Supersymmetric Field Theories on AdS_p x S^q

In this paper we study supersymmetric field theories on an AdS_p x S^q space-time that preserves their full supersymmetry. This is an interesting example of supersymmetry on a non-compact curved space. The supersymmetry algebra on such a space is a (p-1)-dimensional superconformal algebra, and we classify all possible algebras that can arise for p>= 3. In some AdS_3 cases more than one superconformal algebra can arise from the same field theory. We discuss in detail the special case of four dimensional field theories with N=1 and N=2 supersymmetry on AdS_3 x S^1.


Introduction
In the last few years, it has been useful to study supersymmetric field theories on curved manifolds. In many cases exact results in such backgrounds (when they preserve supersymmetry) may be computed (following [1]), including partition functions and some expectation values. In some cases these contain information beyond the usual exactly-computable results in flat space. This allows us to learn more about these field theories, even when they are strongly coupled, providing new windows into strongly coupled field theories. So far, this study has been almost exclusively limited to compact curved manifolds. In this paper we take first steps towards a systematic study of supersymmetric field theories on non-compact curved space-times, by looking at the specific example of d-dimensional supersymmetric (SUSY) field theories on AdS d−q × S q . This example is maximally symmetric, and, for an appropriate definition of the theory and of the boundary conditions, can often preserve the full supersymmetry of the d-dimensional field theory. Another reason for being interested in this specific example is that field theories on AdS d−q × S q often arise on various branes and singularities in string theories on anti-de Sitter (AdS) space, and in some cases they can be decoupled from the full gravitational theory. In a specific example, the 6d N = (2, 0) superconformal field theory on AdS 5 ×S 1 was recently embedded in string theory [2], and this led to surprising results. Four dimensional supersymmetric theories on AdS 4 were studied in detail in the past, for instance in [3][4][5][6][7][8][9][10][11][12] (this example is particularly interesting since it is related by a conformal transformation to four dimensional theories on a half-space). Supersymmetric theories on AdS 3,4,5 were studied in [13][14][15][16][17][18][19][20][21][22], on AdS 5 in [23], and on AdS d and AdS p × S p in [24]. Other examples did not attract much attention. Another example worth mentioning is the N = 4 super-Yang-Mills theory on AdS 3 × S 1 , that can be embedded in string theory by considering D3-branes on AdS 3 × S 1 in type IIB string theory on AdS 5 × S 5 .
We limit our study to anti-de Sitter spaces of at least three dimensions. In section 2 we go over all possible UV-complete supersymmetric field theories on AdS d−q × S q , and classify the possible supersymmetry algebras that they can have that preserve the full supersymmetry. These algebras always contain the isometries of AdS d−q , so if we start from a field theory with n supercharges, we obtain a superconformal algebra in (d − q − 1) dimensions with n total supercharges, n/2 regular and n/2 conformal supercharges. When (d − q) > 3 such a superconformal algebra is unique, so the only questions are whether we can preserve all the supercharges and obtain this algebra or not. For (d − q) = 3 there are several 2d superconformal algebras with the same total number of supercharges, and we often find that more than one possibility can be realized by the same field theory when we put it on AdS d−q × S q .
There are several different methods to perform this analysis. One possibility is to use the formalism of supersymmetry in curved space, by coupling the supersymmetric theory to a background supergravity theory with an appropriate metric (and additional background fields). This formalism is only available for some cases, but when it is available it is straightforward to find that the AdS d−q × S q background can preserve all the supercharges, and to write the supersymmetric actions and Killing spinor equations on AdS d−q × S q . There are actually two variations of this formalism. One can use a "regular" supergravity containing background fields coupled to the usual supersymmetry algebra, or one can use a "conformal" supergravity containing background fields coupling to the superconformal algebra (see, for instance, [13][14][15][16][17][18][19][20][21][22]). When both formalisms are available they are identical, since the "regular" supergravity arises as a particular gauge-fixing of the "conformal" supergravity. In particular they give rise to the same Killing spinor equations, though they may be written in terms of different background fields. In section 3 we use a "regular" supergravity to analyze (following [25]) the case of 4d N = 1 theories on AdS 3 × S 1 , and we use "conformal" supergravity to analyze the case of 4d N = 2 theories on the same space. We analyze these two cases in detail, constructing explicitly the Killing spinors and the supersymmetry transformations. For various different free 4d theories, we analyze in detail the spectrum that we obtain on AdS 3 × S 1 , and the corresponding representations of the 2d superconformal algebra.
In section 2, in order to perform the complete classification, we use a completely different method. This is a more general approach, that does not require knowledge of the precise background supergravity that is relevant (and that is not always available); this method was previously used in [24]. To do this we note that AdS d−q × S q (with equal radii for AdS and the sphere) is related to flat space R d by a conformal transformation. Thus, the resulting (d − q − 1)-dimensional superconformal algebra must be a subalgebra of the d-dimensional superconformal algebra, and we can just classify all such subalgebras (that contain half of the fermionic charges of the d-dimensional superconformal algebra, and the isometries of AdS d−q × S q ). In this method the supercharges that we preserve on AdS d−q × S q are combinations of regular supercharges and conformal supercharges, that arise after performing the conformal transformation from flat space. This is similar to what we obtain by coupling our theory to conformal supergravity, but here we do not need to use any details of this coupling, and it is clear from the discussion above that the results apply to general supersymmetric field theories on AdS d−q × S q (not necessarily superconformal). The embedding of the (d − q − 1)-dimensional superconformal algebra into the d-dimensional superconformal algebra immediately tells us which d-dimensional R-symmetries are required for preserving supersymmetry on AdS d−q × S q . Our analysis is limited to supersymmetric field theories that have a UV-completion as superconformal field theories; we do not discuss in this paper theories with no known field-theoretic UV completion, such as the 6d N = (1, 1) supersymmetric Yang-Mills theory.
It is important to emphasize that a supersymmetric field theory on AdS d−q × S q is not equivalent to a (d − q − 1)-dimensional superconformal theory; for instance it does not contain a graviton that would map to the energy-momentum tensor of such a theory. However, such theories do have a natural action of the (d − q − 1)-dimensional superconformal algebra, such that their states and fields sit in representations of this algebra, and they can arise as decoupled subsectors of full-fledged (d − q − 1)-dimensional superconformal theories [2].
One new aspect which arises for theories on non-compact space-times like AdS is the need to specify boundary conditions, in particular in a way that preserves the full supersymmetry. We do not discuss this issue in general here, assuming that such a choice is always possible. In the cases that we discuss in detail we explicitly discuss some boundary conditions which preserve supersymmetry. Often there are many different choices of maximally supersymmetric boundary conditions, in particular for non-Abelian gauge theories [26].
It would be interesting to understand what are the specific quantities that can be computed exactly for supersymmetric field theories on AdS d−q × S q , by localization or other methods. We leave this to future work.
2 Superconformal field theories on AdS d−q × S q In this paper we study supersymmetric field theories in d = 3, 4, 5, 6, that are put on manifolds of the form AdS p × S q (d = p + q, p ≥ 3) in a way that preserves the full supersymmetry (SUSY). While some partial results are available (for example, for 4d N = 1 theories [27]), a general method for analyzing and constructing supersymmetric theories on curved space is not yet available; in some cases one can use a coupling to background supergravity fields for this. However, on space-times that include anti-de Sitter space, we have the advantage that any supersymmetry algebra must include the isometry algebra SO(p − 1, 2), which means that it must be equivalent to a superconformal algebra in (p − 1) dimensions, with the same total number of supercharges.
We can use the following trick to analyze all possible SUSY algebras that can arise from (p + q)-dimensional supersymmetric theories on AdS p × S q . When the AdS space and the sphere have equal radii of curvature, the space AdS p × S q is conformally equivalent to (p + q)-dimensional flat space. We can use for AdS p × S q the metric where µ = 0, · · · , p − 1, dΩ 2 q is the metric on a unit S q , and the boundary is at z = 0. Then, multiplying the metric by z 2 /L 2 gives the metric on flat space, where z is a radial coordinate in (q + 1) dimensions. Boundary conditions imposed on AdS space may lead to singularities of various fields on the (p − 1)-dimensional subspace z = 0.
Suppose we have a (p+q)-dimensional superconformal theory, whose symmetry algebra includes n supercharges and also n superconformal charges. The fact that AdS p × S q is related to flat space by a conformal transformation, means that the supersymmetry algebra on AdS p × S q must be a subalgebra of the superconformal algebra in (p + q)-dimensions. It is clear that the boundary conditions on AdS p must break at least half of the total number of fermionic generators of the superconformal theory. We will be interested in the cases where exactly half of the fermionic generators are broken, leading to a supersymmetry algebra on AdS p × S q with n fermionic generators. Thus, our goal will be to classify all possible (p − 1) dimensional superconformal algebras with n total supercharges (n/2 regular supercharges and n/2 superconformal charges in (p − 1) dimensions) that arise as subalgebras of (p + q)-dimensional superconformal algebras with 2n total supercharges. The superconformal groups in different dimensions on flat space were classified by Nahm [28], and we will use this classification in our analysis.
Naively this classification is only relevant for superconformal theories in (p + q) dimensions. However, it is clear that if we have a general supersymmetric theory on AdS p × S q , it cannot preserve a larger supersymmetry than that of a superconformal theory on the same space. Thus, the same classification will give us all supersymmetry algebras that can arise when we put a general UV-complete supersymmetric theory on AdS p × S q in a way that preserves the same number of supercharges as we had in (p + q) dimensions. Similarly, it is clear that when we discuss AdS p and S q spaces with different radii of curvature, the supersymmetry algebra cannot be larger than the one which arises for equal radii, so any supersymmetry algebra that arises for such space-times must also be included in our classification. For q = 1 one can show that the same supersymmetry algebra arises for any ratio of the radii.
In this section we will describe this classification of all possible symmetry algebras that preserve all supercharges of a (p + q)-dimensional theory on AdS p × S q , going over all possible p ≥ 3 cases one by one. The results are summarized in appendix F.

