Bootstrapping AdS 2 × S 2 hypermultiplets: hidden four-dimensional conformal symmetry

We bootstrap the 4-point amplitude of N = 2 hypermultiplets in AdS 2 × S 2 at tree-level and for arbitrary external weights. We hereby explicitly demonstrate the existence of a hidden four-dimensional conformal symmetry that was used as an assumption in previous studies to derive this result.


Prologue
Correlation functions of local operators are the most basic and natural observables to study in any (super)conformal field theory.By virtue of the AdS/CFT duality they are dual to on-shell scattering amplitudes in AdS and in the holographic limit these observables are expanded in powers of the inverse central charge.To leading order holographic correlators are given just by generalized free field theory.To extract non-trivial dynamical information we need to consider higher orders in the central charge expansion.The computation of subleading contributions is burdensome from the CFT side owing to the theory being strongly coupled.In the weakly coupled dual description it is possible to perform these calculations, at least in principle.Traditionally one would have to resort to a diagrammatic expansion in AdS.It should be noted, however, that this approach requires the precise knowledge of the effective Lagrangians and due to the proliferation of diagrams and complicated vertices, see for instance [1], it has been rather impractical to use and results were obtained in the early days for a handful of examples [2][3][4][5][6].Furthermore, while this approach is conceptually straightforward, the computations become quickly unwieldy and hence the form of the answer lacks any suggestive structure.
It was only in recent years that we have understood a truly effective approach to compute these holographic correlators and since then we have witnessed a profusion of significant results in these studies in different regimes of the expansion.These new developments are based on a different strategy altogether.In this modern approach, we work directly with the holographic correlators and use superconformal symmetry and other consistency conditions to fix the result.One of the upshots of this bootstrap approach is that it shuns the need of an explicit effective Lagrangian.This method was initiated in [7,8] and led to the complete 4-point functions of 12 -BPS operators with arbitrary Kaluza-Klein (KK) levels at tree-level 1 .This paved the way for an array of very impressive results.From the tree-level correlators we can extract the CFT data for unprotected double-trace operators [12][13][14][15].In turn, we can proceed by considering these as input to obtain results at one-loop [16][17][18][19], subsequently move on to two-loops [20,21], and even use this bootstrap approach to extend the studies beyond the 4-point case [22,23].Not only that, but one can consider stringy corrections to the 4-point correlators [24][25][26][27][28][29][30][31][32].This very beautiful story has been unfolding to different extents in other backgrounds as well.The techniques developed in AdS 5 × S 5 have been used to provide us with a plethora of results in AdS 7 × S 4 [33][34][35][36][37][38] and AdS 4 × S 7 [35,[39][40][41][42][43][44] and AdS 3 × S 3 [45][46][47][48][49][50] supergravities 2 .
Having a panoply of results available allowed the observation of impressive underlying structures in the descriptions of holographic correlators in some specific setups.These structures are hidden in the sense that they are not obvious in any way from the Lagrangian descriptions of the theories.They are interesting not only from a practical point of view allowing one to obtain more compact and suggestive expressions for the correlation functions, but mainly because they are strong indications of new symmetry properties of the bulk theory, hence sharpening our understanding of the theories under examination.These hidden symmetry structures include the Parisi-Sourlas supersymmetry [65], AdS double copy relations [66], and the emergence of a hidden conformal symmetry.The latter was first observed in the context of AdS 5 × S 5 [67] and later in AdS 3 × S 3 [45], and AdS 5 × S 3 [52].
In this work we are interested in hidden conformal symmetry and more specifically its status in the context of AdS 2 × S 2 supergravity.This background arises after a T 7 compactification of M-theory [68] and a further reduction on the S 2 yields gravity-and hypermultiplets [68][69][70].The current state of affairs for AdS 2 × S 2 is the following: in [71] the authors used as a working assumption the existence of a four-dimensional conformal symmetry and managed to derive the unmixed anomalous dimensions of the exchanged double-trace operators.Thereupon, the work of [72] provides evidence for this symmetry being present at the loop-order.
