Pre-potential in the $AdS_{5}\,\times\,S^{5}$ Type IIB superspace

We found the pre-potential in the superspace with $AdS_{5}\,\times\,S^{5}$ background. The pre-potential appears as part of the vielbeins, without derivatives. The space-cone gauge destroys the bulk Lorentz covariance, but still preserves boundary Lorentz covariance (projective superspace) $SO\,(\,3,\,1\,)\,\otimes\,SO\,(\,4\,)$, i.e. symmetries of boundary CFT are manifest.


Introduction
In the paper [18] we obtained the curvature tensor (previously discovered in [20]) in a manifestly T-dual way. In the paper [19] we extended the techniques for the case of three dimensional N = 2 T-dual extended superspace. There we correctly obtained the pre-potential (as a part of vielbein), structure of linearised dilaton and field equations. The aim of this paper was to look at the full ten dimensional N = 2 T-dually extended superspace in the flat and also in AdS 5 × S 5 background, i.e. IIB string theory expanded around AdS 5 × S 5 background. The AdS was earlier analysed in superspace in papers [14], [15], [16] and [17]. In this paper we discovered the projective (and also the chiral) pre-potential to sit in a certain combination of H S S and H D D . This was first obtained in the flat case and later generalised for the AdS 5 × S 5 . We also performed the near horizon limit and derived the equation of motion for the pre-potential in that limit. This limit also picks out the projective pre-potential instead of the chiral pre-potential, even though both pre-potentials are valid bulk solutions. The projective and harmonic superspaces were earlier analysed in [12] and [13].
As indicated above, the algebra for the right generators is the same up to the overall sign. We can assign the canonical dimensions to the generators: dim (S, D, P, Ω, Σ) = (0, 1 2 , 1, 3 2 , 2). The S generators generate the algebra for SO ( 9, 1 ) ⊗ SO( 9, 1 ), i.e. left and right local Lorentz transformations. The D generate left and right SUSY transformation and P left and right translations. The Ω and Σ are the left and right dual currents (corresponding to D and S), see also [20].
The only nonvanishing terms in the metric and structure constants are (as can be guessed by dimensional analysis): where we have lowered the upper index on f with η to take advantage of its total (graded) antisymmetry, and used "schematic" notation, replacing explicit indices with their type: Explicitly these are, for the left-handed algebra: (η) mn = η mn , (η) mn pq = δ mn pq , (η) σ ρ = δ ρ σ (5) f mn p q = − δ mn pq , f mn pq rs = η [m[p δ q]n] rs , f σρ m = 2 ( γ m ) σρ , f mn σ ρ = − 1 2 ( γ mn ) ρ σ (6) To proceed further we introduce the explicit notation for the left and right indices. We will call the left index to be the one indicated in algebra 2 (i.e. without tilde). The right index will be with tilde: (left) ≡ M ≡ (S mn , D µ , P m , Ω µ , Σ mn ) ≡ (S, D, P, Ω, Σ) and (right) ≡ M ≡ (S mn , D µ , P m , Ω µ , Σ mn ) ≡ ( S, D, P , Ω, Σ). (Note, we abuse the notation for indices a bit, compare with general indices definition in (4). From the context it should be clear which definition we are using).

Gamma matrices
The gamma matrices (γ m ) µ ν used in the algebra (2) are the 16 ⊗ 16 block gamma matrices from 10 dimensional 32 ⊗ 32 chiral representation: Moreover the block gamma matrices satisfy: The IIB fermion generators (in algebra 2) are described by 16 ⊕ 16 chiral fermion generators (for left and right generators) with same 10 dimensional chirality. For the future use we need to look closer at the structure of the matrices (γ m ) µν from equation (7). The equation ( For the future reference we will define the following 16 ⊗ 16 matrix Γ 5 : where γ m are block gamma matrices from (7).

Space -cone basis and indices
Because in the future we will use the space -cone (and light -cone) basis we introduce it for the gamma matrices we constructed in previous subsection. We first notice that the block gamma matrices γ m in equation (7) could have either upper indices (γ m ) µ ν or lower indices (γ m ) µ ν .
From the construction it follows that those matrices are equal up to the sign.
In the above introduced 8 ⊕ 8 basis the (9) looks like σ 3 ⊗ 1 where 1 is the 8 ⊗ 8 unit matrix and σ 3 is the Pauli matrix. exponential of) a super 2-form, while the torsion is a super 3-form; similarly, the Bianchi identities are a super 4-form.
To solve the theory (in terms of pre-potential, to get physical fields and equation of motion) orthogonality condition (15) has to be solved explicitly (or at least at linearised level). Moreover there is a huge gauge group invariance that should be fixed: At the top of the orthogonality condition and the gauge invariance, we should include the torsion constraints. The torsion constraints are additional constraints on vielbein. They are imposed by putting some of the torsions in (16) to zero. Of course, not all torsions in (16) are zero.
The relevant torsion constraints have been carefully analysed in [23]. All possible constraints on torsions are coming from curved space version of the A B C D (first class) constraints: A Virasoro, (string world-sheet) diffeomorphism constraints, B and C and D are the first class fermionic κ symmetry constraints, for further details see [7], [8], [9], [10], [11] and [23]. The rule of thumb is that at least the torsions of negative (10 dimensional) engineering dimension should be zero.
We will not try to solve the full nonlinear version of the theory. We linearise the theory around some background. In previous papers [18], [19] we linearised around the flat background.
In this paper however we linearise the theory around the AdS 5 × S 5 background solution of the classical supergravity (reformulated in the language of the doubled algebra).
After the linearisation we rewrite the orthogonality constraints (15) and torsions (16) using ). Let us for simplicity rename the Then the equation (15) is just statement that: H ( C E ] = 0 and the structure of linearised torsion (16): where