Four dimensional theories on AdS 3 × S 1
In four dimensions we have four consistent superconformal algebras that do not contain higher spin conserved currents : N = 1, N = 2, N = 3 and N = 4. Each algebra contains 4N regular supercharges, and 4N superconformal charges. We would like to analyze the possible algebras that can arise for supersymmetric theories on AdS 3 ×S 1 , preserving all 4N supercharges. As discussed above, such algebras should be two dimensional superconformal algebras with 2N regular supercharges, that arise as subalgebras of the four dimensional superconformal algebras. Thus, they must be (p, 2N − p) superconformal algebras in two dimensions. Throughout this section we will follow the notations of [29]; our conventions may be found in appendix A.

N = 1
In this section we describe the algebraic structure of these theories; the construction of some explicit field theories for this case will be discussed in detail in section 3.
The 4d N = 1 superconformal algebra includes the fermionic generators Q α and S β , which we take to be Majorana spinors with α, β = 1, 2, 3, 4, and the bosonic generators: Lorentz transformations M µν of SO(3, 1), translations P µ , special conformal transformations K µ and dilatations D, with µ, ν = 0, · · · , 3. Moreover, it includes a U (1) R-symmetry whose generator we denote by T . The bosonic generators satisfy the SO(4, 2) algebra, and together with the fermionic generators they satisfy the SU (2, 2|1) algebra: Here γ * is the chiral gamma matrix, that exists in any even number of dimensions. In order to have a consistent superalgebra on AdS 3 × S 1 , we need a subalgebra of this superconformal algebra that includes the AdS 3 × S 1 isometries, SO(2, 2) × SO(2) ⊂ SO(4, 2), and it should include half of the fermionic operators Q and S, that must close on this subalgebra. We expect the SO(2) generator of rotations on the circle to include P 3 , and to commute with all other bosonic generators, in particular with the SO(2, 2) generators. The correct choice turns out to be P 3 − c 2 K 3 , where we show in appendix B that the absolute value of c is related to the curvature of AdS 3 by The generator P 3 −c 2 K 3 does not commute with the following eight bosonic generators: P a − c 2 K a , M a3 , D, and P 3 + c 2 K 3 (a = 0, 1, 2), so these generators will not be part of the resulting supersymmetry algebra. Six of the remaining bosonic generators, M ab and P a + c 2 K a , form an SO(2, 2) algebra. In order to see this it is convenient to define the operators and The SO(2, 2) ≃ SO(2, 1) × SO(2, 1) algebra may then be written as: When c > 0, the L a are the right-moving SO(2, 1) charges and theL a are the left-moving SO(2, 1) charges. Without loss of generality we will assume from here on that c > 0; the other sign is related to this by a two dimensional parity transformation. When the SO(2, 2) is embedded into a full Virasoro algebra (which may or may not be the case), these charges are proportional to the L −1,0,1 andL −1,0,1 charges of the Virasoro algebra (the constant c above should not be confused with the central charge of the Virasoro algebra, which can be non-zero when our field theory is embedded into some gravitational theory).
The supercharges that close only on the remaining bosonic generators mentioned above are half of all the fermionic generators, that may be written as Q ≡ Q + icγ * γ 3 S. Their algebra is (2.1.6) We can identify this as part of the two dimensional right-moving N = 2 superconformal algebra, where we identify (P 3 − c 2 K 3 + 2cT ) as the two dimensional R-charge; note that this is a mixture of the isometry of the circle and the 4d R-charge. Together with the other generators written above we have the N = (0, 2) superconformal algebra (when c is positive), with a bosonic subgroup SO(2, 2) × SO(2); in particular In addition to this algebra we have an extra U (1) generator (P 3 − c 2 K 3 − 2cT ) that commutes with all the generators of the superconformal algebra. When this generator is preserved, it gives an extra global U (1) symmetry, but we can also preserve the same amount of supersymmetry on AdS 3 ×S 1 when this generator is broken, and only the specific combination (P 3 − c 2 K 3 + 2cT ) is conserved; we will see examples of both possibilities in section 3. However, the fact that we must preserve the combination (P 3 − c 2 K 3 + 2cT ) which appears on the right-hand side of the supersymmetry algebra on AdS 3 × S 1 means that even when the 4d theory that we start with is not superconformal, it must still have an R-symmetry in order to preserve supersymmetry on AdS 3 × S 1 [25,27]. This will be true also in our subsequent examples -preserving supersymmetry on AdS p × S q requires having in the (p + q)-dimensional field theory all the R-symmetries that appear on the right-hand side of the supersymmetry algebra on AdS p × S q .
Note that even though just the counting of supercharges would have allowed also a 2d N = (1, 1) superconformal algebra in this case, we see that this possibility does not arise.
We will see also in the other cases below that on AdS 3 × S q (q ≥ 1) we always obtain an even number of left-moving and of right-moving supersymmetry generators.