The existence of such a hidden conformal symmetry in a given theory relies heavily on some crucial facts.It is worthwhile stressing that we still lack a formal and thorough understanding of this hidden symmetry, however, we know that its existence simplifies the computations dramatically, and by now we have obtained some intuitive understanding on when to expect that it will be present.Let us briefly review some of the intuition of [67] for AdS 5 × S 5 and see how these statements can be extended to our case of interest.To begin with, the metric of the AdS 5 × S 5 background is conformally equivalent to that of flat space.This is a feature that is common to AdS 5 × S 3 and AdS 3 × S 3 , however it is not true for AdS 4,7 × S 7,4 .Furthermore, the 10-dimensional flat-space amplitude of the type IIB theory contains the dimensionless factor G 10 N δ 16 (Q) that is regarded as a dimensionless coupling.Finally, the 10-dimensional flat-space amplitude of type IIB is conformally invariant and can be considered as the generating function of all KK modes on AdS 5 × S 5 .The AdS 2 × S 2 background draws many similarities with the above.The metric is, in this case as well, that of flat space up to a conformal factor.In addition to that, the G N δ 4 (Q) factor that enters in the expression of the flat-space amplitude is dimensionless in 4 dimensions.And finally, the flat-space amplitude is invariant under the action of the generator of conformal transformations.
The only qualitative difference when comparing to the situation in the AdS 5 × S 5 picture is the existence of two different types of multiplets in our set-up.We have the gravity and hypermultiplets and this is close analogy to the AdS 3 × S 3 that possesses gravity and tensor multiplets.It was observed in [45] that the 4-point function of tensor multiplets enjoys an accidental 6-dimensional conformal symmetry and there is a comment that with the current results in the literature it appears that this will not be true for the gravity multiplet.This, and also the study of [71], leads us to focus on the hypermultiplets in this work.Another is due to the subtleties that arise in a 1-dimensional CFT, one of which concerns the lack of a stress tensor [73].
Extra physics motivation for our work comes from the study of defects in the context of holography.AdS 2 is ubiquitous in the framework of defects and there is much progress to that end with studies of Wilson lines spanning an AdS 2 subsector within the AdS 7 × S 4 [74], AdS 5 × S 5 [75][76][77], and AdS 5 × RP 5 [78] backgrounds.While these setups are not exactly the same as the one we are considering here, they are closely related.
On top of the discussion so far, there is some mathematical motivation as well.More specifically it is interesting to examine how well the position-space bootstrap can work in this simple setup.This is because, while the Mellin-space bootstrap has been very successful in the higher-dimensional cases, in this specific scenario it is not applicable.This is due to the usual problems and complications that arise when trying to define the Mellin transformation in a 1-dimensional theory; for a thorough analysis of these subtleties see [63, section 6] and also [79,80] for related progress to that direction.
On the contrary, the position space bootstrap approach developed in [7,8] can be straightforwardly applied in the case of AdS 2 .In this approach, one has to write an ansatz for the holographic correlator that is a sum of Witten diagrams.To do so, one has to consider the most general selection rules that follow from the structure of the underlying supergravity theory, while having some arbitrary coefficients in the ansatz.These coefficients are then fixed by imposing general consistency conditions on the correlator.Taking all of the above into consideration, in this work we take on ourselves to employ a position-space bootstrap for the computation of the 4-point correlation functions of 1  2 -BPS operators of hypermultiplets in AdS 2 × S 2 for arbitrary external charges.This task is, in some sense, a way to prove the emergence of a hidden 4-dimensional conformal symmetry in this simple setup.To do so, we begin by writing down a general ansatz in terms of contact Witten diagrams with 0-and 2-derivatives corresponding to tree-level supergravity that is consistent with the general selection rules of the AdS 2 × S 2 description.We, then, proceed to impose crossing symmetry, superconformal Ward identities and the bulk-point limit on our ansatz to determine the free coefficients.This fully fixes the answer up to an overall number.Our result agrees with the expectations from hidden conformal symmetry as we explicitly demonstrate.
