr λ=0 =Lâ λ=0 = 0, these equations can be solved
Preliminary
where (M 2 )â b = −θĉfĉrâθdfdbr. If we choose θ to be the standard fermion coordinates of the superspace, this gives the superspace geometry in Wess-Zumino gauge.
Preliminary
If the above background describes the 11-d background, the membrane action can be written as
Preliminary
where BÂBĈ represents 3-form superfield, which should be determined separately. Later we will do this for 11-d plane-wave background in section 3.
Preliminary
Once we obtain the membrane action in 11-d then we carry out the double dimensional reduction to obtain the string action in 10-d plane wave background.
Preliminary
3 Superfields for the eleven dimensional plane wave background
Symmetry superalgebra
The eleven dimensional plane wave background [34] is one of the maximally supersymmetric solutions in eleven dimensional supergravity [35] and is given by
Symmetry superalgebra
where
Symmetry superalgebra
Apart from an interesting solution, this background has an important connection with the AdS type backgrounds such as AdS 4 ×S 7 or AdS 7 ×S 4 through the Penrose limit [15] . In this case, the dimensionful parameter µ is inversely proportional to the radius R of the AdS space, µ ∝ 1/R.
Symmetry superalgebra
The details of the convention are summarized at the appendix. Also P and J denote the translation and the rotation generators respectively and
Symmetry superalgebra
Secondly, the algebra between the bosonic and the fermionic generators is
Symmetry superalgebra
where Γ's are 32 × 32 Dirac gamma matrices, 5) and the supersymmetry generator Q with 32 components has been split into two parts by introducing a projection operator P ± as
Symmetry superalgebra
Finally, the algebra of supercharges is
Symmetry superalgebra
We follow the algebra given in [36] but with the notation of [37] for generators. We also take a rescaling of supercharge as Q → Q/ √ 2 for our convenience. 2 Throughout the paper, we suppress the spinor indices unless there is some confusion.
Symmetry superalgebra
where C is the charge conjugation matrix satisfying CΓrC −1 = −Γr T . Here, it seems that the last anticommutation relation is problematic in the limit of µ → 0, which is nothing but the flat spacetime limit as can be seen from the background (3.1). Although it should be reduced to the superalgebra of flat spacetime in this limit, it is apparently divergent.
Symmetry superalgebra
However this is just an artifact of convention. Indeed, as pointed out explicitly in [37] , a
Symmetry superalgebra
proper rescaling of superalgebra generators cures the problem of flat spacetime limit.
Cartan one-forms
The symmetry superalgebra, (3.2), (3.4), and (3.7), is the full superalgebra of the plane wave background (3.1) and one feature of it is that it has the form of the algebraic structure of coset superspace G/H [36] . (The whole generators are those of the group G and the rotation generators correspond to those of the stability subgroup H.) This implies that, by starting from the full superalgebra and following the general prescription suggested in [33, 38] , we can read off the superspace geometry represented by superfields which is our goal in this section.
Cartan one-forms
In the case where the superspace has the coset structure, the superfields are expressed in terms of the left-invariant Cartan one-forms. If we let G be the coset representative, then
Cartan one-forms
gives the left-invariant Cartan one-forms, Lr, Lˆi * , Lˆi ′ * , LÎĴ , L + , and L − , where the spinorial one-forms L ± are the projections of the spinorial one-form L just like Q ± of Eq. (3.6);
Cartan one-forms
We note that Lˆiĵ ′ = 0 because there is no generator Jˆiĵ′ in the superalgebra of (3.2), (3.4), and (3.7). Now, from the integrability of (3.8) and the superalgebra, the left-invariant
Cartan one-forms
Cartan one-forms turn out to satisfy the Maurer-Cartan equations,
Eleven dimensional superfields
The left-invariant Cartan one-forms are functions of the supercoordinate composed of xr and θâ parametrizing the superspace, and each of them has its own expansion in terms of θ. Since their expanded form is eventually necessary in the construction of the superstring action, we now determine them to all orders in θ. In order to do this, we begin with making a particular choice of the coset representative G in (3.8) as
Eleven dimensional superfields