N = 2
In the 4d N = 2 case, we use a convention where we have eight complex Weyl supercharges Q iα and S j β , with i, j = 1, 2, α, β = 1, 2, 3, 4, and a SU (2) × U (1) R-symmetry generated by a traceless 2 × 2 matrix U j i and by T . The position of the R symmetry index i, j, · · · is used to distinguish between left-handed and right-handed supercharges in the following way: In order to connect the results here to our later results in 5d and 6d, we follow [29] and treat the Weyl spinors as four component spinors, with two components vanishing according to (2.1.8). The spinor indices α, β, · · · can be raised and lowered by the charge conjugation matrix C αβ . In the usual notations of [30], our Q i 's are the Q's, and our Q i 's are the Q's.
Together with the other bosonic generators these satisfy the SU (2, 2|2) superconformal algebra (2.1.9) As in the N = 1 case, we choose a subgroup of the bosonic generators to give the isometry algebra of AdS 3 × S 1 , and we correlate the ordinary supercharges and the superconformal charges so that we get a consistent superalgebra on AdS 3 × S 1 . In the N = 2 case there are two different ways to do so, which are a straightforward generalization of the N = 1 case (see also [31]).
The first option is the diagonal option preserving SU (2) R , where for i = 1, 2 we choose the conserved fermionic generators to be (2.1.10) These satisfy the algebra (2.1.11) Notice that P 3 − c 2 K 3 + 2cT appears on the right-hand side of {Q i , Q j }, but it commutes with all the other generators in the theory. Therefore, k θ ≡ P 3 − c 2 K 3 + 2cT is a central charge and fixed within a representation. This should not be confused with the central charge of the (super)-Virasoro algebra, which does not appear in its global subalgebra that we obtain here. We can rewrite the second line of (2.1.11) in the following way: with L a defined as in (2.1.3), and U i defined by This supersymmetry algebra (together with the commutators of the bosonic generators) is isomorphic to the 'small' N = (0, 4) superconformal algebra in two dimensions, which contains an SU (2) R-symmetry, together with an extra central charge k θ . Note that this central charge is consistent with the algebra we wrote, but not with its extension to a super-Virasoro algebra; thus if we embed our theory into a theory that has this extended algebra, k θ must vanish. The other possible option for N = 2 theories on AdS 3 × S 1 is (2.1.14) It will be convenient to define the following basis for the charges The algebra that includes this specific half of the supercharges is then (2.1.16) Notice that we get two separate subalgebras, each satisfying the N = 1 algebra of section 2.1.1, but with opposite chirality.
These subalgebras contain two independent U (1) generators and each one of them acts on half of the supercharges: (2.1.18) This algebra is the N = (2, 2) superconformal algebra in two dimensions, with left-moving and right-moving U (1) R generators T ± .
As in the previous cases, we define supercharges Q ii ′ ≡ Q i + icγ * γ 3 S i ′ , where we begin with arbitrary and independent i and i ′ . We wish to see which combinations of indices will close on the isometries of AdS 3 × S 1 , chosen as above. Generally, the commutation relations of these supercharges are since then we get the specific combinations of the generators P 3 − c 2 K 3 , P a + c 2 K a . Moreover, as discussed above D cannot appear in the algebra, implying another constraint This condition also ensures that is in the algebra. With these constraints the commutator is simplified to Up to permutations, there are three possible solutions to the constraints, and we will analyze each of them separately : The naive analysis of the R-symmetry in the 2d superconformal algebra would be to check what subgroup of the entire R group is consistent with the supercharges Q ii ′ .
for the cases I, II, III respectively. The naive expectation is that the 2d R-symmetry will be a product of G R with the S 1 isometry (or S q isometries in the general case). As we already saw in the N = 2 analysis, this is not the case. The mixture of the sphere isometries and the R-symmetry generators can modify the symmetry by central charges and U (1) factors. In some cases we will see that not all of the generators appear on the right-hand side of anti-commutators of supercharges, and therefore will not be part of the 2d R-symmetry. For this reason, a more careful analysis needs to be done. I. This case is similar to the N = (0, 4) case above, and the algebra takes the form (2.1.24) Notice that all unitary traceless R-symmetry generators U j i appear in the algebra, and therefore the full SU (4) remains as a two dimensional R-symmetry. The additional generator P 3 − c 2 K 3 , which is the generator of the U (1) symmetry on the circle S 1 , also appears as an R-symmetry. This algebra is isomorphic to the N = (0, 8) superconformal algebra, where the N = 8 algebra appearing is the one with a U (4) R-symmetry.
II. In this case it will be convenient to write the algebra in terms of generators Q 1 , Q 2 and Q ± = 1 √ 2 (Q 3 ± Q 4 ). We will use indices i, j = 1, 2, and write the ± explicitly. The fermionic commutation relations read (2.1.25) The bosonic commutation relations of the Lorentz group with Q i and Q i are as in case I, and with Q ± and Q ± as in (2.1.16). The R-generators that appear in the commutation relations form a SU is the generator of the U (1) symmetry, and it commutes with the other eight generators. This U (1) generator is diagonal in the Q's with eigenvalues of − 1 2 for Q + and Q i and 3 2 for Q − ; 1 2 for Q + and Q i and − 3 2 for Q − . Together with the additional U (1) generator but not with the rest. To summarize, we get an algebra in which the generators Q 1 , Q 2 , Q + , Q 1 , Q 2 , Q + , together with U (3) R-symmetry generators, obey non-trivial commutation relations among themselves and (anti-)commute with Q − , Q − . The full algebra is isomorphic to the N = (2, 6) superconformal algebra in two dimensions, with R-symmetry group III. In this case it will be convenient to use fermionic generators Q (1) , for which we get the following algebra: with the bosonic commutation relations as in (2.1.16). We see that the algebra splits into two commuting sectors with individual SU (2) R-symmetry generators. There is also a central charge, , appearing on the right-hand side, which commutes with all the algebra and with the SU (2) + × SU (2) − R-symmetry. The supercharges Q ±(1) and Q ±(2) are doublets of SU (2) ± respectively. This algebra is isomorphic to the two dimensional 'small' N = (4, 4) superconformal algebra, with an additional central charge k θ (that cannot appear in the super-Virasoro extension of this algebra).

N = 3
No superconformal field theories with N = 3 that do not have N = 4 are known, but we still include this algebraically consistent case for completeness. By straightforward generalizations of the previous cases, we can get here the N = (0, 6), N = (2, 4), N = (4,2) or N = (6, 0) two dimensional superconformal algebras.