The structure of this work is as follows: in section 2 we briefly review some basic facts about AdS 2 ×S 2 supergravity, the kinematics of 4-point correlation functions in 1-dimensional CFTs, fermionic Witten diagrams and the flat-space limit in the position-space approach, namely the bulk-point limit.Subsequently in section 3 we demonstrate our algorithm in great detail for the lowest-lying holographic correlator, the ⟨O 1 O 1 O 1 O 1 ⟩.We proceed to the discussion of more general charges in section 4. Section 5 contains a review of some basic statements about hidden conformal symmetry and we explicitly show the agreement of our results with expectations of this hidden structure.We conclude and offer some suggestions for future research in section 6.In appendices A and B we provide the characteristic relations governing D-and D-functions and the explicit form of D-functions used in the case, for the reader's convenience, respectively.

AdS 2 × S 2 supergravity in a nutshell
In this work we are studying correlation functions in a 1-dimensional theory dual to scattering in the AdS 2 × S 2 background.The supergravity Kaluza-Klein spectrum in AdS 2 × S 2 has been obtained in [68][69][70], see also [81] for more recent related work.
More specifically, this background can be derived from 11-dimensional supergravity starting from AdS 2 ×S 2 ×T 7 and reducing the theory on T 7 , while considering only the zero-modes on the torus.In terms of the bulk description, this approximation holds true when the radius of the torus is parametrically smaller than the radii of the AdS 2 and the S 2 with the latter two being equal in this instance.After the T 7 a further compactification on AdS 2 × S 2 yields 4-dimensional, N = 2 supergravity.Upon reduction on S 2 one obtains an infinite tower Kaluza-Klein states that are organised into representations of the su(1, 1|2) superalgebra.
For illustrative purposes, we present the brane-scan of the AdS 2 × S 2 × T 7 theory in table 1: The supersymmetric brane intersection.In the above notation -denotes that a brane extends along that particular direction, while • means that the coordinate is transverse to the brane.
It is worthwhile pointing out that the brane configuration presented in table 1 was originally discovered in [82] as a connection to 4-dimensional black holes.The authors in [82] considered the dimensional reduction of the 11-dimensional supergravity background in type IIA and subsequently performed a T-duality transformation to derive a type IIB supergravity background with differently arranged stacks of D3-branes.
We proceed to describe some basic facts about the spectrum of the theory.We will mainly follow [81].
The matter content of 4-dimensional, N = 2 supergravity contains 1 graviton, 6 gravitinos, 15 vector and 10 (complex) hypermultiplets.The fields in AdS 2 × S 2 are organised in terms of two quantum numbers, h and j, with the former being the lowest eigenvalue of the generator of the SL(2, R) and the latter the relevant number for the SU(2).Hence, an (h, j)-representation has a (2j + 1)-degeneracy from the SU(2) and an infinite tower of states with eigenvalues h, h + 1, h + 2, . . .from the SL(2, R).All fields are organised in chiral multiplets that assume the form: with k in the above taking values k = 1 2 , 1, 3 2 , 2, . ... Note that the case k = 1 2 is special and the final term of equation (2.1) should be understood as the empty representation.The chiral multiplets in equation (2.1) are short multiplets.As it turns out, there is a unique way to organise the matter content of the theory into chiral multiplets as described by equation (2.1): where k = 0, 1, 2, . . . in the above.

Kinematics of four-point functions
We are interested in the 4-point functions of 1 2 -BPS operators in a 1-dimensional CFT.We will briefly review the formalism here following the discussion in [71,72].
We start by noting that it is not possible to define a stress-energy tensor in a 1-dimensional theory, since that would be just a constant.Therefore, we are examining bulk theories with no gravitational degrees of freedom.In these theories, however, we can construct correlation functions as they can be thought of as arising purely from the symmetries of the bulk picture and hence we can formally consider them as correlators of a CFT on the boundary of the space.These 4-point correlation functions admit a large central charge expansion, c.In the large-c limit, the 1  2 -BPS operators are dual to scalars in the bulk AdS 2 that follow from the infinite KK tower of modes on the S 2 .We, furthermore, want to address the lowenergy limit of the theory.In this limit, the theory is 4-dimensional, N = 2 supergravity.However, contrary to [71], we will not deal with sub-leading contributions that come as higher-derivative corrections.To account for this, we introduce a small parameter, α, for which the α → 0 limit is the strict low-energy limit.