which is known as the Wess-Zumino type parametrization. The bosonic factor g(x) is for the purely bosonic part of the superspace and is left unspecified. The fermionic coordinates in the exponential factor are defined by, using the projection operator in (3.6),
Eleven dimensional superfields
As a next step, we take the rescaling θ → λθ with an auxiliary parameter λ [33, 38] and put a subscript λ for rescaled quantities such as, for example, G λ = G(x, λθ) and
Eleven dimensional superfields
. Then the differentiation with respect to λ of (3.8) with (3.11) and the superalgebra, (3.2), (3.4), and (3.7), lead us to have
Eleven dimensional superfields
As pointed out in [33] , these first-order differential equations have the structure of coupled harmonic oscillators, and thus can be solved exactly. To solve these equations, we first impose the initial conditions for the Cartan one-forms as 14) whereêr is the elfbein andωˆiĵ,ωˆi ′ĵ′ are the spin connections of the plane wave geometry. We note thatωˆiĵ ′ = 0 automatically due to the fact that Lˆiĵ ′ = 0 as alluded to previously. As for the remaining Cartan one-forms, one may be tempted to take Lˆi * λ=0 =ωˆi 0 and Lˆi
Eleven dimensional superfields
from the definition of (3.3). However, this is naive expectation. Indeed, if we compare the purely bosonic part of the Maurer-Cartan equation (3.10) with the usual Cartan structure equation, we can see that the correct initial conditions are
Eleven dimensional superfields
together with a consistency conditionω −r = 0. The initial conditions, (3.14) and (3.15),
Eleven dimensional superfields
form an enough set of data for solving the differential equations of (3.13). Furthermore, they give us an information about non-vanishing derivative at λ = 0 [32] , which is identified with the eleven dimensional covariant derivative for the fermionic coordinate denoted by
Eleven dimensional superfields
If we consider the covariant derivatives for θ + and θ − separately, their explicit expressions are obtained as
Eleven dimensional superfields
Now it is straightforward to solve the equations (3.13) with the initial conditions (3.14)
Eleven dimensional superfields
and (3.15) . After setting λ = 1, that is, L = L λ=1 , we finally have
Eleven dimensional superfields
where Dθ = Dθ + + Dθ − and M 2 is the 32 × 32 matrix given by
Eleven dimensional superfields
The left-invariant Cartan one-forms of (3.18) are the superfields describing the superspace geometry. On the other hand, there is one more ingredient in the superspace for the eleven dimensional plane wave background. It is the three-form superfield B, which forms the Wess-Zumino part of the supermembrane action. Basically, the problem is to find the closed four-form superfield H from a certain combination of various products of the superfields (3.18) and relate it to B through the local equation d B = H. In the present case, there are two possible candidates for H, which areL∧ΓrŝL∧Lr ∧Lŝ and Lr ∧Lŝ ∧Lt ∧LûFrŝtû. 3 We take a linear combination of these terms for H and fix the relative coefficient by requiring the closedness of H, that is, dH = 0. This process is performed by using the Maurer-Cartan equation (3.10) and the eleven dimensional Fierz identity (CΓrŝ) (âb (CΓŝ)ĉd ) = 0, and the resulting expression is obtained as
Eleven dimensional superfields
20) 3 We note that two terms are the same with those for the supermembrane in AdS 4 × S 7 or AdS 7 × S 4 [32] . We may think that this is natural since the plane wave background is the Penrose limit of these two AdS type backgrounds and hence the formal structure of the Wess-Zumino part is expected to be unchanged although the details are different. For more comprehensive study on the Wess-Zumino part of the supermembrane as well as the super fivebrane in various backgrounds, see for example Ref. [39] where the systematic ChevalleyEilenberg cohomology [40] has been used.
Eleven dimensional superfields
where the overall multiplicative constant has been fixed such that it leads to the standard form of the Wess-Zumino term, for example as in [32] .
Eleven dimensional superfields
Having the expression of H, we first apply the trick of rescaling θ → λθ to H; H λ = H(x, λθ). Then the following identity provides the equation for finding B.
Eleven dimensional superfields
By using the differential equations (3.13), we find that
Eleven dimensional superfields
If we plug this into (3.21) and use the local relation d B = H = H λ=1 , then we obtain 23) where the first term on the right hand side has resulted from H λ=0 and Crŝt is three-form gauge field whose field strength is Frŝtû = 4∂ [r Cŝtû ] .