AdS 5 × S 5 background
In the expansion (18) we need to have the concrete structure constants f A B C (i.e. vacuum values of torsions). We are interested in solving the theory (at least identifying the pre-potential) around this AdS 5 × S 5 background. The relevant structure constants for the T-dually extended superspace in the context of the AdS 5 × S 5 background were analysed in the last section of the paper [23]. We are repeating them here for the convenience. In the next section we will embed this AdS 5 × S 5 version of T-dual algebra (see [23]) to a certain larger algebra that will be actually used in computations. The relevant non-vanishing AdS 5 × S 5 torsions from [23] are: dim 0 : T Ω Ω P = R Ω Ω P || T Ω Ω P = R Ω Ω P || T P P Σ = R P P Σ dim 3 : note the left and right index notation introduced below (6).
The f A B C in (19) are usual flat superspace structure constants for the (flat) T-dually extended superspace with the (common) local Lorentz group SO ( 5 ) ⊗ SO ( 4, 1 ). The nontrivial curvatures from table (19): R D D Σ , R P D Ω . . . are defined using the dimension 1 torsion T P D Ω .
The T P D Ω ≡ T a α β = γ a α σ F β σ , where the R-R field strength F Ω Ω ≡ F α β = 1 r AdS ( Γ 5 ) α β . Note that the Γ 5 was defined in (9) and the new parameter r AdS is the AdS 5 space radius (note, r AdS = r S i.e. the radius of S 5 is the same of AdS 5 , so that Ricci scalar R = 0). More specifically some of the table (19) curvatures: dim 1 : where the dim 2 curvature is proportional with the constant 2 − D 2 + 1 and the dim 3 curvature is proportional with a constant D (where D is 10 in our case).
All the other curvatures (i.e. torsions) in (20) are obtained using the appropriate Bianchi identities (one can obtain basically all curvatures from T P D Ω using Bianchi identities). We note that the torsions (19) and curvatures (20) are consistent with torsions and curvatures given in the [5].

Extended AdS 5 × S 5 T-dual algebra
To identify the pre-potential in the generalised vielbeins, i.e. solving the spectrum of the theory (on linearised level) we want to proceed as described in earlier papers [23], [18], [19]. There the vielbeins were introduced as in (12) and linearised as above equation (18). This procedure means the expansion of generally curved superspace around some (in those papers just a flat) background. Moreover the gauge invariance was completely fixed (in referenced papers the covariant gauge was considered) and after that the pre-potential was identified as a part of vielbein (acting on by derivatives, the physical spectrum is produced).
Here we want to proceed in similar way. We want to introduce the vielbeins and linearise the theory around the AdS 5 × S 5 background. We have tried to use solely the algebra described in the previous sub-section, i.e. to take the algebra (19) and introduce the vielbeins, gauge fix and linearise. Even though we still believe that the pre-potential is sitting in that theory in some combination of vielbeins, it was not easy to identify it. The reason was that to identify the pre-potential we need to find a scalar contraction of some linear combination of vielbeins that is anihilated by the D v and Dv operators. The D v and Dv are certain combinations of D α and D α (see the index notation in (4) and above (10), i.e. they are a particular chiral part of the SO ( 8 ) chiral decomposition of the 16 SUSY translations D α defined at the beginning, see (2) and section above (10)).
Because the metric is in H P P vielbein, we expected the pre-potential to be (at least a part of it) in Tr H P P . The problem with Tr H P P is that it already has dimension 0. To show that it is annihilated by D v and Dv operators we would need to use torsion constraints of dimension 1 2 . Moreover we also know that the pre-potential has to be annihilated by the suitably defined P + operator (P + ∝ ( P + + P + ) where P + ≡ D P + and P + ≡ D P + ), in a "light cone" basis introduced in (10) and in the "near horizon limit" (defined later)). The high dimensionality of Tr H P P then requires to use at least dimension 1 torsion constraints to prove that P + vanishes (in the near horizon limit). That seemed to be problematic to analyse in the theory based just on the algebra (19) and (20) because of the degauging procedure. The degauging appears since the theory coming from algebras (19) and (20) is really coming from the full SO ( 10 ) ⊗ SO ( 10 ) so there are some missing Lorentz connections. By simple dimensional analysis its evident that the missing Lorentz connections are first appearing at dimension 1 (for example the appearance of the full H P S in the dim 1 torsion T P P P ∝ . . . + H [ P | S f P P ] Σ + . . . ).
For that reason we extended the algebra (19) to include the original (Wick rotated) local Lorentz group SO ( 10 ) ⊗ SO ( 10 ). All the other structure constants and curvatures in (20) and (19) [22]. Then introduce the curved version of that space via vielbeins and then linearise around some background as described earlier. The extension of (19) and (20)  We still have present the full D S ≡ S generators. Those derivatives could hit the (non-invariant) curvatures. The solution of this is to keep the explicit mixed curvature dependence (as generic mixed curvatures) till the S derivatives are not being explicitly evaluated. We will describe this procedure in detail later.