Taking the above into consideration, we can define the double expansion to be given by α k−1 c −m , with 2k derivatives in the bulk scalar interaction and k and m being non-negative integers.More explicitly the expansion is The chiral primary fields have protected conformal dimension ∆ and SU(2) representation of spin-j given by j = ∆ with ∆ = 1 2 , 3 2 , 5 2 , . ... To keep a track of the R-symmetry structures it is useful to introduce 2-component polarisation spinors v I such that the chiral primary fields are: where we can set the first component to 1 such that v I = (1, y).
We are interested in the 4-point correlation function of 1 2 -BPS operators that are described by equation (2.4).This 4-point function is a correlator of fermionic primary fields, ψ ∆ , with half-integer conformal dimensions and R-symmetry representations.It is convenient and useful to exchange this fermionic label for a bosonic one by considering the shift, ∆ → p − 1 2 , with p = 1, 2, . ... Having done so, we label the primaries as O p .In this notation the 1  2 -BPS operators have dimensions and R-symmetry representations given by p− 1 2 and the correlator is written as: (2.5) The 4-point correlators of chiral primaries of the theory, equation (2.5), can be written as functions of the conformal and R-symmetry cross-ratios.In a 1-dimensional CFT there is only one conformal cross-ratio given by: that is related to the conformal cross-ratios in higher-dimensional theories via: and by setting we can see that the conformal cross-ratio of the 1-dimensional theory corresponds to the holomorphic limit of the usual cross-ratios from higher-dimensions.We have used the abbreviation x ab for various quantities, which is defined as x ab = x a − x b , unless otherwise stated.
Similarly for SU(2) R-symmetry we can define a cross-ratio y: y = y 12 y 34 y 13 y 24 . (2.9) We note that the x and y can be understood as the bosonic components of the super-Grassmannian Gr(1|1, 2|2) matrix of coordinates [83] that is relevant to the description of the correlation functions in analytic superspace.
Having exchanged fermionic labels in favour of bosonic ones and introduced the appropriate cross-ratios above given by equations (2.6) and (2.9), we are able to re-write the 4-point correlation function in equation (2.5) in the following manner: with and we remind the reader that p ij = p i − p j in equation (2.11).
The fermionic charges of the su(1, 1|2) impose more constraints, which are the superconformal Ward identities.In this notation they assume the form [71]: The solution to the superconformal Ward identities, equation (2.13), yields: where in the above G 0,{p i } denotes the protected piece, R is determined by superconformal symmetry to be: and H {p i } is the reduced correlator that carries the non-trivial dynamical information.
We find it useful to re-write the above solution with all the kinematic factors being restored as: The R in equation (2.16) is related to R given by equation (2.15) in the following way: R = x 13 x 24 y 12 y 34 R , (2.17) and from the above we can see that R is crossing anti-symmetric reflecting properly the fermionic statistics.Finally, we can, also, work out the relation between the interacting parts with and without the kinematic factors.It reads: Before closing this section and to set up concrete conventions, we mention that in this work, we will be assuming that the charges are in ascending order, namely without loss of generality and we distinguish between two cases: Correlators are characterised by their extremality, which we denote by E, and for the two cases that we have distinguished above is given by: These definitions will become useful at a later stage when discussing the bulk-point limit in section 2.4.

Fermionic Witten diagrams
For the purposes of our analysis, a pivotal role is played by contact Witten diagrams.
These are depicted in figure 1: The tree-level contact Witten diagam of external scalars, shown in figure 1c, that carry dimensions ∆ 1 , . . ., ∆ 4 and with no derivatives is represented in terms of the so-called Dfunction, which is given by: with the scalar bulk-to-boundary propagator being equal to: (2.22) In [84] the author studied various classes of Witten diagrams that contain different number of external fermions, see for instance figures 1a and 1b.We are interested in the case  In figure 1c we draw a scalar contact Witten diagram.Note that in the case of fermions, unlike the associated scalar Witten diagrams, there is no t-channel contribution.
that schematically is written as ⟨ ψψ ψψ⟩.The main result related to our purposes here, is that these diagrams are essentially proportional to the associated scalar Witten diagrams.