Kaluza-Klein reduction
Having the eleven dimensional superfields for the eleven dimensional plane wave background in hand, we carry out the KK reduction along a spatial isometry direction of the plane wave geometry and obtain the superfields in ten dimensions. Since the Cartan one-forms, Lr and L, and the three-form superfield B are the necessary elements for constructing the IIA superstring action, we will focus only on them.
Ten dimensional bosonic geometry
As we can see from the eleven dimensional plane wave geometry (3.1), there is no explicit spatial isometry direction. However, it has been shown [21, 22] that the geometry can be made to have such an isometry under a suitable coordinate transformation, which is taken by following the prescription suggested in [41] . In our convention, if x 9 is taken to be the desired isometry direction, the transformation is given by
Ten dimensional bosonic geometry
, and x 9 → sin(µx − /6)x 4 + cos(µx − /6)x 9 without changing other coordinates. From the transformed geometry, we may choose the elfbein aŝ
Ten dimensional bosonic geometry
where The above elfbein (4.1) is taken to be a parametrization of the purely bosonic part of the super elfbein, that is, the Cartan one-form Lr of (3.18). For its validity, we would like to note that it satisfies the two consistency conditions of the last section for the spin connection stemming from the symmetry superalgebra, which areωˆiĵ ′ = 0 andω −r = 0 (just below of Eqs. (3.14) and (3.15)). Indeed, through the actual computation of the spin connection with the elfbein (4.1), we see that
Ten dimensional bosonic geometry
and thus our parametrization is valid one.
Ten dimensional bosonic geometry
Following the standard rule of KK reduction, we can directly read off from the elfbein (4.1) the ten dimensional quantities, that is, the zehnbein, the dilaton, and the RamondRamond (R-R) one-form gauge field. In the string frame, the zehnbein is
Ten dimensional bosonic geometry
from which the corresponding non-vanishing spin connection is obtained as
Ten dimensional bosonic geometry
The dilaton is trivially zero, φ = 0, and the field strength of the R-R one-form gauge field is
Ten dimensional bosonic geometry
Together with the R-R four-form field strength F −123 = µ descending from the eleven dimensional plane wave background (3.1), (4.5) and (4.7) form the ten dimensional IIA plane wave background.
Ten dimensional bosonic geometry
As shown in [22, 23] explicitly, the IIA plane wave background is not maximally supersymmetric and preserves 24 supersymmetries. To see this, let us consider the Killing spinor equation for our ten dimensional background,
Ten dimensional bosonic geometry
If we write the 32 component ǫ as ǫ = ǫ + + ǫ − where ǫ ± ≡ P ± ǫ with the P ± of (3.6), and introduce a new projection operator defined by 
Ten dimensional superfields
Now we turn to the KK reduction of eleven dimensional superfields. Similar to the previous bosonic case, there is a condition (the KK ansatz) that the super elfbein Lr should satisfy for the consistent KK reduction. It is L r 9 = 0 regarded as the superspace extension of (4.3) [42] . To check this condition, let us rewrite the eleven dimensional super-covariant derivative Dθ appearing in Lr of (3.18) in terms of the ten dimensional quantities of the previous subsection. Then, from the expression of Dθ given in (3.17), we have
Ten dimensional superfields
where h − is the projection operator defined in (4.9) and Dθ is the ten dimensional supercovariant derivative identified as
Ten dimensional superfields
From (4.11) and the expressions of M 2 andê 9 given in (3.19) and (4.1) respectively, it turns out that Lr in (3.18) has non-vanishing component in x 9 , especially, L r 9 = 0 except for r = −. Thus, the super elfbein does not satisfy the KK ansatz.
Ten dimensional superfields
We would like to note that the obstacle for the dimensional reduction depends on h − θ − (or h − P − θ) which precisely corresponds to the components of eight broken supersymmetries. This is also the case in the construction of the superspace for the 24 supersymmetric AdS 4 × CP 3 background through the dimensional reduction [13] . In some sense, this kind of structural similarity may be expected naturally because the IIA plane wave background is related to AdS 4 × CP 3 via the Penrose limit. We may guess that, if some fraction of the supersymmetry was not broken along the direction of compactification, the super elfbein would not lead to any problem in going down to ten dimensions.