Gauge fixing
We want to fix the space-cone gauge (T-dual super space-cone gauge) for the first fluctuation H A B , i.e. like in the usual light-cone we have D P − ≡ P − operator invertible, now we have P − and P − invertible (where D P − ≡ P − ).
First we look at the gauge variation (17) more closely and at the linearised level: In the light-cone gauge we in general pick a vielbein with an P − or P − index, put that vielbein to zero. In order to maintain that gauge we need to fix the particular gauge parameter. For simplicity we call P − ≡ − and P − ≡ − then: Note that there are more possibilities to fix the particular gauge parameters λ A . To fix λ A we could also put H −A = 0 and use the invertibility of P − . Of course we can not fix some gauge parameter twice. We have to decide which vielbeins we are going to fix in this "double" light-cone gauge.
We picked the approach where we used the mixed vielbeins to vanish by the double light-cone where we used our mixed light-cone gauge and r AdS → ∞ of extended algebra in (24).
The same as in (23)  Let's take an example, from the equation (23) we can see that in the AdS 5 × S 5 case in the mixed light-cone gauge the H AB is determined as: In many instances in this paper we use similar relation as in (25) to fix some particular vielbein in terms of another vielbeins. If all structure constants f would be SO ( 10 ) ⊗ SO ( 10 ) invariant tensors then there is not a problem and we can treat the f structure constants as genuine constants also with respect to the S derivatives. In our case however the mixed f structure constants (that we call also the curvatures) in (20) explicitly break the SO ( 10 ) ⊗ SO ( 10 ) local Lorentz invariance down to the SO ( 5 ) ⊗ SO ( 5 ) local Lorentz (as it should be in the AdS 5 × S 5 case). One possibility is to restrict our superspace local Lorentz invariance (the S derivatives) to SO ( 5 ) ⊗ SO ( 5 ). Then we would return back to the P SU ( 2, 2 | 4 ) that we wanted to avoid in the first place (in order to have H +a +b instead of H a b ). The alternative, that we picked, is to work with the full SO ( 10 ) ⊗ SO ( 10 ) local Lorentz group. But then the structure constants that are breaking that invariance are not invariant tensors and so the action of those S derivatives on the mixed structure constants has to be accounted for. So we should keep the mixed f structure constants and when needed explicitly act by the S derivatives on them. We will evaluate them as the very last step in our calculations. Let's look at the example in (25) and look at the action of S +a , where a ∈ { 1 . . . 8}: We will evaluate the f B a N in the second equation just after all the (possibly future) S derivatives have already acted. We also should bear in mind that whenever we are acting by the S derivative on some vielbein (that is determined by another vielbeins) there might be the above described issue. The second term can (and it will) nontrivially contribute to our calculations.

Torsion constraints and H S S vielbein
The torsion constraints are (mainly) given by the curved version of the A B C D (first class) constraints, see [7] and [23]. There are further constraints called T A = 0 coming from requirement of partial integration in the presence of the dilaton measure, see [20], and [19]. There is also a strong constraint: on every field in the double field theory one has to require D A D A = 0. There are also a dimensional reduction constraints, as we see later.
Our aim is to analyse the necessary constraints consistent with the above constraints by which we can identify the pre-potential. Following the analysis given in [6], we identify the pre-potential as a scalar super-field (given by some super-trace of possibly a combination of vielbeins), that is annihilated by certain combination of the D ν and D ν . The precise combination of D ν and D ν is also going to be determined from the constraints.
As usual, we start to eliminate the lowest dimensional vielbeins. The vielbeins of the lowest dimension are H S S , H S S , H S S . They are of the dimension − 2 (we mean the ten dimensional dimension). Using equations (23) and (24) for indices A = S and B = S (also after change left ↔ right) we immediately get that H S S = 0 = H S S . Note, that even in the extended We mention an important observation that will help us simplify future calculations. As we saw in previous sub-chapter, we should keep the mixed structure constants generic and evaluate them at the end. Note however, that the mixed structure constants with the S indices are always zero (like the one we considered here: f S AB ). The action of S derivatives on them results in the mixed structure constants again with the S index and such are zero after the evaluation. So specifically, we can evaluate the mixed structure constants with the S indices to zero even before acting by S derivatives on them.
The mixed vielbein H S S is not all zero and the claim is that the part of the pre-potential is in this particular vielbein. To see which parts are possibly nonzero we rewrite the H S S in the double light-cone components: where we might swap left index with the right index in (27). Also note that in all previous we have a ∈ { 1, . . . , 8 }. We remind that P + ≡ + ∝ 10 + 9 and P − ≡ − ∝ 10 − 9.
We want to use analog of equations (23) and (24) for the mixed H S S and r AdS = ∞: In equation (28) we have term H − S zero by gauge choice. The vielbein H S − is proportional to We see that this term is zero after evaluating f S − N = 0 by (20). Using the (23) and (24) for A = S and B = − and keeping f − − S nonzero, we get and similarly for H S − and so the third term in (28) is also fixed. The (28) is then: Using equations (23) All the other components of H S S = 0 by (29) and table (1) after evaluation. This is a first hint that we are on the right track. The H +a +c = 0 is the only nonzero part (after evaluation of mixed structure constants) of H S S piece, it has a scalar trace and we can easily relate it to H a b ≡ H P P , where we expect the part of the pre-potential to be (the symmetric part corresponds to the metric).
To see how H P P is related to H S S consider the third relation from table (1) and after evaluation of mixed structure constants: We want to reduce H a +b further to get H a b . One can naïvely expect to just hit H a +b with P − to get rid of the S index (or alternatively hit by P − the H +a b ). It works but one has to be more careful since in the AdS 5 × S 5 space one has the mixed structure constant To see what is this structure constant (after the mixed structure constants evaluation) we remind that in (20) we saw that dimension 2 structure constant is given as We have to be careful with the indices in the R a b cd . The Σ indices cd in (20)  Moreover, the previous definitions give precise embedding of those operators.
Defining S D ij and S D kl we can see that the structure constant f a b N is either 0 or given by the appropriate R P P Σ D . We included a small subindex to the Σ coordinate just to remind us that the Σ coordinate is now for the SO ( 5 ) ⊗ SO ( 5 ) diagonal subgroup of SO ( 10 ) D only.
Finally, taking definitions of R a b cd and table (20) and our definitions we can see that f a b N is With the previous definition and with P − ≡ − = 1 2 ( 10 − 9 ) and b ∈ { 5, 6, 7, 8, 9 } we will get the [ P − , P b ] ∝ S D 9 b = 1 2 ( S 9 b + S 9 b ) we can also use P 9 = ( P + − P − ) and so [ P − , . Knowing the last relation we can proceed and hit the result of (30) by P − and relate H +a +b with H a b for the evaluated version. For non-evaluated version we need to do the same for the non-evaluated version of (30), that is the third top equation in table (1). For vielbein H +b a needed in this procedure we get: The term H − a vanishes because of mixed light-cone gauge, the term H − +b is fixed by the torsion T − − +b = 0 and use of mixed light-cone gauge similarly as in equation (A.1). By that we get: The last term in (31) is just η − + η a N , the analog term as in (30). The extra mixed term (after Plugging (32) into (31) and then the result (that is the fixed vielbein H +b a ) into the third top equation in table (1) we obtain the non-evaluated relation between H +a +b : after evaluation of the mixed structure constants in (33) we get:

Torsion constraints and H D S vielbein
To identify what combination of vielbeins gives the pre-potential, we first repeat the properties we are looking for. We are looking for combination of vielbeins (of the low dimension), that has a scalar contraction and is annihilated by certain combination of D α and D α (see indices defined above (10)). Moreover the combination has to be annihilated by the properly defined P + operator in the R → 0 limit (still to be defined).
To start, we have one nontrivial hint. We showed that the vielbein H +a +b is nonzero and is related to the H a b . So we can examine what is the action of the D α on H +a +b , i.e. we look at the torsion constraint: In the (36) we can see various terms with the S derivatives. If we could evaluate mixed structure constants before an action of S derivatives, those S terms in (36) would vanish (because the relevant vielbeins are proportional to vanishing mixed constants as we will see). Now they will nontrivially contribute. We note again that we still have the The vielbeins H α +a and H α +b are fixed by torsion constraints T P D S = T − α +a = 0 = T P D S = T − α +b and some few other torsion constraints, as will be shown. We note that as in (23) and (24) we almost always have the strategy to use invertibility of P − and P − together with our mixed left-right light cone gauge to eliminate/fix vielbeins. Sometimes its not enough and we need to explore some further torsion constraints. Let's look at the already mentioned set of torsion constraints: in (37) we used just the mixed light cone gauge and f S P N = 0 (in flat case and also in AdS). To evaluate the last term in (38) that is present only in AdS case we have to take the AdS curvature (20). For our specific indices we have 1 r AdS (γ − ) α ν Γ 5 ν β but the (γ − ) α ν = 0 as we can see in the construction of the light cone basis for the gamma matrices in (10). The vielbein H α −a can be fixed in almost the same set of equations as H α +b . Fixing the vielbein H α +b (and similarly H α −b ) is also similar but a bit more profound. For that we first examine torsion analogous to (35) but for H α +b vielbein: The (40) structure constant f − α N ∝ (γ − ) α ν but as before that particular piece of gamma matrix is zero (remember the non-mixed structure constants are not breaking the SO ( 10 ) ⊗ SO ( 10 ) so we can evaluate them without any concern). The other term in (40) is H − α . That is fixed by the dim 1 2 torsion constraint T − − α = 0: We used in (42) where f and g are defined as follows: Changing left ↔ right in (45) we get the equation for D α H +a +b . There is one simplification we can make in equations (45). Because only half of the block diagonal γ + matrix is nonzero and is proportional to the δ for the nonzero part.
The observation from (45) and its left ↔ right swap is that the action of the D α and D α on H +a +b is producing two new vielbeins H β +a and H β +a . This hints that we need some another vielbein, such that the action of D α and D α on it will effectively subtract the fields H β +a and H β +a . We found such a vielbein, but before giving it we will look at the flat case superspace first to give a motivation. After that we will generalise it to the AdS 5 × S 5 background.