We briefly review some basic features leading to that conclusion and refer the interested reader to [84, section 2.1] for a more thorough exposition3 .Before proceeding, however, we feel it necessary to make some remarks on the embedding of spinors; see [85,86] for details.
Let us start by passing from the physical 1-dimensional spacetime to the 3-dimensional embedding space.Working in embedding space is very convenient.The action of the special conformal transformations is realised non-linearly by the coordinates, x µ , of the phyiscal spacetime.However, the embedding space coordinates P A transform linearly under the special conformal transformations.We consider M to be a 1-dimensional Euclidean spacetime with metric η µν and we call the embedding space M endowed with the flat metric The embedding of M into M is realised as the null hypersurface P 2 = η AB P A P B = 0. We can introduce the light-cone coordinates as P ± = P 2 ± P 1 , such that we write the 3-dimensional coordinates in embedding space as P ≡ (P µ , P + , P − ) = (x µ , 1, x 2 ).Now, we wish to consider a spinor, ψ(x) in the physical space that is a primary field.
Formally speaking, the spinor representation is Majorana.We can take the γ-matrices to be real and hence the Majorana spinor has real components.For the ψ(x) we can consider a spurionic field s that is position-independent such that we form ψ(x, s) = sψ(x), that is now a spacetime scalar 4 .Working in the same vein, we can form a spacetime scalar starting from a spinor field in the embedding space as Ψ(P, S) = SΨ(P ).The relation between the two spacetime scalars formed out of spinors in the physical and embedding spaces is: while the relation of the polarisation spinors in the two spaces reads5 : (2.24) The 2-point function of the spinors is given by [85,86]: . (2.25) Using equation (2.24) we can obtain (2.26) as expected.Now, we turn our attention to the 4-point function of 4 external fermions with scaling dimensions ∆ i with i = 1, . . ., 4: that arises from contact interactions in AdS.This fermionic 4-point function can be constructed by the fermionic bulk-to-boundary propagators which are given by: where in the above b stands for bulk and ∂ denotes the boundary, and we have introduced the polarisation spinor in the bulk similarly to the discussion above.The answer of the 4-point function contains 2 pieces, the s-and u-channel parts; see figures 1a and 1b For concreteness, we focus on the s-channel part, since the answer for the u-channel follows straightforwardly.We have [84] with the integral being over AdS 2 .Using the identity [84,87] 6 that relates the fermionic bulk-to-boundary propagators to their scalar counterparts we are able to express equation (2.30) as: having defined δ i = ∆ i + 1 2 .Hence, we observe that equation (2.30) is re-written completely in terms of scalar propagators with shifted conformal dimensions; namely a D-function with shifted weights.

The bulk point limit
Another important ingredient that we are going to utilize in our bootstrap algorithm is the flat-space limit of an AdS amplitude as described by the relevant correlator in the dual CFT.In the position-space representation that we have employed in this work, it amounts to considering the so-called bulk-point limit7 .The bulk-point limit in its essence is the statement that if one considers a sufficiently localised AdS wave-packet, one can focus on a point in the bulk.Effectively, by doing so one cannot see any effects of the curvature, and thus this recovers the scattering amplitude in flat-space.We will closely follow [21] in taking the bulk-point limit of AdS amplitudes.
More precisely the bulk-point limit requires to analytically continue from Euclidean to Lorentzian signature, which in terms of cross-ratios amounts to considering an analytic continuation of z around 0 counter-clockwise and z around 1, also counter-clockwise.After that, taking the limit z → z gives a singularity of the schematic form: (2.33)Such a behaviour is the expected one for any holographic correlation function possessing a local bulk-dual description [93][94][95][96].The residue of the singularity is related to the 4dimensional scattering amplitude of hypermultiplets in flat-space, A (4) as: where k, l, m are integers and are related to the dimension of the bulk interaction vertex; see for example [21,94].The parameter z is dimensionless and is defined in terms of the scattering angle θ or in terms of the Mandelstam parameters via: We wish to pause for a moment, in order to make a clarifying comment.In a 1-dimensional theory there is only one conformal cross-ratio, and hence considering the analytic continuation is a bit tricky.However, in this theory we can still use the higher-dimensional prescription described above.Our correlator is a sum made of D-functions, or equivalently D-functions using equation (A.4).The correct thing to do is to first perform the analytic continuations independently in terms of the higher-dimensional cross-ratios, prior to taking the limit z → z.