Ten dimensional superfields
For the consistent KK reduction, the non-vanishing L The problem is to determine the transformation matrix Λŝr in such a way that the component E 9 r of Er vanishes, that is, E 9 r = 0. It is not so difficult to solve this. By the aid of the orthogonality condition, 14) and the requirement of proper Lorentz transformation, det Λrŝ = +1, Λŝr is uniquely determined as
Ten dimensional superfields
where we have defined
Ten dimensional superfields
Here L r 9 = 0 and L The transformation matrix S is derived by making use of the relation between the vector and the spinor representations of the Lorentz group,
Ten dimensional superfields
If we take the standard expression S T = exp 1 4 Γrŝǫrŝ with the transformation parameter ǫrŝ and investigate the infinitesimal transformation, it follows that Γ r9 generates the Lorentz transformation for the spinorial quantities. Based on this, we can obtain the explicit expression of S for finite transformation as
Ten dimensional superfields
where 1 32 is the 32 × 32 unit matrix.
Ten dimensional superfields
The transformed super elfbein, Er and Eâ, has the required form for the KK reduction.
Ten dimensional superfields
Thus, we are now ready to get the ten dimensional superfields by following the relation between the eleven and ten dimensional quantities [42] . First of all, the dilaton and dilatino superfield are obtained as
Ten dimensional superfields
We note that, when the eleven dimensional spinorial quantity is related to the ten dimensional one, a dilaton factor e φ/6 should be multiplied for each spinor index such as
Ten dimensional superfields
θâ. This is basically due to the necessity for having the canonical supersymmetry transformation rule in ten dimensions. However, the dilaton field is trivial in the IIA plane wave background, and thus the dilaton factor does not appear in the above expression.
Ten dimensional superfields
As for the super zehnbein, we get
Ten dimensional superfields
where Λ s r and S b a are the Lorentz transformation matrices of (4.15) and (4.19). We note that, although it is clear from the form dZ M L A M , the ten dimensional super-covariant derivative (4.12) is used in super elfbein instead of the eleven dimensional one (4.11) . That is, we use the super elfbein given in (3.18) but with the replacement of Dθ by Dθ. One may think thatê 9 appearing in L
9
M and the relation between the eleven and ten dimensional supercovariant derivatives (4.11) somehow contributes to super zehnbein because we haveê 9 − = 0 from (4.1) which corresponds to the non-vanishing R-R one-form gauge field. However, an explicit calculation shows that such part does not contribute to the super zehnbein. In fact, this should be the case because the super zehnbein is neutral under the gauge transformation associated with the R-R one-form gauge field. Thus, it is understood that, in the actual evaluation of (4.21),ê 9 µ is ignored and the super-covariant derivative is ten dimensional one. We turn to the three-form superfield B. As one can see from its expression (3.23), it is a Lorentz scalar because it does not contain any index in tangent space. Therefore, the effect of the local Lorentz transformation is just to replace the quantities in its expression with the transformed ones, and what we get after the transformation is
9
where the subscript λ means the rescaling θ → λθ in the superfields as in (3.23) andθ ′ is the transformed fermionic coordinate given by
9
with S T λ = S T | θ→λθ . The first term on the right hand side remains intact under the Lorentz transformation because it is originated from the purely bosonic part of the closed four-form 6 We have used SCS T = C, the property of the charge conjugation matrix C under the Lorentz transformation.
9
H, H λ=0 , as can be seen from (3.21) , and the effect of Lorentz transformation with the transformation matrix Λŝr disappears at λ = 0.
7
Under the KK reduction, the three-form superfield gives R-R three-form and NS-NS two-form gauge superfields in ten dimensions. Here, we restrict our attention to the NS-NS superfield because it is relevant in constructing the superstring action. If we denote it as
7
and B is given by
7
where the three-form gauge field Crŝt does not contribute to B because its field strength does not span along x 9 direction in the plane wave background (3.1).
Type IIA superstring action on IIA plane wave background
We have obtained all the ten dimensional superfields necessary for the construction of type IIA superstring action on the IIA plane wave background, and are now ready to write down the action containing all the 32 fermionic components.