Five dimensional theories on AdS 4 × S 1
As classified by Nahm [28], the only possible five dimensional superconformal algebra has N = 1 supersymmetry and is called F (4). We will use the real form F 2 (4) as in [32] to write the algebra: In this algebra the supersymmetry generators Q iα and the superconformal generators S iα (i = 1, 2, α = 1, 2, 3, 4) are symplectic Majorana spinors, with a total of eight real components. The U ij generators form a SU (2) R-symmetry algebra, and they are anti-hermitian and symmetric, Unlike our conventions in four dimensions, here the indices i, j = 1, 2 are raised and lowered by ǫ ij and ǫ ij which satisfy The charge conjugation matrix C, and also the matrices Cγ a , are anti-symmetric. The algebra (2.2.1) is quite similar to the N = 2 superconformal algebra in four dimensions, where we saw that the algebra closes on the isometries of AdS d−1 × S 1 for the following choices of supercharges: We may expect to have the same two options here, but it turns out that only one of them is consistent: the twisted choice with As in the four dimensional case it will be convenient to work in the basis in which the algebra takes the form (2.2.8) Our choice of supercharges breaks the SU (2) R-symmetry to a U (1) with the generator U 11 − U 22 . As in section 2.1.1, one linear combination of the isometry of S 1 and this unbroken U (1) R appears on the right-hand side algebra and acts as a 3d SO(2) R generator: The other combination is a global symmetry that may or may not be broken. The full algebra that we find is equivalent to the N = 2 three dimensional superconformal algebra OSp(2|2, R) [33]. Indeed, based on the amount of supersymmetry, this is the only possibility.

Six dimensional theories on
The largest space-time dimension consistent with superconformal symmetry is six [28]. Assuming no higher spin conserved charges, there are two possibilities in this case, which are both chiral: the minimal N = (1, 0) superconformal algebra, and the extended N = (2, 0) superconformal algebra. Putting them on AdS 5 × S 1 should lead to a four dimensional superconformal algebra with half of the number of fermionic generators.
The 6d algebra in this case is given in terms of symplectic Majorana-Weyl spinors by [34] [M µν , where i, j = 1, 2, and α, β = 1, · · · , 8. As in the five dimensional case, the spinor indices are raised and lowered by ǫ ij and ǫ ij , respectively. The U j i 's are generators of the SU (2) R-symmetry.
Again, this algebra is very similar in its structure to the N = 2 superconformal algebra in four dimensions. As in the five dimensional case of section 2.2, only the twisted combinations Q 1 = Q 1 + icγ 5 S 2 and Q 2 = Q 2 + icγ 5 S 1 form a consistent algebra. The R-symmetry that is preserved is U (1), and we also have the U (1) isometry of S 1 . The remaining subalgebra is isomorphic to the N = 1 superconformal algebra in four dimensions, SU (2, 2|1). One combination of U (1)'s P 5 − c 2 K 5 − 4ic(U 1 2 + U 2 1 ) appears in this algebra as the U (1) R generator, and the other one may or may not be a global symmetry.

N = (2, 0)
This algebra has an U Sp(4) ≃ SO(5) R-symmetry, generated by the symmetric 2-form U ij (i, j = 1, 2, 3, 4). The indices are raised and lowered by the antisymmetric invariant tensors Ω ij and Ω ij , which satisfy Ω ij Ω jk = δ i k . In order to perform computations, we choose a specific representation for these matrices: The algebra then takes the form Naively there are three choices of combining the supercharges to get a superconformal algebra on AdS 5 × S 1 , just as in the N = 4 case in four dimensions. But it turns out that only for two of them the algebra closes; these are 3.4) and (2.3.5) These two options turn out to give the same algebra, in which the 6d R-symmetry breaks to a U (2) symmetry with (2.3.6) U + , U − and U z satisfy the SU (2) algebra, and T commutes with them. Defining combinations of the Q's of (2.3.4) as in case I of section 2.1.3, they act on the supercharges in the following way We also have the extra U (1) generator ∂ θ = P 5 − c 2 K 5 . One combination of this with T acts as the U (1) R-generator in the four dimensional superconformal algebra, and the other may or may not be a U (1) global symmetry. The algebra we find is equivalent to the four dimensional N = 2 superconformal algebra SU (2, 2|2); again this is the only possibility based on the counting of supercharges. Theories realizing this construction were discussed in [2].

Field theories on AdS
in this case we wish our symmetries to commute with the SO(3) isometry group of the S 2 factor. Using similar manipulations to the AdS d−1 × S 1 cases, we can single out the last two space-time dimensions by using two gamma matrices in the form of the conserved supercharges on This ansatz turns out to give a superconformal algebra that preserves the isometries of AdS d−2 × S 2 . Since we are not studying AdS 2 here, we can choose d = 5 or d = 6; however d = 6 turns out to be impossible (see Appendix D) so we only have one case.

Five dimensional field theories on AdS 3 × S 2
As in (2.2.1), we consider the F (4) superconformal algebra. This time we choose as conserved supercharges Q i = Q i + icγ 34 S i , which turns out to be the only consistent choice giving a closed subalgebra. We will denote the coordinates µ = 3, 4 by A, B, · · · and µ = 0, 1, 2 by a, b, · · · . The fermionic part of the algebra is The SO(2, 2) = SL(2, R) × SL(2, R) isometry group of AdS 3 is now generated by γ a (P a − c 2 K a ) − cM ab γ ab γ 34 = γ a L a , γ a (P a − c 2 K a ) + cM ab γ ab γ 34 = γ aL a . (2.4.2) Only the L a appear on the right-hand side of (2.4.1), therefore the two dimensional superconformal algebra is chiral. P A + c 2 K A and M 34 generate the SO(3) isometry group of the sphere, and U 2 1 , U 1 2 and U 1 1 = −U 2 2 generate an SU (2) R symmetry. In the two dimensional superconformal algebra, the sphere generators join with the R-symmetry generators to form an SU (2) × SU (2) R-symmetry. This algebra turns out to be the 'large' N = (0, 4) superconformal algebra [35] 1 .

Field theories on AdS d−3 × S 3
Here the only examples are the six dimensional ones. We use the same conventions for the 6d superconformal algebras as above. Following the previous sections we propose the conserved supercharges to be of the form Q i + icγ (d−3)(d−2)(d−1) S i ′ , and we denote the indices µ = 3, 4, 5 by A, B, · · · , and the indices µ = 0, 1, 2 by a, b, · · · .

Six dimensional N = (1, 0) theories
The only consistent choice for combining the supercharges is the diagonal choice, (2.5.1) These obey the algebra The generators (P A + c 2 K A ) + icǫ ABC M BC and U j i form an SU (2) × SU (2) R-symmetry in the two dimensional superconformal algebra. As in the previous case, the full algebra turns out to be the 'large' N = (0, 4) superconformal algebra in two dimensions. The other three S 3 rotation generators P A + c 2 K A − icǫ ABC M AB commute with the supercharges and may or may not be a global symmetry.