Since we are dealing with a sum of D-functions essentially, we need to understand what the individual contribution of a given D-function is in the bulk-point limit 8 .In order to take the z → z limit, we recall that any D-function can be uniquely decomposed as: where in the above the various R denote rational functions of z and z and ϕ (1) (z, z) is the well-known scalar one-loop box integral in four dimensions which is evaluated in terms of dilogarithms, see equation (A.8) for its precise form.Upon taking the bulk-point limit in equation (2.36) the contributions of the {log U, log V, 1} are sub-leading compared to the ϕ (1) (z, z) part.Hence, we can write the schematic, but suggestive: In backgrounds of the form AdS × S we, also, have to account properly for the higher KK excitations in the internal manifold.The discussion so far applies to the lowest-lying KK mode.For the higher KK states, the result of the flat-space limit is that of the AdS times a factor that accounts for the KK modes and depends on the polarisations.
To write down the B {p i } factor we need to regroup the spinors appearing in the definition of the chiral primary spinors fields, equation (2.4), in such a way that we form SO(3) null The result is then given by the Wick contractions of the p i − 1 null vectors, t i , of the SO(3) and is proportional to [52]: where t ij = t i • t j = (y ij ) 2 and with the exponents being given by: (2.40) In terms of the R-symmetry cross-ratio equation (2.39) is given by 9 : (2.41) We remind the readers that the meaning of Cases I and II is related to the ordering of the external charges and was spelled out in equation (2.19), while our definition of extremality for these two cases is given by equation (2.20).
3 The simplest bootstrap: the We begin by considering the AdS scattering of the lowest-lying states, the O 1 operators, and making an ansatz in terms of contact Witten diagrams for the In the ansatz we allow for all structures that can appear with 0-and 2-derivatives.The 9 To facilitate the interested reader we note that in the language of [52] their internal cross-ratios σ and τ can be realised in terms of the y in our setup as σ = 1 ansatz is the following10 : In the above, equation (3.1), the first line of Υ i contains all the 0-derivative terms, while the remaining ones are the 2-derivative structures.
Before we proceed to bootstrap equation (3.1) we wish to explain the R-symmetry structures, the factors of y, and how they arise.R-symmetry requires that the polarisation spinors can only appear as polynomials with i and j being particle numbers, and a ij being symmetric, a ij = a ji .Additionally, all the diagonal elements are being given by a ii = 0. Further, the exponents are non-negative, a ij ≥ 0. Furthermore, the a ij need to satisfy The integer solutions to the above constraints give all the R-symmetry structures in equation (3.1) and all subsequent examples we consider in section 4.
Having sufficiently discussed all the terms that enter the ansatz we wrote above for the , we start by counting how many free coefficients enter in the ansatz and we observe that it comes with 36 unfixed parameters.
We are now at a position to implement crossing symmetry.There are 6 ways to cross the correlator, however, only 3 of them are independent.In the ansatz written in terms of the x i , y i and the various D-functions, we consider the conditions: where in the above 1111 is the ansatz when considering crossing 1 ↔ 2 and likewise for the rest.Note, also, that the ansatz is minus itself after crossing reflecting the fermionic statistics.
To implement the crossing conditions, one extracts a kinematic factor to re-write it as a functions of cross-ratios x and y: Furthermore, one needs to use the explicit expressions for the D-functions.Note that in higher dimensions any D-function can be uniquely decomposed in the basis of {ϕ (1) (z, z), log U, log V, 1}.In the 1-dimensional case one further needs to take the limit z = z = x, and the basis reduces to {log x, log(1 − x), 1}.The crossing conditions should hold for the coefficients of each element of the basis and for any values of y.The solution to the crossing equations, provides us with 10 conditions on the free coefficients that we had in the ansatz.Note that these 10 conditions relate the various free parameters amongst themselves.