Type IIA superstring action on IIA plane wave background
The general form of type IIA Green-Schwarz superstring action is
Type IIA superstring action on IIA plane wave background
where Π A m and B 2 are the pullback of the super zehnbein and the NS-NS two-form gauge superfield onto the string worldsheet given by
Type IIA superstring action on IIA plane wave background
As for the worldsheet quantities, σ m (m = 0, 1) is the worldsheet coordinate with the usual
Type IIA superstring action on IIA plane wave background
7 Here, Λŝr is also understood as the rescaled one through θ → λθ.
Type IIA superstring action on IIA plane wave background
h mn is the worldsheet metric, and the anti-symmetric tensor ǫ mn follows the convention
Type IIA superstring action on IIA plane wave background
If we now plug the expression of (4.24) for the NS-NS two-form superfield into the superstring action, then we have The superstring action is composed of the kinetic and the Wess-Zumino term:
Type IIA superstring action on IIA plane wave background
Let us first consider the kinetic term. Then (4.20) and (4.21) allow us to express it as follows.
Type IIA superstring action on IIA plane wave background
where the expressions of 
Type IIA superstring action on IIA plane wave background
where the wedge product is understood as the pullback version, that is, for example
Type IIA superstring action on IIA plane wave background
Explicit expressions for the various quantities appearing in the action are given by 9) where the ten dimensional super-covariant one form Dθ and the 32 × 32 matrix M 2 are
Type IIA superstring action on IIA plane wave background
given in (4.12) and (3.19) respectively.
Light-cone gauge fixed action
The superstring action (5.5) is a complete action containing all the 32 fermionic coordinates. We now take the fermionic and the bosonic light-cone gauge choices and obtain the superstring action in the light-cone gauge.
Light-cone gauge fixed action
We first fix the fermionic κ-symmetry by taking the usual κ-symmetry light cone gauge
Light-cone gauge fixed action
Under this choice, it follows immediately that
Light-cone gauge fixed action
as can be seen from (5.9). The super-covariant one form Dθ of (4.12) is simplified as
Light-cone gauge fixed action
As for the matrix M 2 of (3.19), it simply vanishes due to the presence of projection operator P − in every terms. This fact leads to a pretty much simplification for the remaining superfields of (5.9) as follows.
Light-cone gauge fixed action
with Dθ of (5.12).
Light-cone gauge fixed action
If we plug the above expressions from (5.11) to (5.13) into the superstring action (5.5) and use the bosonic zehnbein of (4.5), then we obtain the κ-symmetry fixed superstring action as
Light-cone gauge fixed action
We now turn to the bosonic light-cone gauge. The equation of motion for X − is harmonic, and thus the light-cone gauge, X − ∝ τ , is allowed. Let us take the following light-cone gauge choice
Light-cone gauge fixed action
where p − is the total momentum conjugate to X + . With this choice, the worldsheet diffeomorphism can be consistently fixed as √ −h = 1, h στ = 0, which allow us to fix other worldsheet metric components as h τ τ = −1 and h σσ = 1. Then the κ-symmetry fixed superstring action (5.14) is further simplified, and the superstring action in the light-cone gauge, S LC , is obtained finally as 16) where the fermionic coordinate has been rescaled as θ → θ/ √ 2α ′ p − and m defined as m ≡ µα ′ p − is a mass parameter for the worldsheet variables. This is the action considered previously [21, 22] , and thus shows that our complete superstring action of (5.5) satisfies a basic consistency check.
Conclusion
We have constructed the complete type IIA Green-Schwarz superstring action on the ten dimensional IIA plane wave background with 24 supersymmetries. As a consistency check, we have obtained the superstring action in the light-cone gauge and shown that it is exactly the same as that considered previously.
Conclusion
Having the complete action containing all the 32 fermionic components, we can study the various possible superstring configurations by taking an appropriate κ-symmetry fixing condition. Especially interesting thing is the configuration whose correct quantum description requires the fermionic components corresponding to the broken supersymmetries. Another interesting issue that may be considered with the complete superstring action is the realization of worldsheet supersymmetry. As has been shown in [22] , 
A Notation
The eleven dimensional quantities and indices are denoted basically by using hat to distinguish from those in ten dimensions.
A Notation
In The metric ηrŝ for the tangent space-time follows the most plus convention. Since the light-cone coordinate is defined as 

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