Flat space diagram
To see what could be possibly a missing vielbein that will subtract vielbeins in (45) (and its left ↔ right change) we first solve the same problem in flat space background. That is the extended superspace with r AdS → ∞. Note that in flat superspace (r AdS → ∞) the relation (45) simplifies significantly, because there are no r AdS dependent parts. The surviving part after r AdS → ∞ is just the second term on the right hand side of (45) with g = 1, i.e. 1 2 (γ +a ) α β H β +b .
Let us therefore further examine an action of D α and D α on H β +a and H β +a respectively: The mixed terms in the f part of (47) are zero (note they are zero also in the AdS background).
The structure constant f α β N = 2 (γ a ) α β δ a N (the same is in the AdS background). The vielbein H +a α 0 by the table (2). Then the equation (47) can be rewritten as: To evaluate the only S derivative term in (48) Similarly, we can also look at the equations (45)  and together with (49) and (50) we can observe the following interesting flat space diagram: The scheme (51) is nice and actually tells us what we should do next. Recall that the nodes H +a +b and H c +b and H +a c could be identified by the use of invertible operators P − and P − see (30). Our original aim was find a field that has a scalar trace and could possibly subtract actions of D α and D α on H +a +b . We will see that the missing field is exactly H β β (in the flat space, the only nonzero part of H D D ). The diagram (51) suggests what to do. We calculate the remaining arrows and fill the square.
To fill the remaining arrows we need to calculate the action of D α and D α on H α β together with some another arrows that will be discussed later. We consider the dimension 1 2 torsion constraint T D D D ≡ T α β σ = 0. We note again that for now on we are working in the flat space.
Later we will generalise the procedure for the AdS space: where we used that H α β = H α σ = 0 in flat space (see (A.40) and (A.45) and do flat space limit). We also have a left ↔ right swap of (53). The vielbein H σ c in (53) is related to H +c σ .
For that consider torsion constraint T P S D ≡ T − +c σ = 0: where we used left-right light-cone gauge, together with H − +c = 0 that is shown in the Appendix and holds even in AdS, see (A.20).
To fill the diagram (51) we need to calculate two more torsion constraints that are providing the actions of D α on H +a c and on H σ c . We first consider T D S P ≡ T α +a c = 0: where we used that H α +a = 0 (holds even in the AdS, see table (2)). We also used that H α c = 0 (that is enough in a flat space to have S +a H α c = 0). To see that H α c = 0 we use the torsion T P S D ≡ T − +c α = 0: and previously we saw that H +c α = 0 (even in the AdS case, see table (2)). From (60) in the flat case follows that H α c = 0. Examining the (60) in the AdS case one also finds that H α c = 0 (after evaluation). The (58) however could have some additional term in the AdS case.
The structure constant f α c N = 0 and so the term proportional to that structure constant in the AdS case is 1 r AdS (γ c ) α σ ( Γ 5 ) σ ν H +a ν . That term is nonzero in the AdS case. Moreover, in the (56) one finds one more AdS term, coming from evaluated action S +a H α c . Those terms are not of a big concern right now (doing the flat space first), we will see them later in the section where we generalise to AdS case.
Last torsion constraint to examine in order to fill the (51) is the one that determines the action of D α on H β c . Consider therefore the dimension 1 2 torsion T D D P ≡ T α β c = 0: where we used the H α β = 0 (holds also in AdS after the evaluation) and H c α = 0 (also holds in AdS after the evaluation). In the AdS case in the equation (62) we have two additional terms.
They come from f β c N = 0 and also f α c N = 0. Those terms will be further analysed in future sections, let just write their structure as 1 The vielbein H σ α = 0 (in flat case and also in AdS after the evaluation) as can be calculated from torsion constraints T − σ α = 0 and T − − σ = 0 and the use of the double light-cone gauge.
The term H σ β is nonzero (H σ β vielbein is a part of a pre-potential).
We can add results of (flat space) equations (58) and (62) together with (53) and their left ↔ right swaps to the diagram (51) and find the following square diagram: As we saw before the (63) nodes { H +a +b , H c +b , H +a c , H c c } should be identified (as one node). We proved that using various torsion constraints, mixed light-cone gauge and invertibility of P − and P − . The same way the nodes { H +a β , H c β } and independently nodes { H β +b , H β c } should be identified (as two independent nodes). The vielbein H β β is then just a single node.
After the described identifications the diagram (63) could be rewritten in the simpler and more informative form.
Note, the dashed arrows stand for action of D α and solid arrows stand for action of D α . From the nice flat space diagram (64) it is obvious that in order to have a vanishing derivative we have to combine D α with D α and that combination should act on the combination of traces of H +a +b with H β β .

The H matrix
The diagram (64) could be rewritten in the matrix form. The observation is that each action of the derivatives in the (64) is given by some matrix. The derivatives are mixing fields just as in (64). Let us introduce the 2 ⊗ 2 block matrix H: The action of D α is then given as the left action of some constant (up to P − operator) block off diagonal matrix Γ α : The action of D α on H is given as a right action of similar matrix Γ α : Now we will proceed to the main step. We arbitrarily linearly combine D α and D α , i.e. we multiply the D α with some unknown nonsingular matrix M α β : Moreover we impose that in the matrix version of D α action the matrix M acts as follows: for some nonsingular matrices A and B. Combining (66) and (68) together with (70) and (71) we get: by the Str we mean the super-trace. We put to zero the Str ( is not that simple since Γ α = Γ α . Therefore some more involved solution has to be found.

Solution via the gamma matrix identity
We solve the equation (74) using the following identity: has two SO ( 4 ) invariant solutions:  (78) is defined by the (76) to fix the signs. Note that the indices a in (76) have a range: a ∈ { 1, . . . , 8}.
Next, we look explicitly at the equation: let's rename the members of the matrix H: Now we want the λ and ρ and ∆ to satisfy: The (87) is just a matrix equation:  There is one last piece in the equation (90) that we did not relate to the fields in the H matrix.
The field H σ a . As we saw in the flat case, that field should be related to H σ +a via P − . We have seen however (for example in (34)) that such relations are a bit changed in the AdS case.
Consider the following torsion constraint (and use mixed light-cone and H − +a 0): In the table (3) we derived the relation between H +a ν and H +a ν . That result together with (92) we get: With the equation (93) From the above diagram is obvious that we again have to combine H +a +b and H α β and derivatives D α and D α to get a vanishing derivative on some scalar. r AdS ). We could write those equations in such explicit matrix form, but resulting equations are complicated and unnecessary for our purpose. We instead summarise the right hand side of D α H and D α H using two new matrices X α and Y α respectively. We propose matrix from of the AdS equations: The matrices X α and Y α are fully fixed by (45)