Having obtained the solutions to the crossing symmetry equations, we wish to examine the implications of the superconformal Ward identities on the correlator.Having already the ansatz in terms of cross-ratios equation (3.5), we can work as we did for crossing symmetry and write the ansatz in the basis spanned by {log x, log(1 − x), 1}, and we wish to impose the superconformal Ward identities given by equation (2.13).Note that the Ward identities should hold for the coefficients of each element of the basis independently.This gives us one and final condition on the undetermined parameters.
After imposing this last condition on top of the previous ones coming from crossing symmetry on our ansatz, we obtain the answer: and hence we have fully fixed the correlator up to an overall number.This result agrees with the one derived in [71].Note that the protected part of the correlator G 0,1111 = 0 which is a non-trivial statement.
cancel.The second one is that it uniquely determines the R-symmetry structures to be given by the B {p i } -factor, given by equation (2.41).The final result is: We note that the protected part of the correlator G 0,2222 = 0 as was the case for the We have checked explicitly that our position-space algorithm agrees with the all examples that are listed in [71, equation.(74)].We have explicitly bootstrapped, in addition to these, In all these examples we find the same structure.In particular the protected part of the correlator vanishes and the final result can be conveniently written as: As we will see, this structure is an implication of the underlying hidden conformal symmetry.

Hidden conformal symmetry
In [7,8] the authors obtained the 4-point function of 1 2 -BPS operators with arbitrary external weights in the 4d, N = 4 super Yang-Mills theory in the limit N → ∞ and λ = g 2 N ≫ 1.This result was obtained by solving an algebraic bootstrap problem.The remarkable simplicity of the formula hinted for some underlying principle governing this structure.In addition to this, there was further suggestive evidence for a hidden conformal symmetry based on the work of [14].This work studied the matrix of anomalous dimensions describing the mixing of double-trace operators constructed from different harmonics in the S 5 .The eigenvalues of the problem are simple rational numbers for which a general formula was obtained.
The status of hidden conformal symmetry was further elaborated and made precise in [67].
The authors conjectured the existence of a 10-dimensional conformal symmetry, in terms of which the 4-point function of all 1  2 -BPS operators can be organized into one generating function.The latter is obtained by promoting the distances in 4 dimensions to 10-dimensional distances in the lowest-weight correlator.While we should mention that to this day we still lack a very rigorous explanation pertaining to the origin of such a symmetry, several intuitive arguments we provided in [67].We briefly review some of the basic facts and then make the connection to the AdS 2 × S 2 background.
We begin with the simple observation that the AdS 5 × S 5 is conformally equivalent to 10dimensional flat space, R 1,9 .However, the SO(2, 10) symmetry can be naturally interpreted as the conformal group in R 1,9 .The same statement can, also, be made for the AdS 2 × S 2 , with the SO(2, 4) symmetry being interpreted as the conformal group of the flat R 1,3 .Furthermore, the type IIB amplitude in flat-space is given by: A ∼ G N δ 16 (Q), the stripped expression is annihilated by the generator of special conformal transformations: Similarly, in AdS 2 × S 2 the flat-space amplitude of hypermultiplets is given by Upon dividing by the dimensionless combination G N δ 4 (Q) which is regarded as the effective coupling in this case, we obtain a stripped amplitude that is invariant under the action of equation (5.2).
A final observation coming from [67] is that the form of the unmixed anomalous dimensions of double-trace operators concurs with the coefficients of the partial-wave decomposition of the 10-dimensional amplitude.We do not investigate the relevant situation in this work, since the counterpart of this reasoning in AdS 2 × S 2 was thoroughly scrutinized in [71].
Owing to the above similarities we proceed to extract the prediction of a hidden SO (2,4) symmetry in the AdS 2 × S 2 .This hidden symmetry is the statement that the lowest-lying correlator, H 1111 = D 1111 , is promoted into a generating function.This is done by replacing the distances in AdS with distances in higher dimensions where x 2 ij − t ij is the conformally invariant distance.The generating function is: To

Epilogue
In this work we have bootstrapped the 4-point correlation function of hypermultiplets in AdS 2 × S 2 supergravity.The approach we undertook relied only on crossing symmetry, the superconformal Ward identities and the bulk-point limit.Having explicitly derived the result for the 4-point function, we proceeded to demonstrate that there is an exact agreement of our approach with the predictions of a hidden 4-dimensional conformal symmetry.In this sense, we have provided a proof for the existence of this underlying structure in this simple setup.