Six dimensional N = (2, 0) theories
In this case there are two options to form a consistent algebra. The first case is the diagonal case The R-symmetry in this case consists of an SU (2) subgroup of the sphere SO(4) isometries generated by P A + c 2 K A + icǫ ABC M BC , and of the full 6d R-symmetry U Sp(4) ∼ SO(5), and we have 8 chiral supercharges. We obtain a N = (0, 8) superconformal algebra that is different from the one we encountered before; this algebra is classified as case (III) in [36]. The other three SO(4) generators again commute with the supercharges and may or may not be a global symmetry. The second option is to split the generators into two pairs. For the representation of Ω that we chose in section 2.3.2 they are Here the supercharges close on the entire SO(4) sphere isometries, and our choice preserves an SO(4) subgroup of the U Sp(4) ≃ SO(5) R-symmetry. Altogether we obtain the 'large' N = (4, 4) superconformal algebra with SO L (4)×SO R (4) R-symmetry. Each one of the Rsymmetry groups SO(4) L/R acts only on left/right-handed supercharges, and is generated by three out of the S 3 isometries and three out of the preserved six dimensional R-symmetry generators.

Field theories on AdS d
This is the final possibility, which is related by the conformal transformation discussed above to flat space with a codimension one boundary. In this case our ansatz for the conserved supercharges is simply Q = Q + icγ * S. The supersymmetry algebra should be a (d−1)-dimensional superconformal algebra. For the d = 6 N = (2, 0) case it is clear just by counting supercharges that this is not possible, and in fact it is easy to see (essentially by chirality arguments) that one cannot preserve half of the supersymmetry for any 6d theory with chiral supersymmetry on AdS 6 (see Appendix D). So, we will analyze the three, four and five dimensional cases.

Four dimensional N = 1
Four dimensional supersymmetric theories on AdS 4 were discussed in the past, for instance in [3][4][5][6][7][8][9], and also more recently in [10][11][12]. For the N = 1 case the algebra we obtain must be the three dimensional N = 1 superconformal algebra, with no R-symmetry, and indeed we find The extra U (1) R symmetry that we have in four dimensions is broken by the choice of the combination of supercharges that appears in (2.6.1); this must happen because a codimension one boundary reflects left-handed fermions into right-handed ones.

Four dimensional N = 2
As we saw in previous similar cases, also here there are two options, the diagonal case Q i = Q i + icγ * S i and the twisted case Q 1 = Q 1 + icγ * S 2 , Q 2 = Q 2 + icγ * S 1 . As can be seen by a change of basis, the two options turn out to be equivalent and obey the algebra This algebra is isomorphic to the three dimensional N = 2 superconformal algebra with an SO(2) R-symmetry, with U 2 1 − U 1 2 as its generator; the other generators of the 4d U (2) R-symmetry are broken.

Four dimensional N = 4
Also in this case all three options of combining the supercharges Q i with the superconformal charges S i ′ turn out to give the same algebra This is equivalent to the three dimensional N = 4 superconformal algebra with SO(4) Rsymmetry. Again the 4d SU (4) R-symmetry is broken to SO(4). Similarly, the 4d N = 3 case leads to a 3d N = 3 superconformal algebra.

Five dimensional theories on AdS 5
In the five dimensional case only one out of the two natural options is consistent. This is the twisted choice, Q 1 = Q 1 + icS 2 and Q 2 = Q 2 + icS 1 . The algebra is then the N = 1 four dimensional superconformal algebra, with a U (1) R-symmetry generated by U 11 − U 22 .

Three dimensional theories on AdS 3
The three dimensional superconformal algebra is (2.6.4) Here A ij are SO(N ) generators, i, j = 1, · · · , N . For more conventions see [33].
As in previous cases, in order to close the algebra on the AdS 3 isometries, we define the supercharge Q a = Q a + cS a ′ that gives the commutator ). (2.6.5) For some N , we can choose n diagonal and 2m twisted supercharges where n + 2m = N , such that we define Q i = Q i + cS i , i = 1, · · · , n Q a = Q a + cS a+m , a = n + 1, · · · , n + m Q a ′ = Q a+m + cS a , a = n + 1, · · · , n + m. (2.6.6) The algebra is then (2.6.7) From chirality analysis, we get the N = (m, n + m) 2d superconformal algebra. The n + m right handed spinors are Q i , 1  generators, that give an SO(n + m) × SO(m) R-symmetry algebra which acts on the left-handed and right-handed spinors.
The N = (0, n + m) subgroup involves the n + m right handed spinors, the three P µ + c 2 K µ − icǫ µνρ M νρ SL(2) R isometries, and an SO(n + m) R symmetry made out of The N = (m, 0) subgroup involve the m left handed spinors, the three P µ + c 2 K µ + icǫ µνρ M νρ SL(2) L isometries, and an SO(m) R-symmetry made out of (A a b+m + A a+m b − A a b − A a+m b+m ). The two subgroups (anti-)commute with each other.
We will show explicitly how to write the actions and transformation rules for different 4d multiplets. For N = 1 we use the simple notations of new minimal supergravity (SUGRA) and the results of [27]. For N = 2, we build the actions and transformation rules using the superconformal approach discussed in section 2, starting from an N = 2 superconformal theory and coupling it to superconformal gravity. The different choices for the supercharges Q i correspond to relations η(ζ), where ζ, η are parameters related to the Q and S transformations, respectively. By starting from superconformal field theories on flat space and plugging in the relations η(ζ), we get the correct Killing spinor equations, action, and transformation rules (see appendix B for more details).
We focus on free theories for which we will explicitly construct the action and boundary conditions on AdS 3 × S 1 that preserve all of the supercharges, and study the spectrum of the 2d superconformal algebras. Unlike in the previous section, here we allow for different radii for AdS 3 and S 1 , in order to show that one can still preserve the same supersymmetry algebras also in this case. We also allow for 4d field theories that are not necessarily conformal, though most of our examples will be conformal.

Four dimensional N = 1 theories on AdS 3 × S 1
In the previous section we analyzed the supersymmetry of N = 1 theories on AdS 3 × S 1 algebraically. Another general way to study such theories is to couple them to background fields of new minimal supergravity, and to use the results of [27]. We will show explicitly that the two consistent values for the background supergravity fields result in 2d N = (0, 2) and N = (2, 0) superconformal algebras. We use the metric where θ is the coordinate on S 1 with θ ∼ θ + 2π, R and L are the radii of the S 1 and AdS 3 respectively, and for r → 0 we reach the boundary of AdS. The curved space sigma matrices are related to the flat ones by For spinors and sigma matrices conventions, we follow [30]. The classification of geometries preserving different numbers of supercharges for four dimensional N = 1 theories on various manifolds can be found in [25,27]. Following their work we couple the theory to the new minimal supergravity multiplet [37] which contains, in addition to the physical graviton g µν and gravitino Ψ α µ , the following auxiliary fields: a U (1) R gauge field A µ and the 1-form V µ = 1 4 ǫ µνρλ ∂ ν B ρλ . The conditions for preserving all four supercharges are given by where W µνκλ , R µν are the Weyl and Ricci tensors of the metric g µν , respectively. When these conditions are satisfied, there are four independent solutions to the Killing spinor equations The superalgebra then will be where δ K is the R-covariant Lie derivative along the Killing vector K µ = ζσ µζ , andq is the generator of the U (1) R symmetry. We begin by determining the values of the background fields V µ and A µ . From (3.1.3) we find In order to preserve the isometries of our spacetime, V µ and A µ must take values in the S 1 direction. We then get two solutions From the Killing spinor equations and the requirement that the spinors should be singlevalued (see appendix C), we find that the allowed values for A µ are The parameter n here corresponds to a large gauge transformation of the background U (1) R field around the circle, which is essentially the same as shifting the momentum generator around the circle (normalized to be an integer) by n times the R-charge. The effect of this is discussed in appendix C. It has no effect on the supersymmetry algebra, so from here on we will set n = 0. Note that this construction only works when we have a U (1) R symmetry; for superconformal theories this is guaranteed, but for other theories it is a necessary condition for preserving all supercharges on AdS 3 × S 1 .
We can now solve (3.1.4) to get an explicit form for the Killing spinors ζ andζ: Here a,ā, b andb are Grassmanian parametrizations of the components of the Killing spinors which correspond to the four independent supercharges, z ≡ x + t andz ≡ x − t are coordinates in the spatial directions of the boundary of AdS 3 , and the subscripts L, R denote left/right-handed solutions.
From now on we will focus on the right-handed solution ζ R ,ζ R ; a similar analysis can be done for the left-handed one. Using the explicit form of the spinors, we can compute the Lie derivative δ K acting on different fields.
For example, when acting on a scalar (which can have some non-zero R-charge as the eigenvalue ofq), the Lie derivative takes the simple form L K = K µ ∂ µ , resulting in the following commutators of the generators: (3.1.11) Here ∆ = h L + h R is the sum of the left and right dimensions (which are equal for scalars), andR is the U (1) R generator of the two dimensional superconformal algebra. We can repeat the procedure for higher spin fields and find also the spin, s = h R − h L . We find that for a bulk Weyl spinor with ψ ± =    1 ±1   , the 2d spin value is ∓ 1 2 , and for a vector v x ± v t , the 2d spin value is ±1. Thus, we find also in the explicit field theory language the same algebraic structure as in the previous section. Note in particular that, as in section 2.1.1, the 2d R-chargeR is a linear combination of the KK momentum and the 4d R-charge; the specific combination we had in section 2.1.1 arises here for L = R (which we assumed in the previous section). If we choose the opposite sign for V µ , we similarly get the N = (2, 0) superconformal algebra.

Free field theory on AdS 3 × S 1
In this section we analyze the spectrum and boundary conditions of different fields on AdS 3 × S 1 . For related work see, for example, [38] and references within. We begin with free fields, and later join them into supersymmetry multiplets.

Scalar
The bulk action for a free massless scalar of R-charge q (which is the bottom component of a chiral multiplet) coupled to the new minimal supergravity auxiliary fields is We now expand φ in Kaluza-Klein (KK) modes around the S 1 by defining In terms of these modes, the action (3.2.1) on AdS 3 takes the form The asymptotic solution near the boundary is given by the standard formula The physical modes must have ∆ > 0. We will fix the other modes on the boundary and find the correct boundary action to yield a well-defined variational principle. The variation of the bulk action is where h is the induced metric on the boudnary of AdS space (say with some cutoff on the radial direction), d 2 x denotes an integration over the variables x and t which label this boundary, and n is a vector transverse to the boundary. The bulk terms vanish on shell. In order for the boundary term to vanish, we add a boundary action S bndy,φ to S φ such that the total variation vanishes when the physical boundary conditions are held. If we choose we get the variation The operators n µ g µν ∂ ν ± i R D θ annihilate φ k + ,φ k + for every k. Therefore, by fixing the φ k− modes, δφ k − = δφ k − = 0, the variation of the action vanishes. If, on the other hand, we take the boundary action to be (3.2.11) In this case, the variation vanishes when fixing δφ k + = δφ k + = 0.
In order to satisfy ∆ > 0 for all the fluctuating modes, we need to have a mixed kdependent boundary action. For the KK modes with 0 < ∆ < 2 we have two possibilities for the boundary action, while otherwise we have just one choice. The full boundary action should then be with k * such that for k < k * , ∆ k − > 0 and for k ≥ k * , ∆ k + > 0 (in some cases there may be more than one possible choice for this k * , giving different theories on AdS 3 × S 1 ).

Fermion
The bulk action for a fermion in a chiral multiplet is a kinetic term and a term that couples the fermions to the background field, (3.2.13) Here we took the fermion to have R-charge (q − 1), consistent with putting it in the same multiplet as the scalar with R-charge q.
As in the case of the scalar, we expand in KK modes

2.14)
and solve the equations of motion near the boundary. We take the fermion to be periodic around the circle, anticipating that this will be required for preserving supersymmetry. The asymptotic solution is given by (3.

2.15)
As explained in section 3.1, the 2d conformal dimensions of ψ ± satisfy h L − h R = ± 1 2 . The spectrum can be written as (3.2.16) As before, in order to have a well defined variational principle, we need to split our boundary action. If we take the boundary action to be then δS ψ + δS bndy,ψ± = 0, when we keep the modes ψ k,± .
Note that because ∆ f,k ± = ∆ s,k ± ± 1 2 , the constraints on k * for the fermion are the same as we found for the scalar. This is of course important for SUSY, as will be shown in the next section.

Gauge field
The action for a free U (1) gauge field v µ is the Maxwell term When expanded in KK modes,

2.20)
it takes the following form  i . The normalizable modes are a scalar whose dimension in the 2d conformal algebra is ∆ = 2, and a U (1) gauge field on the boundary, while the nonnormalizable modes that couple to them are a scalar of dimension ∆ = 0 and a conserved current. Note that from the point of view of the 2d superconformal algebra this means that (for the action (3.2.19)) we do not get a conserved current representation, but rather a representation that contains the 2d field strength arising from the value of v µ on the boundary. For the k = 0 modes, we have a complex massive vector field v (k) i . The asymptotic solution to the equations of motion gives the following dimensions for the k'th KK mode of v i , Similar to the previous cases, we demand that the variation of the total action should vanish when the physical boundary conditions are satisfied. The boundary action that we need to add is The generalization to non-Abelian gauge fields is straightforward.

A free N = 1 chiral multiplet
The on-shell supersymmetric chiral multiplet consists a complex scalar φ and a Weyl fermion ψ. Following the previous subsections, the action for the free massless chiral multiplet with R-charge q on AdS 3 × S 1 is given by and the covariant derivatives are The SUSY variations of the fields are The full action S chiral accompanied with the boundary conditions specified in section 3.2 is invariant under all four supercharges. Each scalar k ± mode with dimension (3.2.6) has a superpartner fermion k ± mode with dimension (3.2.15), such that By comparing to the known N = (0, 2) multiplets, we see that the (φ k + , ψ k + ) form chiral multiplets, and (φ k − , ψ k − ) form Fermi multiplets.

BF bound saturation
According to Breitenlohner-Freedman [39,40], the minimal mass of a scalar on AdS 3 can be otherwise we get complex dimensions from the point of view of the 2d conformal algebra. Supersymmetry guarantees that m 2 ≥ m 2 BF (see (3.2.4)), but we should discuss the special case where this bound is saturated. This happens if there exists an integer k such that kL R + q = 1, and then for this k ∆ k± coincide. In this case, the asymptotic solution to the Klein-Gordon equation is If we fix the non-normalizable mode φ − on the boundary, the analysis is similar to the one done in the previous sections and all supercharges are preserved. The other boundary condition breaks the conformal symmetry.

A free massive chiral multiplet
We can add a mass in a supersymmetric way by adding a superpotential W = 1 2 mφ 2 and taking the R-charge of the scalar to be q = 1. After integrating out the auxilliary field F , we get the bulk Lagrangian (3.3.8) The spectrum is modified due to the mass. The equation of motion of the scalar is The dimension of the k'th KK mode is given by For the Fermion, we have the coupled equations Asymptotically, we get Plugging one into the other, we get 3.13) and the dimensions 3.14) with the asymptotic expansions Now the SUSY transformations mix the fields φ k ,φ −k , ψ k ,ψ −k but they can be diagonalized such that they split into four multiplets with dimensions The boundary conditions fix two of them -either the first and fourth, or the second and third -such that only two give operators in the 2d superconformal algebra, one chiral and one Fermi multiplet for every k.

Breaking the S 1 isometry
As was discussed in section 2, in order to have supersymmetry on AdS p × S q , some mixture of the R-symmetry and the S q isometries that appears on the right-hand side of the supercharges anti-commutator must be preserved. Specifically, for the case of N = 1 on AdS 3 × S 1 , from the algebraic analysis we know that we must preserve a specific combination of the S 1 isometry and the U (1) R generator, but can break each one of them separately. A simple field theory realization for this is to add a θ-dependent mass term to the free chiral multiplet. This is done by adding the superpotential W = me 2i(q−1)Aµx µ φ 2 ,W = me −2i(q−1)Aµx µφ 2 . By taking q = 1 we get the regular massive chiral multiplet discussed in the previous section, but we will keep q arbitrary such that the Lagrangian ) breaks the S 1 isometry and the U (1) R symmetry. The theory is still supersymmetric as before, with the dimensions modified to 3.20) but it no longer has an extra U (1) global symmetry.

A free N = 1 vector multiplet
The bulk action for a U (1) gauge multiplet is The KK modes are defined as above, and we can compute their variations. We will do it explicitly for δ b , δb, which is enough to understand to structure of the multiplet. Some of the transformations are: For k = 0, the − modes are the physical ones. They contain a 2d gauge field v i , a dimension 1 2 , 1 fermion λ − and a dimension (1, 1) scalar v θ . Its dual multiplet (containing the couplings to these operators) contains a conserved current, a dimension 1 2 , 0 fermion and a dimension 0 scalar.
For k > 0, the + modes are physical. They contain a vector of dimension 1 + kL 2R , kL 2R , and a fermion of dimension 1 2 + kL 2R , kL 2R . For k < 0, the − modes are physical. They contain a vector of dimension − kL 2R , 1 − kL 2R and a fermion of dimension 3.5 A free N = 2 hypermultiplet in the N = (0, 4) case For N = 2 theories we can either just guess the form of the Killing spinors and supersymmetric actions, based on the results for the N = 1 case and on requiring their consistency with the full N = 2 supersymmetry, or derive them by coupling to a background N = 2 superconformal gravity, as described in appendix B.
We begin in this subsection by studying the N = 2 hypermultiplet using the supersymmetry that gives a N = (0, 4) superconformal algebra. Denoting the scalar fields by φ 1,2 and the fermions by ψ 1,2 , the supersymmetry transformations of the different fields are (3.5.1) Here V µ is the same as in the N = 1 case above. The bulk action, equations of motion and Killing spinor equations are with the covariant derivatives defined as

5.4)
and their complex conjugates. A µ here is a background field for the U (1) R-symmetry of the N = 2 theory; this symmetry must exist in order to use this construction, based on our algebraic discussion. We take the R-charge of the scalars in the hypermultiplet to be q. As in our discussion of the N = 1 case in section 3.1, we can perform shifts in the background value of A µ corresponding to large gauge transformations. In our conventions of this section, in order for the Killing spinors to be periodic, we need to perform a specific shift which amounts to setting A µ = 0, and we will use this value below. The µ = θ part of the Killing spinor equations here seems different than the one we used for the N = 1 case. This happened because here we used the conformal supergravity approach, rather than coupling to regular supergravity. As discussed in the introduction, the two approaches must give the same answer. This is consistent because the N = 1 Killing spinor equations can be brought to the form we found here by redefinitions of the background fields that have no effect on the physics (they still describe AdS 3 × S 1 with the same parameters). The solution to the Killing spinor equation is Each representation is characterized by the central charge −iD θ = k. At first sight this is confusing because in our discussion of section 2.1.2 we found that the central charge was a combination of the KK momentum and the U (1) R charge -this is related to the conventions we used above for defining A µ , which are equivalent to shifting the KK momentum by the R-charge. The supersymmetry algebra that we find from the transformations above is precisely the N = (0, 4) supersymmetry algebra with this central charge, as in section 2.1.2. We have two kinds of representations, the + and the −. The computations are the same as in the N = (0, 2) case. The + representations form a N = (0, 4) hypermultiplet, and the − representations form a N = (0, 4) Fermi multiplet. In particular, the scalars are charged under the SU (2) R symmetry in the N = (0, 4) superconformal algebra, as can be shown from 2 (3.5.7) 3.6 A free N = 2 hypermultiplet in the N = (2, 2) case In this section we discuss the same theory when we put it on AdS 3 × S 1 with the N = (2, 2) superconformal algebra. As opposed to the N = (0, 4) case, here the Killing spinors can have a relative phase (i.e. different θ-dependence). As discussed in appendix B, we will work with θ-independent Killing spinors, such that the SUSY transformations don't mix different KK modes and we can have a supersymmetric theory for every ratio R L . For that, we choose A µ = 0 and the Killing spinor equations for two spinors of opposite chirality are ∂ µ ζ 1 = −iV ρ σ µρ ζ 1 , ∂ µ ζ 2 = +iV ρ σ µρ ζ 2 . (3.6.1) The Lagarangian and the equations of motion are (3.6.2) with the transformation rules (3.6.3) 2 From the Killing spinors, we can construct Vµζiσ µζ j = 2 L aibj − biāj which is coordinate independent, and is therefore a good R-symmetry generator.
The asymptotic dimensions of the fields are (3.6.4) The modes that sit in the same multiplets are (3.6.5) Using (3.6.4), we find that in the same multiplet we have (3.6.6) We see that we can have a consistent supersymmetric theory for any value of R L . For the two signs we find either an N = (2, 2) multiplet that is made out of a N = (0, 2) chiral multiplet and a N = (2, 0) Fermi multiplet, or vice versa.
The bulk action is (3.7. 2) The SUSY variations are In this case the fermions are charged under SU (2) R , as can be seen for example from The dimensions of the scalar and fermions are This N = (0, 4) multiplet is the combination of N = (0, 2) chiral and vector multiplets.
3.8 A free N = 2 vector multiplet in the N = (2, 2) case The transformations in this case are for some constants γ 1,2 , and the Killing spinor equations are as before We can fix the constants γ 1,2 from the demand that We have The transformation is a pure gauge transformation iff The off-diagonal variations of the fermions are (3.8.7) Therefore the equations of motion arē This is also in agreement with current conservation, and without any other conditions. These imply the equation of motion for the scalar (3.8.10) and the action (3.8.11) The dimensions of the fermions and scalar are then (3.8.12) For k = 0, the N = (2, 2) multiplets contain a fermion of dimension 3 2 ± kL R , a scalar and a vector of dimension 1 ± kL R , and a fermion of dimension 1 2 ± kL R . For k = 0, the BF bound of φ is saturated. The physical multiplet contains a 2d gauge field, two dimension 3 2 fermions and two scalars of dimensions 1 and 2 (The last one comes from v θ ). The logarithmic scalar mode is part of the dual (non physical) multiplet and therefore the physical boundary conditions preserve the full superconformal symmetry.

Chern Simons action from AdS 3 × S 1
As we saw in section 3.2.3, the three dimensional action coming from the KK reduction of the four dimensional Maxwell term gives a three dimensional Maxwell term for the k = 0 KK mode. It is well-known [38] that the physical boundary conditions for a gauge field on AdS 3 with just a Maxwell term give a 2d gauge field on the boundary, as we indeed found above. On the other hand, if there is a Chern-Simons term on AdS 3 , the physical boundary conditions will give a conserved current associated with a U (1) global symmetry in the 2d superconformal algebra.
Therefore, it is an interesting question whether we can write a four dimensional theory that will give a Chern-Simons action on the three dimensional AdS space. One way to do this (for flat space times a circle) is to note that [41] showed how to couple 4d vector and two-form multiplets in a supersymmetric and parity-odd way. Denote the superfields (3.9.1) whereG µ = 1 2 ǫ µνρτ ∂ ν B ρτ , B ρτ is a two form, and V, W a are the regular gauge field and field strength multiplets in the Wess-Zumino gauge. The Maxwell-Chern-Simons action in four dimensions is given by In components, the action reads When reducing the theory on S 1 , we get another (one form) gauge field from w i = B (k=0) iθ . The KK zero mode then contains two vector fields A i , w i with Maxwell terms and a mixed Chern Simons term with coefficient m. Therefore, when putting this theory on AdS 3 × S 1 , the physical boundary conditions will lead to conserved currents and their superconformal partners as part of the two dimensional spectrum.
In the same way we construct the supersymmetric multiplets from the superconformal multiplets. For example, we can take a superconformal hypermultiplet with the transformation rules (B.0.12) and plug in the diagonal decomposition rule We get the supersymmetry transformation rules on AdS 3 × S 1 (B.0.14) In the same way, this is done for the other multiplets and the other supersymmetries.

C Comments about R-gauge transformations
As we saw in section 3, there is a freedom in choosing the background R-symmetry gauge field, which results in the Killing spinor in a phase ζ ∼ e inθ . Above we always chose n = 0, but let us see what happens in the more general case. In the four dimensional N = 1 case on AdS 3 × S 1 , is we use such a more general background A µ and represent some scalar field with 4d R-chargeq using its KK modes around the S 1 , the U (1) R charge of the k'th KK mode in the two dimensional superconformal algebra is kL R +q + nqL R (this can be seen from (3.1.11)). We wish to understand if different choices for n lead to different physical theories.
• Ifq is an integer, n can always be absorbed by shifting the definition of k, and therefore it has no consequences on the physics.
• Ifq is irrational, different choices of n will lead to a different spectrum of the theory with different dimensions and two-dimensional R-charges, and thus to a different theory on AdS 3 × S 1 .
• Ifq = M N is rational (where M and N are coprime integers), there are N different physical theories with different spectra that can be obtained by changing n, while two theories with n and n ′ = n + N are physically equivalent.
The Killing spinors should clearly be periodic in θ for the supersymmetry generators to be well-defined. In particular, if the Killing spinors have a phase ζ ∼ e inθ , the supersymmetry transformations relate different KK modes with k f ermion − k boson = ±n, which exist only if n is an integer.
D Six dimensional superconformal algebras on AdS 6 and AdS 4 × S 2 We claim that the six dimensional N = (1, 0) and N = (2, 0) superconformal (SC) algebras do not have subalgebras that close on the isometries of AdS 6 and AdS 4 ×S 2 . One argument for this (say in the AdS 6 case) is that AdS 6 is conformally related to flat space with a boundary, and the boundary conditions on a spinor necessarily modify its chirality, which cannot be done in a supersymmetric way when the supercharges are chiral.
We can also see this in a purely algebraic way. The six dimensional algebra on AdS 6 must be equivalent to the F (4) SC algebra in 5 dimensions which is the only 5 dimensional superconformal algebra. This algebra includes 16 supercharges and SU (2) as its R-symmetry. The N = (2, 0) 6 dimensional superconformal algebra has 32 supercharges, so the numbers fit (after breaking half) but there is no choice of supercharges that will break the R-symmetry to SU (2) (but to a larger group). If we discuss AdS 4 × S 2 , then by counting supercharges, the 6 dimensional N = (1, 0), (2, 0) should correspond to the 3 dimensional N = 2, 4 superconformal algebras with R-symmetries of SO(2), SO(4) respectively. These R-symmetry groups must be a mixture of the SO(3) S 2 isometries and a subgroup of the six dimensional R-symmetry group. For the two cases this cannot be done. More explicitly, in order to form such a subalgebra, there is a limited amount of options to connect the supercharges Q i and S j . Specifically, for AdS 6 , we can choose either Q i + icS i ′ or Q i + icγ * S i ′ . In both cases, the anti-commutators of these charges contain the dilatation operator D, which does not keep us in the bosonic subalgebra we want, but rather brings us back to the full superconformal algebra. The same arguments apply in the case of AdS 4 × S 2 , with the options Q i + icγ 45 S i ′ and Q i + icγ * γ 45 S i ′ .
E Super-Virasoro algebra on AdS 3 × S d−3 In the analysis of section 2 we found the possible 2d superconformal algebras arising in field theories on AdS 3 × S d−3 . The conformal generators that are dual to the AdS isometries are just the global ones (which in the language of Virasoro generators are denoted by L ±1,0 , L ±1,0 ), and the superconformal algebras we found contain just the global superconformal algebra and not the full super-Virasoro algebra. As was shown in [43], when there is a fluctuating gravity theory on asymptotically AdS 3 spacetimes, the symmetry group includes the entire Virasoro group. Here we studied curved manifolds without discussing gravity at all, and we can ask whether our field theories can come from some G N → 0 limit of a gravitational theory.
One basic requirement is that the superconformal algebras we found can be extended into a super-Virasoro algebra. We encounter problems in two cases, the N = (0, 4) and N = (4, 4) superconformal algebras that come from 4d N = 2, 4 supersymmetric field theories on AdS 3 × S 1 . These algebras contain a central extension beyond the known superconformal algebras. As far as we know, these central charges are inconsistent with the extension to a super-Virasoro algebra. Thus, we claim that such field theories cannot appear as decoupled sectors of gravitational theories on AdS 3 , though they could arise as decoupled sectors of gravitational theories in higher dimensions (e.g. on D3-branes wrapping AdS 3 × S 1 inside AdS 5 × S 5 ). It would be interesting to understand how this is consistent with string theories on AdS 3 × S 3 × T 4 , which give rise to such super-Virasoro algebras.