Chiral and projective solutions for AdS 5 × S 5
In the next step we repeat the argument we gave in the flat case section but for the AdS equations (97) and (98). We define: now we act by (99) on H: we multiply by B and apply Str: We will further analyse the structure of (101) in next discussion but before we note one change with respect to (70). In (70) we used B −1 here we are using (yet to be determined) matrix B, the difference is purely conventional. As in the flat case, we want to put the right hand side of (101) to zero and by that obtain vanishing D v α on some scalar field Str ( H B ), that will be called pre-potential. In the flat space it was crucial that we had the identity (76). It was used in the relation (71). Similarly in the AdS case the identity (76) will also be crucial.
In the solution of the vanishing (101) right hand side we still want to maintain the SO ( 4 ) ⊗ SO ( 4 ) invariance. Therefore the B matrix has a block-diagonal form: Let us also simplify the notation for the constants appearing in the equations (45) and (94) and similarly for their left-right conjugates. In (45) we redefine: where the f and g were defined in (46). In (94) we define: With the definitions (102), (103) and (104) let us rewrite the right hand side of (101) explicitly: into the form: The equation (107)  AdS section we ask for the SO ( 4 ) ⊗ SO ( 4 ) invariance. With that requirement we get two branches for the M matrix (actually we get four, as we will see, but the ± is not very important to us): We first notice few nice properties of (109). The solution is actually the same as in the flat case, see (88). We are in the AdS space but the matrix M that combines D α and D α does not depend on the r AdS . Unfortunately we got the same not very nice square root factor in (109). We would need to find some way to deal with it.
Having solved one half of equations (105) and (106). We solve the second half, that is just relation (105): The claim is that given solution M the block matrices b +a +b and b α β are fixed (up to the overall constant). We will again use the same identity (76) as in the flat case. We expect the solutions (we have two branches) will be certain r AdS dependent deformation of the original flat space solutions. We also require to maintain the SO( 4 ) ⊗ SO ( 4 ) invariance of the solution so the most general ansatz for the equation (110) is: Because of later importance we will first solve the ( Γ 5 ) branch of the M solution (109). Later we will also provide solution for the δ branch of the (109). We plug M and (111) into (110) and solve for A, B, C and D using the identity (76), we remind that q := ± P − P − . We get the following solutions: We can again see that as we do r AdS → ∞ limit in (112) we will get the flat solution (88), keeping the D ( or A in δ branch ) r AdS independent in that limit.

Near horizon limit
In the previous section we found the structure of the linearised pre-potential (112) and (113) and also the construction of D v α that vanishes on the pre-potential (99) and (109). We will now introduce the complementary derivative D w α that is constructed after picking the D v α derivative (i.e. picking the matrix M in (109)) and changing the sign in front of the M (the second linearly independent combination). Thus we have: The notation for the upper indices v and w in ( (99) and (114)) comes from equivalent notation for d v and d w derivatives used in [6], (and also for d u and dū, whose analogies are to be defined later). In analogy with the paper [6] we want to define the P + operator that has D v α and D w α as eigenvectors with nonzero eigenvalues. We can solve for P + in full generality, i.e. keeping the non-local square root factors in derivatives D v α and D w α . This would introduce the non-local square root factors also into the definition of P + and would cause further problems. What we will do instead is to restrict the coordinate dependence of the pre-potential V to be just the P SU (2, 2 | 4). This is the same algebra we wanted to use at the beginning of this project, but we were forced to extend it to the full SO (10) ⊗ SO (10) T-dually extended super-algebra. Now, we want to restrict just the coordinate dependence of the pre-potential. Doing so the P − = P − on pre-potential, not everywhere. That is enough to get rid of the non-local factors in D v α and D w α as they act on pre-potential. Then we can redefine (99) and (114) by saying that the new square root free D v and D w to be our new definitions. With this it is easy to see that the good definition of P + is: where the last equality holds on pre-potential.
Following the definitions in [6] of the AdS boundary limit we propose that any operator K which is an eigenvector of P + operator, i.e. [ P + , K ] = c K, scales as R c as we approach the boundary, i.e. R → 0 limit, where R is a radial coordinate on the Poincaré patch. Another way how to state the limit is that by putting the R → 0 we contract the isometry groups SO ( 4, 1 ) and SO ( 4, 1 ) to ISO ( 3, 1 ) and ISO ( 3, 1 ) (we Wick rotated the S 5 isometry group for the purpose of this limit). For more details on this limit (that can be stated also through the explicit coordinates on AdS 5 and S 5 ) see notes [6].
Using the previous definitions of the AdS boundary limit we can analyse the different branches of the D v solutions (109). Let's first pick the δ α β branch (let's work with both ± sub-branches at once). Note that even on pre-potential the D α = D α as can be seen from the explicit construction of those derivatives in [23] in the section 5. Then the commutator is: The . . . part correspond to the current that vanishes in the super-gravity limit (i.e. we do not see string parameter σ) and on pre-potential. We also used the commutators from (2) and the mixed AdS commutators from (20). We also used the explicit solution for the P SU ( 2, 2 | 4 ) (we are on pre-potential) derivatives in terms of τ and σ currents, see section 5 in [23]. More specifically we where the ω α is the current proportional to σ derivative and it has to vanish in the super-gravity limit. The equation (116) is very interesting.
It tells us how the D v scales for the ± δ α β branch of (109). We also notice that the scaling constant is r AdS dependent and vanishes for r AdS → ∞. More importantly because of the ( Γ 5 ) for fixed r AdS and for fixed sub-branch of ± δ α β the scaling constant c is either + ( 1 r AdS ) for one half of SO (8) chiral index α or − ( 1 r AdS ) for second half. And this is not good because by [6] the D v derivative should scale like 1 R and D w should scale like R (put r AdS = 1 for simplicity). In (116) we can see that just 1 2 of derivatives scale properly. This boundary limit then distinguishes between two branches of (109). In the following we will see that the ( Γ 5 ) branch has exactly right scaling properties so it corresponds to the right solution. Without this boundary limit we did not have a way how to pick a branch in (109). In the case of ( Γ 5 ) branch we have one more ( Γ 5 ) matrix in (116) thus we get: The equation (117) will give us the correct solution. From (117) we can see that for fixed r AdS and for fixed ( Γ 5 ) sub-branch we will have proper scaling for full SO ( 8 ) chiral index α . Because we require D v to scale like 1 R and D w scale like R we have D v α = (D α − ( Γ 5 ) α ρ D ρ ) and The positive news is that the blowing-up derivative D v is zero on the pre-potential by our construction, so there is no possible singularity arising as we approach the boundary.
Its easy to see how the derivatives D α and D α scale. Because the (γ + ) α β = 0 the [ P + , D α ] = [ P + , D α ] = 0. So they scale like 1. Those derivatives are building up the D u and Dū, analogous derivatives to paper [6] derivatives d u and dū. The explicit forms of D u and Dū won't be needed in this paper so we do not provide them.