At the level of computing the holographic correlators, the take-home message is that hidden conformal symmetry is equivalent to imposing superconformal symmetry, the constraints of crossing symmetry and the consequences of the flat-space/bulk-point limit.
The answer for the 4-point correlator with arbitrary external weights in AdS 2 × S 2 is given by: with w being a positive number and the t i the null vectors on the S 2 .
There are several fascinating avenues for future work: • It would be very interesting and useful to establish an appropriate formalism of Mellin amplitudes for this setup, perhaps along the lines of work of [79,80].The Mellin approach has been proved to be extremely useful in the higher-dimensional backgrounds, particularly in revealing hidden properties of holographic correlators.
• Owing to the simplicity of this setup, it would also be very desirable to extend the position-space bootstrap to higher-points, see the works [22,23,56,58] for recent progress in bootstrapping high-point correlators in different backgrounds.The simplicity of the answers in AdS 2 × S 2 might be indicative of simplifications and suggestive for how hidden conformal symmetry works in the higher-dimensional cases that are not obvious directly in the higher-dimensional picture.
• We stress, once more, that the approach utilized here can be employed, in addition to some input from string theory, in the study of holographic defects when the codimension surface spans an AdS 2 subspace in the ambient geometry.This has already been exploited very successfully in [74] for defects in the six-dimensional (2, 0) theory.
Extending the logic to other theories, like the ABJM, should be straightforward.
We hope to report to some of these aspects in the near future.this class of special functions is given by

A Properties of D-functions
where in the above K ∆ i (z, x i ) is the bulk-to-boundary propagator.
These are n-point contact Witten diagrams in AdS d+1 without derivatives.Note that we can represent contact diagrams with derivatives as D-functions, also, with shifted weights using that: It is very convenient to re-write to write the D-functions as functions of the conformal cross-ratios.This is achieved by extracting a kinematic factor.For the special case of n = 4, D-functions can be written as D-functions defined by: where we have used the shorthand Σ ∆ to denote the sum of the dimensions.
Another particularly useful parameterisation of D-functions is provided by the use of the Feynman parameter.This leads to: Of course we can construct the holomorphic limit of a D-function with any charges in this way.We refrain, though, from providing more explicit expressions, as the formulae become quite lengthy.
Tree-level contact Witten diagram of external scalars.

Figure 1 :
Figure 1: Tree-level contact Witten diagrams.In figure 1a we depict the s-channel diagram of external fermions, and in figure 1b the relevant u-channel diagram.In figure1cwe draw a scalar contact Witten diagram.Note that in the case of fermions, unlike the associated scalar Witten diagrams, there is no t-channel contribution.

1 )
When we divide the flat-space amplitude given by equation (5.1) by the dimensionless effective coupling, G

. 4 )
This is indicative of a 4-dimensional conformal symmetry.Since the two sub-manifolds of AdS 2 × S 2 are of equal radius, the background can be conformally mapped to the flat R1,3

A
D-function, denoted by D ∆ 1 ...∆n , represents a contact Witten diagram where the external operators have dimensions given by ∆ i .Working in Euclidean AdS d+1 with unit radius and in the Poincaré coordinates
obtain a correlator with general charges H p 1 p 2 p 3 p 4 , we only need perform a Taylor expansion of H(x i , t i ) in powers of t ij , and subsequently collect all the possible monomials 34 D 11nn , H 22nn ∝ (t 12 t 34 + (n − 1)t 13 t 24 + (n − 1)t 14 t 23 ) D 22nn , H nnnn ∝ (t 12 t 34 + t 13 t 24 + t 14 t 23 ) n−1 D nnnn .We can see that the results derived in equation (4.3) agree with the prediction of hidden conformal symmetry given by equation (5.6) and one can check more examples explicitly.
i<j(t ij ) γ ij that can appear in the correlator.There is only a finite number of such monomials for a given H p 1 p 2 p 3 p 4 .We provide some examples below:H 11nn ∝ t n−1