Near horizon limit and field equations
Comparing result with [6] we want to see that the field equations for the pre-potential in the near horizon limit (i.e. in the R → 0) is just of the form P + V = 0 + O( R ). This will be our final confirmation that we discovered the right pre-potential. We first notice that the Lorentz generator scales like O( 1 ), this can be seen from commutator [ S +a , To see what is P + on pre-potential we could directly use some appropriate torsions (remember pre-potential is a linear combination of fields). We found it easier however to use a different approach. Let's look at the torsion constraint (35) but for the α index instead of α (the α index is one of the SO (8) chiral indices): First notice that the structure of (119) is very different than the structure of (35). There is no f term in (119) and there is the full derivative term present. Even in the AdS case the f term is missing. This can be seen as follows. The f +a +b N = 0 in AdS and also in flat case and also f α +b N = 0. The only possibly nonzero f term is coming from f α +a N . The The vielbein H ν +b is zero (also in the AdS) as was shown in the analysis under (35). Next, we can recognise the term H α +a as a part of H matrix (65). The vielbein H α +a has also been analysed in table (3). It is related to H +a α , see table (3). We need to be more careful with that relation because in (119) we again discover the S derivative peculiarity, we saw earlier.
In general all fields in H (now better viewed as their irreducible pieces) could be obtained from the pre-potential V by an action of appropriate (irreducible) combination of D w on the pre-potential. One could analyse in full detail what is the exact structure of those pieces and reproduce famous field content of AdS 5 × S 5 supergravity first discovered in [1] and later used in [2]. This would lead us away from this paper real aim, so we postpone this analysis to next paper. The aim of this section is to show that on pre-potential V the operator P + vanishes in the near horizon limit.
For this reason we notice following expansions: where factors c 0 , c 2 . . . , d 1 , d 3 . . . and e 1 , e 3 . . . are constant factors with appropriate index structure. Note that the c 0 is non-zero. The important observation is that for each term in (120), (121) and (122) we know how it scales in the R → 0 limit, because we know that D w scales like R.
Next, we want to combine (119) with known scalings of all (119) objects to get an information how D α V scales. On one hand it should scale like O (1) on the other hand the relation (119) relates it to different fields. What we obtain is a nontrivial relation that D α V = O ( R ) as we go to the boundary. It just means that D α V = 0 (and so also D α V = 0) in the near horizon limit. Because of the anti-commutator { D α , D β } = 2 (γ − ) α β P + . This is enough to see that P + V ≡ P + V = 0 in the near horizon limit. There are two crucial steps. One is to relate the (119) term S +b H α +a to H +a α . This is relatively straightforward using table (3) and explicit S +b derivative. Second step is to plug expansions (120), (121) and (122) and the scalings of particular pieces into (119). Doing that we get the following:

Conclusion 37
The equation (124) contains all the right expressions to establish the near horizon limit. By the discussion below (117) the D w α derivative scales like O ( R ) (for the projective branch), we also have the scaling of S +a and S +b that goes like a constant. Applying that knowledge we get the equation (124) in the near horizon limit: The c 0 is nonzero constant (tensor) so it follows that D α V = 0 at the AdS boundary. From { D α , D β } = (γ − ) α β P + we get the field equation for the pre-potential in the near horizon limit:

Conclusion
We outline results we have obtained: starting from the 10 dimensional IIB string theory. We embedded the AdS 5 × S 5 background and expanded the theory around this background (we also considered a flat background, i.e. AdS 5 × S 5 with r AdS → ∞ ). Our aim was to obtain (linearised) pre-potential with desired properties in the case of AdS 5 × S 5 (also in the flat case).
We succeeded and obtained pre-potential construction for flat and AdS 5 × S 5 background. We Dū on pre-potential in the near horizon limit. This near horizon limit picks out the projective pre-potential instead of chiral pre-potential (both were obtained as valid bulk solutions).
The vanishing of P + at the boundary fixes the difference between the conformal weights Next, examine: the H − −a is fixed by T P P S ≡ T − − −a = 0: plugging (A.8) into the (A.5) we get: We notice that combining the result (A.10) with (A.3) we get after the evaluation of the mixed structure constants that H −a α 0 and so also H −a α 0. Similar analysis can be made for the rest of the vielbeins (we mean those from table (2), except α switched with α). Thus we get the following table (3): Table 3: H D S vielbein Let us repeat our goal. We wanted to determine the actions of S +b and S +a on H α +a and H +b α respectively. We wanted to do that because then the (36) gives the action of D α on H +a +b (where at least the part of the pre-potential sits). The action of S +b on H α +a is easily computed using our table (2). Taking the second top relation from the table (2) and by explicitly applying the S +b derivative we get: To evaluate S +a H +b α we need to work a bit more. One can directly use the last relation in the table (2). We found an easier way however. For that we need an alternative fixing of the vielbein H +b α . This alternative fixing seems to be more suited for an explicit evaluation of the S +a action (and S −a action). An alternative way how to fix H α +a is to use T P D S ≡ T − α +a instead of one that we used in (40) and (41). Similarly it will be useful to find an alternative fixing for H α −a . Again that could be done by considering torsion T P D S ≡ T − α −a . Let's look at this alternative fixing more closely: The H −a − type of vielbein has been fixed in (A.8). Plugging the fixing into (A.15) we get an alternative H α −a fixing: again we can see the behaviour of the H α −a in (A.17) as we evaluate the theory, as it should be comparing with its behaviour from the fixing in the table (2). The alternative fixing for the vielbein H +b α is calculated similarly: The H +a − is fixed similarly to (A.8) resulting in: The H α a is fixed by the dim 1 2 torsion constraint T P D P ≡ T − α a = 0: The last vielbein we need to fix is the H − α , that is again fixed by the dim 1 2 torsion constraint T P P D ≡ T − − α = 0: Plugging (A.25) into (A.22) and that into (A.19) we finally get an alternative fixing for the H α +a : Now we are ready to calculate an action of S −a and S −a and S +a and S +a on table (2) vielbeins (with the exception of S +a H +b α that we want to calculate in the end of this paragraph). We summarise those S (and S) actions in the next tables: action on H α +a is straightforward. It's done using the relation (A.26) and applying S −b , thus we get: Note, the ± in last line in (A.28) is explained in the section above (31). According to the table (4) all actions of S −b in the second line of (A.28) are evaluated to zero except of the S −b H α +a that we want to determine. Then the (A.28) could be rewritten in a way that determines S −b H α +a (after evaluation): We remind that the H +a vielbein is related to the H +a vielbein by the second top line in the The result is: and after evaluation, where we again use the results from table (4): Finally the action of S +b H α +a is calculated as in (A.27). Now we know that by an analogy with the (A.27) and its evaluation we would need to know analogy of the and the (partially) evaluate version of (A.33): We can see why we just partially evaluated the equation (A.33). The reason is that last term leads to an action of S +b . Fortunately for us we already computed all those actions in tables (5) and (6) except S +b H α +a that we want to calculate. Therefore the (A.34) leads to the evaluated version of the S +b H α +a : where we used results of tables (5) and (6). For completeness we provide (just the evaluated version) the last remaining part of tables (5) and (6), i.e. S +b H α +a : We repeat the first important relation we derived by the above analysis from (35) where we add results from (A.13) and (A.35): where f and g are defined as follows: f := ( 1 − In (A.39) we again used the mixed light-cone gauge. In the flat case the mixed f terms are zero so is H α β . In the AdS case (after evaluation), term proportional to f α − N is zero because of (γ − ) α β = 0. But the term proportional to f β − N is nonzero. Luckily for us the f β − N ∝ 1 r AdS (γ − ) β σ ( Γ 5 ) σ ν η ν N . That structure constant just eats up the β index and returns ν index with some fixed constant dependence. The torsion constraint T P D D ≡ T − σ α = 0 relates H σ α back to H σ α (after the evaluation). From that and assuming some wider invertibility (P − and P − are bigger than some constant lower bound in AdS) we get also in the AdS space H α β 0 (after the evaluation).
We apply S +a on the result of non-evaluated (A.40), thus get: The last term in (A.42) does not bother us too much (it will be a part of the pre-potential), the first term in (A.42) is actually something we need to evaluate. For that we need to fix H α σ .
That could be done by the torsion constraint T P D D ≡ T − α σ = 0: moreover the H − α has been fixed in (A.25), plugging that into (A.44) we get fixing of H α σ : We are ready to calculate S +a H α σ i.e. the term needed in (A.42): Now, we can evaluate (A.46), for clearness we include terms that we already know are evaluated to zero or are zero by the mixed light-cone gauge: where we simplified (A.55) by using the explicit property of γ − and γ + being the unit or zero matrix (depending on specific indices), see (10). We used this simplification in another equations as well (for example in equation (45)). In the equation (A.56) we also used new matrix ( Γ 5 ) d g , that was be introduced in (78). We also used the identity (76) to simplify (A.56). Plugging the evaluated expression (A.56) into (48) we will get the action of D α H β +a : where h is defined as: The equations (A.57) and (A.58) are very interesting since after applying the D α (or D α ) derivatives we are getting terms like H +a c that is basically our original H +a +c , see (30). Moreover we got also term H β σ that is a new term and was important in chapters where we constructed the pre-potential.
Another important derivatives are D α on H β +a and D α on H β +a . We will look at those closer: H +a α is evaluated to 0, see table (2). Evaluating everything in (A.62) we get: