Dual Pair Correspondence in Physics: Oscillator Realizations and Representations

We study general aspects of the reductive dual pair correspondence, also known as Howe duality. We make an explicit and systematic treatment, where we first derive the oscillator realizations of all irreducible dual pairs: $(GL(M,\mathbb R), GL(N,\mathbb R))$, $(GL(M,\mathbb C), GL(N,\mathbb C))$, $(U^*(2M), U^*(2N))$, $(U(M_+,M_-), U(N_+,N_-))$, $(O(N_+,N_-),Sp(2M,\mathbb R))$, $(O(N,\mathbb C), Sp(2M,\mathbb C))$ and $(O^*(2N), Sp(M_+,M_-))$. Then, we decompose the Fock space into irreducible representations of each group in the dual pairs for the cases where one member of the pair is compact as well as the first non-trivial cases of where it is non-compact. We discuss the relevance of these representations in several physical applications throughout this analysis. In particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.


Introduction
The reductive dual pair correspondence, also known as Howe duality [1,2], provides a useful mathematical framework to study a physical system, allowing for a straightforward analysis of its symmetries and spectrum. It finds applications in a wide range of subjects from low energy physics -such as in condensed matter physics, quantum optics and quantum information theory -to high energy physics -such as in twistor theory, supergravity, conformal field theories, scattering amplitudes and higher spin field theories. Despite its importance, the dual pair correspondence, as we will simply refer to it, is not among the most familiar mathematical concepts in the theoretical physics community, and particularly not in those of field and string theory, where it nevertheless appears quite often, either implicitly, or as an outcome of the analyses. For this reason, many times when the dual pair correspondence makes its occurrence in the physics literature, its role often goes unnoticed or is not being emphasized, albeit it is actually acting as an underlying governing principle.
In a nutshell, the dual pair correspondence is about the oscillator realization of Lie algebras. In quantum physics, oscillators realizations -that is, the description of physical systems in terms of creation and annihilation operators, a and a † -are ubiquitous, and their use in algebraically solving the quantum harmonic oscillator is a central and standard part of any undergraduate physics curriculum. Oscillator realizations are more generally encountered in many contemporary physical problems. It can therefore be valuable to study their properties in a more general and broader mathematical framework. The dual pair correspondence provides exactly such a framework, applicable to many unrelated physical problems.
The first oscillator realization of a Lie algebra was given by Jordan in 1935 [3] to describe the relation between the symmetric and the linear groups. This so-called Jordan-Schwinger map was used much later by Schwinger to realize the su(2) obeying angular momentum operators 1 in terms of two pairs of ladder operators, a, a † and b, b † , as follows the U (1) representations; i.e. those labelled by the oscillator number 2j. This interplay between the two groups SU (2) and U (1) is the first example of the dual pair correspondence. A generalization of this mechanism to the duality between U (4) and U (N ) was made by Wigner [4] to model a system in nuclear physics. Replacing the bosonic oscillators by fermionic ones, Racah extended the duality to the one between Sp(M ) and Sp(N ) in [5]. Subsequently, the method was widely used in various models of nuclear physics, such as the nuclear shell model and the interacting boson model (see e.g. [6,7]).
Concerning non-compact Lie groups, an oscillator-like representation, named expansor, was introduced for the Lorentz group by Dirac in 1944 [8], and its fermionic counterpart, the expinor, was constructed by Harish-Chandra [9] (see also [10] and an historical note [11]). 2 In 1963, Dirac also introduced an oscillator representation for the 3d conformal group, which he referred to as a "remarkable representation" [15]. The analogous representation for the 4d conformal group was constructed in [16] (see also [17,18]). The oscillator realization and the dual pair correspondence are also closely related to twistor theory [19][20][21], where the dual group of the 4d conformal group was interpreted as an internal symmetry group in the 'naive twistor particle theory' (see e.g. the first chapter of [22]).
Around the same time, oscillator representations were studied in mathematics by Segal [23], Shale [24] and by Weil [25] (hence, often referred to as Weil representations). Based on these earlier works (and others that we omit to mention), Howe eventually came up with the reductive dual pair correspondence in 1976 (published much later in [1,2]). In short, he showed that there exists a one-to-one correspondence between oscillator representations of two mutually commuting subgroups of the symplectic group. Since then, many mathematicians contributed to the development of the subject (see e.g. reviews [26][27][28] and references therein). The dual pair correspondence is also referred to as the (local) theta correspondence and it has an intimate connection to the theory of automorphic forms (see, e.g., [29,30]).
In physics, oscillator realizations were studied also in the context of coherent states [31][32][33]. In the 80-90's, Günaydin and collaborators extensively used oscillators to realize various super Lie algebras arising in supergravity theories; i.e. those of SU (2, 2|N ) [34,35], OSp(N |4, R) [36,37] and OSp(8 * |N ) [38,39] (see also [40][41][42]). Part of the analysis of representations contained in this paper can be found already in these early references. However, in those papers, the (role of the) dual group was not (explicitly) considered, 3 even though it implicitly appears in the tensor product decompositions. (Its relevance was, however, alluded to in [42].) Higher spin field theory is another area where the oscillator realizations and the dual 2 These representations are closely related to Majorana's equation for infinite component spinor carrying a unitary representation of the Lorentz group, published in 1932 [12]. The complete classification of unitary and irreducible representations of the four-dimensional Lorentz group was later obtained in [9,13,14]. 3 Note that the isometry and R-symmetry groups follow a similar pairing pattern, but they are not reductive dual pairs. pair correspondence were fruitfully employed. Spinor oscillators were used to construct 4d higher spin algebra by Fradkin and Vasiliev [43] and to identify the underlying representations by Konstein and Vasiliev [44]. Fradkin and Linetsky found that the extension of the work to 4d conformal higher spin theory, or equivalently 5d higher spin theory, requires a dual u(1) algebra [45,46]. The same mechanism was used by Sezgin and Sundell in extending Vasiliev's 4d theory [47][48][49] to 5d [50] and 7d [51], with the dual algebras u(1) and su(2) (as we shall later see, the latter algebra is more appropriately interpreted as sp (1)). In 2003, Vasiliev generalized his theory to any dimensions using vector oscillators [52], and revisited its representation theory in [53]. In both of these works, the dual pair correspondence played a crucial role, and since then, it has been used several times within the context of higher spin theories. 4 Despite the abundance of relevant works in physics, and reviews in mathematics, it is hard to find accessible references on the dual pair correspondence for physicists, with a notable exception of [70]. 5 In physics heuristic approaches rather than systematic ones are common, whereas in the mathematics literature, the treatment is formal and explicit examples are rare. The current work is an attempt to close the gap between the physics and math literature. Specifically, we provide a systematic derivation of the oscillator realizations for essentially all dual pairs, as well as, in the relatively simple cases, an explicit decomposition of the corresponding Fock spaces into irreducible representations of each group of the pairs. To be self-contained, we included many basic elements of representation theory related to oscillators. Thus, there will be a considerable overlap with many earlier works mentioned previously. Nevertheless, we believe that our treatment could help in filling several gaps in the current understanding of the use and role of the dual pair correspondence in physics. As the subject of the dual pair correspondence is rich, we will not cover all that we intend to in a single paper. We instead complete our program in (at least) two follow-up papers; one of the same kind as this, but on more advanced issues, and another dedicated to physical applications. Although physical applications will be visited in depth in a sequel, we include brief comments on them throughout the current paper, as summarized below.

Brief summary of the paper
In Section 2, we review generalities of the dual pair correspondence, starting with the precise statement of the duality. In Section 2.1, we recall the definition of the metaplectic representation. In Section 2.2, we outline the classification of irreducible reductive dual pairs. In Section 2.3, we introduce the concept of seesaw pairs, which will be crucial in the rest of the paper. 4 In fact, almost all of the higher spin literature is related to the dual pair correspondence in one way or another. For this reason, we mention just a few papers in which the duality is directly and explicitly used [53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69]. 5 The historical accounts given here are indebted to this review.
In Section 5, we derive the correspondence for "compact dual pairs", i.e. the dual pairs in which at least one member is compact. In doing so, we will recover the familiar oscillator realizations of finite-dimensional representations of compact Lie groups, as well as the lowest weight representations of non-compact Lie groups. We also briefly comment on the application of this correspondence to AdS d+1 /CFT d for d = 3, 4 and 6 (which will be discussed in more details in a follow-up paper).
In Section 6, we derive the correspondence between representations of what will be referred to as "exceptionally compact pairs", i.e. dual pairs in which one member becomes either compact or discrete due to an exceptional isomorphism. We briefly comment on its role in dS d representations for d = 3 and 4.
In Section 7, we derive the correspondence between representations of dual pairs where both groups are non-compact, but one is (almost) Abelian. These cases are different from the compact ones in that the representations are of non-polynomial excitations type. We also present the Schrödinger realizations for these representations (i.e. their realizations on L 2+ǫ spaces).
In Section 8, we discuss interesting aspects of the branching rules of the representations appearing in the dual pair correspondence. We comment on the special cases where one of the groups is the three-or four-dimensional conformal group. In such cases, these representations are known as "singletons" and correspond to free conformal fields.
In Section 9, we derive the relation between the Casimir operators (of arbitrary order) of the two groups of a dual pair.
We leave concluding remarks for the sequel papers. In order to be as self-contained as possible, we included three appendices detailing textbook material, used in the bulk of the paper, as well as one detailing the derivations of the Casimir relations: In Appendix A, we review the realization of finite-dimensional representations of the compact form of the classical groups on spaces of tensors. In Appendix B, we summarize the definition of the various real forms of the classical groups. In Appendix C, we provide the seesaw pair diagrams involving maximal compact subgroups and their dual for all irreducible dual pairs. Finally, Appendix D contains the technical details on the derivations of the Casimir relations.

Conventions
Classical Lie groups and their real forms. We denote the complex classical Lie groups by GL N , O N and Sp 2N , i.e. the general linear, the orthogonal and the symplectic group, respectively. When these Lie groups are viewed as real Lie groups, they are denoted by GL(N, C), O(N, C) and Sp(2N, C). The real forms of GL N and O N will be denoted by GL(N, R), U * (2N ), U (N + , N − ), O(N + , N − ), O(N, C) and O * (2N ) with standard definitions and notations. There is no standard notation for the symplectic groups; we use the notation Sp(2N, R) and Sp(N + , N − ) for the groups with rank N, 2N and N + + N − , respectively. See Appendix B for the details. The multiplicative group of non-zero real, complex and quaternionic numbers are denoted by R × , C × and H × , whereas the additive group of real numbers, isomorphic to the multiplicative group of strictly positive real numbers, is denoted by R + . For brevity, the phrase "generators of the Lie group" (also in symbolic form) will be used when actually referring to generators of the associated Lie algebra.
Young diagrams. A Young diagram of the form, i.e. ℓ k ∈ N corresponds to the length of the k-th row of the diagram and p is the number of rows. It will sometimes be useful to present a Young diagram in terms of its columns instead of its rows. In this case, we use the same bold symbol ℓ, but write its components within square brackets and with upper indices, i.e. ℓ = [ℓ 1 , . . . , ℓ q ] , (1.5) where ℓ k ∈ N corresponds to the height of the k-th column, and q is the number of columns. Clearly, the two descriptions are equivalent. In particular, q = ℓ 1 and p = ℓ 1 . For instance, the diagram, can be presented as ℓ = (5, 3, 2, 2, 1, 1) and ℓ = [6, 4, 2, 1, 1] . (1.7) Tensors and indices. We will use round and square brackets for symmetrization and antisymmetrization of indices, respectively. For instance, T (a 1 a 2 ) = 1 2 (T a 1 a 2 + T a 2 a 1 ) , T [a 1 a 2 ] = 1 2 (T a 1 a 2 − T a 2 a 1 ) , (1.8) and more generally T (a 1 ...an) = 1 n! σ∈Sn T a σ (1) ...a σ(n) , T [a 1 ...an] = 1 n! σ∈Sn sgn(σ) T a σ (1) ...a σ(n) , (1.9) where sgn(σ) is the signature of the permutation σ, which is equal to +1 if σ is an even permutation and −1 if it is odd.
Given a non-degenerate antisymmetric matrix Ω with components satisfying the indices of Sp 2N tensors will be raised and lowered as follows (1.11) In particular, this implies that Oscillators. In this paper, annihilation operators will be denoted by a A or a I A with one or two indices. Their Hermitian conjugate, i.e. the creation operators, will be denoted with a tilde and opposite index position, i.e.
(a I A ) † =ã A I . (1.13) We will refer to the creation and annihilation operators as oscillators. The letters b, c and d will also be used to denote (additional) oscillators.

Representations.
A generic irreducible representation of a Lie group G will be denoted by π G (ζ 1 , . . . , ζ p ) , (1.14) where ζ 1 , . . . , ζ p are the parameters which label the representation. 6 For two particular types of representations, the following specific notation will be used: 6 In general, a representation of G can be induced from a representation of one of its parabolic subgroups P . For this reason, a generic irreducible representation is usually denoted by Ind G P (ζ1, . . . , ζp) .
(1. 15) The parameters ζi indicate a representation of P which induces the representation of G. The couple P and (ζ1, . . . , ζp) are called the Langlands parameters.
• Finite-dimensional irreducible representations of the (double-cover of the) compact Lie groups G = U (N ), O(N ) or Sp(N ), will be denoted by [ℓ, δ] G , where ℓ is a Young diagram, and δ ∈ 1 2 Z is a half-integer such that (ℓ 1 + δ, . . . , ℓ N + δ) , (1.16) is the highest weight of the representation. Here, ℓ p+1 = · · · = ℓ N = 0 for the Young diagram ℓ with height p. The correspondence between these highest weight representations and tensors with symmetry of the Young diagram ℓ is reviewed in Appendix A.
• Infinite-dimensional irreducible representations of a non-compact Lie group G of lowest weight type will be denoted by 17) where ℓ k are the components of the lowest weight defining the representation. If the group G has the same rank as its maximal compact subgroup K, we will combine this notation with the previous one as A central result due to Howe [1,2] is that the restriction of the metaplectic representation W of Sp(2N, R), to be discussed below, to G × G ′ establishes a bijection between representations of G and those of G ′ appearing in the decomposition of W. More precisely, W can be decomposed as where π G (ζ) and π G ′ θ(ζ) are irreducible representations (irreps) of G and G ′ labeled by ζ and θ(ζ) , respectively. The map, defines a bijection between a set Σ G W of G irreps and the corresponding set Σ G ′ W of G ′ irreps. In other words, each G representation appears only once in Σ G W and is paired with a unique G ′ representation in Σ G ′ W , and vice-versa. As a consequence, the representation π G (ζ) occurs with multiplicity dim π G ′ θ(ζ) , and similarly for π G ′ θ(ζ) : More details can be found, e.g. in the review articles [28,70], or in the textbooks [71,72] (where Howe duality is presented within the broader context of duality in representation theory).

Metaplectic representation
The metaplectic representation W of Sp(2N, R) is known under different names, such as the harmonic representation, the Segal-Shale-Weil representation or simply the oscillator representation [23][24][25]. This representation can be realized as a Fock space, which is generated by the free action of creation operators a † i with i = 1, . . . , N on a vacuum state |0 , which, by definition, is annihilated by the annihilation operators a i : This infinite-dimensional space carries an irreducible and unitary representation of the order N Heisenberg algebra, whose generators are represented by the N pairs of creation and annihilation operators, as well as the identity (representing its center). Moreover, the operators bilinear in a i and a † i provide a representation of the Lie algebra sp(2N, R) of the symplectic group Sp(2N, R). More precisely, the operators, satisfy the commutation relations of sp(2N, R), i.e.
In this basis, the operators K i i (where no summation is implied) generate a Cartan subalgebra while the operators K ij and K k l with k < l correspond to lowering operators, and the remaining ones to raising operators. The metaplectic representation is a direct sum of two sp(2N, R) lowest weight representations, The vacuum |0 and a † 1 |0 are the lowest weight vectors of the above two representations with weight ( 1 2 , 1 2 , . . . , 1 2 ) and ( 3 2 , 1 2 , . . . , 1 2 ), respectively. Repeated action of raising operators on these vectors makes up the representations D Sp(2N,R) ( 1 2 , 1 2 , . . . , 1 2 ) and D Sp(2N,R) ( 3 2 , 1 2 , . . . , 1 2 ), which are therefore composed of the states with even and odd excitation numbers, respectively.
The metaplectic representation can also be decomposed in terms of the maximal compact subalgebra u(N ) ∼ = u(1) ⊕ su(N ), where the u(1) and su(N ) subalgebras are generated respectively by it has the u(1) eigenvalue N 2 and carries the trivial representations of su(N )), whereas the N dimensional space generated by a † i |0 carries the irrep [(1), 1 2 ] U (N ) (i.e. it has the u(1) eigenvalue N 2 + 1 and carries the fundamental representation of su(N )). These u(N ) representations are annihilated by the lowering operators K ij and they induce the sp(2N, 1 2 ) by the action of the raising operators K ij .
At this point, we have seen that W carries both a representation of the order N Heisenberg algebra and of the symplectic algebra sp(2N, R). A natural question is then to ask whether this can be uplifted to a group representation in both cases. For the Heisenberg group, the answer is positive: a direct computation shows that the operators The above reproduces the group multiplication of the Heisenberg group H N and hence provides its representation on W.
The situation is a bit more subtle for the symplectic group. It turns out that the exponentiation of the sp(2N, R) representation (2.5) is double-valued. To see whether a representation is uni-valued or not, we need to examine the representation along a loop in the group manifold. Since the loops contractible to a point will always give uni-valued representations, we must consider only the other kind of loops, which are classified by the fundamental group of the group manifold. The fundamental group of Sp(2N, R) is that of its maximal compact subgroup U (N ) ∼ = U (1) ⋉ SU (N ), where the U (1) can be chosen to have an element diag(e i φ , 1, . . . , 1 Since SU (N ) has a trivial first fundamental group, we can just consider this U (1) subgroup, generated by subgroup cannot be taken as the diagonal one, In fact, as a group, and exponentiating K 1 1 , one finds the representation, Since a † 1 a 1 has integer eigenvalues, the U (1) representation (2.12) has the property, that is, the representation map U W is double-valued. One can replace the U (1) subgroup by its double cover to render the representation map U W uni-valued. This would require to replace the symplectic group Sp(2N, R) by its double cover, namely the metaplectic group, often denoted by Sp(2N, R). In principle, of course, one can also consider an arbitrary even cover of Sp(2N, R).
As we discussed just above, the essential point resides in the U (1) part of Sp(2N, R), but it will still be instructive to consider the action of the entire Sp(2N, R) group. For simplicity, let us focus on the N = 1 case, Sp(2, R), whose element can be parameterized as g(φ, ψ, τ ) = cos φ cosh τ − sin ψ sinh τ sin φ cosh τ + cos ψ sinh τ − sin φ cosh τ + cos ψ sinh τ cos φ cosh τ + sin ψ sinh τ , (2.14) with g(φ + 2π, ψ, τ ) = g(φ, ψ, τ ) and g(φ, ψ + π, τ ) = g(φ, ψ, −τ ) and τ ∈ R . The oscillator representation for the group element g(φ, ψ, τ ) can be obtained again by exponentiation and expressed as where β = 1 2 e i (ψ−φ) τ . Again, one can check the double-valued-ness of the map U W : We may avoid this problem by viewing U W as a representation of the double cover Sp(2, R) ; it is parametrized by the same variables as Sp(2, R), but the period of φ is extended to 4π. Since U W is faithful for the double cover (but not for other even covers) of Sp(2, R), it can serve as a definition of Sp(2, R), which does not admit any matrix realization. Instead of considering the double cover, one may also try to cure the problem (2.16) by modifying the representation to which is uni-valued. But, then the above has a modified composition rule, where φ 3 , ψ 3 , τ 3 satisfy g(φ 1 , ψ 1 , τ 1 ) g(φ 2 ψ 2 τ 2 ) = g(φ 3 , ψ 3 , τ 3 ) . Therefore, V W is a projective representation of Sp(2, R) .
A more standard way to introduce the metaplectic representation is as follows: given a pair of creation and annihilation operators (a † , a), one can define a new pair (b † , b) through a Bogoliubov transformation, which is simply a linear transformation, preserving the canonical commutation relation, [b, b † ] = 1. The latter condition implies that A ∈ SU (1, 1), which can be parametrized by Since SU (1, 1) ∼ = Sp(2, R), the two-dimensional symplectic group Sp(2, R) is isomorphic to the center-preserving automorphism group SU (1, 1) of the Heisenberg algebra. Therefore, the Bogoliubov transformation (2.19) defines an action of Sp(2, R) on W , which can be expressed as where U W is given in (2.15) , and A and g(φ, ψ, τ ) in (2.14) are related by Since the formula (2.21) is invariant under redefinition of U W with a phase factor, it provides a projective representation of Sp(2, R).
To conclude, let us consider the reflection subgroup of Sp(2N, R), which will be used in identifying representations of dual pairs. They are contained again in the maximal compact subgroup U (N ) as the diagonal elements, 23) where no summation for the i index is implied. The reflection R i flips the sign of the i-th pair of oscillators: so it has a Z 2 action on the oscillators. Notice, however, that the reflection does not form a Z 2 group, but a Z 4 : This is because we are considering the metaplectic group where the maximal compact subgroup U (N ) should be understood as the double cover of the latter. Since these double cover groups are not matrix groups, "reflections" do not work in the way we know for the matrix group. Viewed as an automorphism group of Heisenberg algebra H N , the reflections form Z 2 groups. Besides the aforementioned reflections, there are many other Z 2 or Z 4 automorphism groups of H N . Obviously, the permutation group S N lying in the subgroup of O(N ) ⊂ Sp(2N, R) contains many S 2 ∼ = Z 2 groups, i.e. the permutations of two oscillators: Less obvious Z 2 automorphisms are which can be decomposed as Here, P i acts on H N as forming a Z 4 automorphism of H N , though it is not compatible with the reality structure of H N . However, one can still view P i as Z 2 automorphisms of Sp(2N, R) preserving the reality condition of the latter. Basically, it interchanges the raising and lowering operators and flips the signs of all generators of the maximal compact subalgebra u(N ), and hence corresponds to the composition of the Cartan and Chevalley involutions. From the identity, we can realize P i as an exponentiation of quadratic oscillators, (2.31) Note however that such a realization of P i is not unitary, and hence they are outerautomorphism of Sp(2N, R). In analyzing dual subgroups of Sp(2N, R), it will be often useful to consider the automorphisms S ij , P ij and P i .
As we have discussed, the metaplectic representation can be easily understood using the Fock realization (also referred to as the Fock model). Another simple description of it is the wave function in the configuration space -the Schrödinger realization. The relation between oscillator realization and wave function realization is nothing but the typical relation between (x i ,p i ) and (a i , a † i ) of the quantum harmonic oscillator, Even though the above relation is a very familiar one, it helps to simplify the description of representations in several cases: a certain complicated state |Ψ in the Fock space becomes simple as wave function x|Ψ , and vice-versa. Another useful description is provided by the Bargmann-Segal realization, wherein the states are holomorphic functions in C N , which are square integrable with respect to a Gaussian measure. In this realization, the creation operators simply act as multiplication z while the annihilation operators act as derivatives with respect to these variables, i.e.
The connection to the Schrödinger realization is made by an integral transform, called the Bargmann-Segal transform [73,74].

Irreducible reductive dual pairs
For a given embedding group Sp(2N, R), we can find many inequivalent dual pairs. Any reductive dual pair (G, G ′ ) has the form, where each couple (G k , G ′ k ) is one of the irreducible dual pairs listed in Table 1 and embedded in a symplectic group Sp(2N k , R) such that N 1 + · · · + N p = N . Moreover, one can distinguish between two types of dual pairs: the first one is a real form of (GL M , GL N ) ⊂ Sp(2M N, R) while the second one is a real form of (O N , Sp 2M ) ⊂ Sp(2M N, R). For instance, in Sp(2N, R) one can find the irreducible pair U (N ), U (1) , which is an example of the first type, or the irreducible pair O(N ), Sp(2, R) , which is an example of the second type. Accordingly, the pair O(N − m) × U (m), Sp(2, R) × U (1) is an example of a reducible dual pair in Sp(2N, R) for m = 1, . . . , N . Table 1. List of all possible irreducible reductive dual pairs (G, G ′ ) embedded in metaplectic groups and their respective maximal compact subgroups (K, K ′ ).
Often irreducible dual pairs are classified, using another criterion, again into two types: the type II if there exists a Lagrangian subspace in R 2N left invariant under both of G and G ′ , and the type I otherwise. In the above table, the first three cases are type II, while the rest are type I.
We can extend the idea of the dual pair correspondence to other groups / representations than the symplectic group / metaplectic representation. For instance, one can also consider supersymmetric dual pairs embedded in OSp(N |2M, R) (see e.g. [72,75] and references therein). Another possible extension is to replace the symplectic group with any simple group G and the metaplectic representation with the minimal representation 8 The classification of such pairs was obtained in [77] (see also [78] for complementary work).

Seesaw pairs
Let us consider the situation wherein we have two reductive dual pairs (G, G ′ ) and (G,G ′ ) in the same symplectic group, say Sp(2N, R). If the groups forming the second pair (G,G ′ ) satisfyG ⊂ G and G ′ ⊂G ′ , then the pair of dual pairs (G, G ′ ) and (G,G ′ ) is called a "seesaw pair", and the situation is depicted as Seesaw pairs satisfy the property, which means the space ofG-equivariant linear maps (or intertwiners) from πG to π G |G, namely HomG πG, π G |G , is isomorphic to the space of G ′ -equivariant linear maps from π G ′ to πG ′ | G ′ , namely Hom G ′ π G ′ , πG ′ | G ′ . Here, |G (resp. | G ′ ) denotes the restriction tõ G (resp. G ′ ). In particular, this implies i.e. the multiplicity of theG-representation πG in the branching rule of π G is the same as the multiplicity of its dual π G ′ in the branching rule of theG ′ -representation πG ′ . This situation is usually depicted as where the downward arrows denote the restriction of a G-irrep (resp.G ′ -irrep) to aG-irrep (resp. G ′ -irrep). Seesaw pairs are particularly useful when trying to derive the explicit correspondence between representations of G and G ′ , assuming that the correspondence is 8 The minimal representation of a simple group G is the representation whose annihilator in the universal enveloping algebra U(g) is the Joseph ideal, which is the maximal primitive and completely prime ideal of U(g) (see e.g. [60,76]).
known for representations of the pair (G,G ′ ), or to obtain the decomposition of a representation into its maximal compact subgroup.
Let us illustrate this with a simple example. Both Sp(2, R), O(2) and U (1), U (2) are dual pairs in Sp(4, R), which form a seesaw pair: . (2.39) Note that the generator of the O(2) subgroup of U (2) correspond to the U (1) generator of SU (2), and not that of the diagonal U (1). We will see later in Section 5 the following correspondences, D Sp(2,R) (ℓ + 1) , (2.40) where ℓ and m are non-negative integers. We can use the seesaw pair property to find the restriction of D Sp(2,R) (ℓ + 1) under its maximal compact subgroup U (1): these representations will be dual to the U (2) representations whose restriction under O(2) contains [ℓ] O (2) . The branching rule U (2) ↓ O(2) amounts to finding those SU (2) irreps which contain a pair of states with weight ±ℓ. As the irrep [m] SU (2) contains states with weight k = −m, −m + 2, . . . , m, we can conclude that the relevant U (2) irreps are those labelled by m = ℓ + 2n with n ∈ N. As a consequence, we find which is indeed the correct decomposition.
We will see in Section 5 that an important subgroup of G to consider in the context of seesaw pairs is the maximal compact subgroup K ⊂ G. Indeed, one can show that the centralizer of K in Sp(2N, R), that we will denote M ′ , contains G ′ so that (K, M ′ ) is a dual pair and it forms a seesaw pair with (G, G ′ ). The same property is true if we reverse G and G ′ : the centralizer M of a maximal compact subgroup K ′ ⊂ G ′ also contains G. As a consequence, (G, G ′ ) and (K ′ , M ) also form a seesaw pair. In general, the groups M and M ′ are not compact, in which case one can also consider their maximal compact subgroup, denoted by K M and K M ′ , respectively. It turns out that also they form a dual pair. We therefore end up with four dual pairs. The situation is depicted as The application of the above diagram to each irreducible dual pair in Table 1 is collected in Appendix C.
Seesaw pairs are also useful for tensor products of the relevant representations and their plethysms. Say, we have a dual pair (G, G ′ ) ⊂ Sp(2N, R) and their representations π G and π G ′ . The tensor product of the representation π G and its contragredient oneπ G can be obtained by considering the seesaw pair, Here, the two pairs (G, G ′ p,q ) and (G p+q , G ′p+q ) are embedded in Sp(2(p + q)N, R). By examining the seesaw pairs for each irreducible pair (G, G ′ ), one can find that the groups G ′ p,q can be taken as in Table 2.
Sp(m, n) Sp(pm + qn, pn + qm) From the seesaw duality, we find that tensor product decompositions of G representations can be obtained by the decomposition of G ′ p,q representations: where the plethysm can be also controlled by a discrete group Γ p,q which is a subgroup of the symmetric group S p × S q permuting the same representations. In particular, every irreducible dual pair (G, G ′ ), wherein G ′ is not the smallest group dual to G, can be used for the tensor product of the irreps of G dual to the smallest G ′ . which obey the canonical commutation relations The GL M and GL N groups are generated respectively by where U W is the metaplectic representation, and X A B and R I J are the matrices with com- Notice that when GL M and GL N are realized as above, a specific ordering of ω and ω is chosen, namely the Weyl ordering. If we do not embed the pair (GL M , GL N ) in Sp(2M N, R), one could take different orderings in each of the groups, leading to different factors of δ A B and δ J I (shift constants) in (3.3). Here, the embedding of the pair in Sp(2M N, R) singles out a unique ordering, which fixes the shift constants, such that the diagonal GL 1 subgroups of the GL M and GL N coincide; i.e.
which is a number operator with a constant shift.
Various real forms of GL M and GL N can be chosen by assigning different reality conditions to the operators ω I A andω A I . Such conditions can be straightforwardly deduced from the anti-involution σ associated with each real form by asking that which can be realized by requiring the ω-operators to obey the reality conditions, Let us define In terms of these creation and annihilation operators, the generators of GL(M, R) and GL(N, R) now read and 11) and the common Abelian subgroup GL(1, R) ∼ = R × is generated by Note that in the above expressions, the repeated indices are summed over even when they are positioned both up or both down. Since GL(M, R) and GL(N, R) are disconnected, 9 we need also to take into account the reflection group elements R A and R I , where R I A are the reflections in Sp(2M N, R) introduced in Section 2.1, (3.14) Let us remark also that the elements, respectively. Note that the generators of these compact groups take the form (creation) × (annihilation), so that they preserves the total oscillator number. The real Lie groups GL(M, C) and GL(N, C) are obtained by starting with complex operators ω andω with complex conjugates ω * andω * , obeying the commutation relations, We require the reality conditions, so that the generators, Notice that X A + B and R +I J generate the GL(M, R) and GL(N, R) subgroup of GL(M, C) and GL(N, C) , respectively. For the oscillator realization, we introduce 22) and the only non-zero commutators are Note that under complex conjugation, the a and b oscillators are mapped to one another: In terms of these oscillators, the generators X A B and R I J read and From the above, one can easily find the generators X A ± B and R ±I J using (3.19) and (3.24).
The common Abelian subgroup GL(1, which can be interpreted as the radial and angular part of the complex plane without origin C × , respectively. The maximal compact subgroups U (M ) and U (N ) of GL(M, C) and GL(N, C) are generated respectively by X A B − (X B A ) * and R I J − (R J I ) * , and they have the form (creation)×(annihilation). Their common U (1) generator coincides with Z − .

3.3
To single out the real forms U * (2M ) and U * (2N ), 10 we require X A B and R I J to satisfy This can be realized by requiring the ω operators to obey the reality conditions where the A, B, and I, J indices take respectively 2M and 2N values. The two matrices Ω AB and Ω IJ are both antisymmetric and square to minus the identity matrix, and Ω AB and Ω IJ are their respective inverses: Ω AB Ω BC = δ A C , Ω IJ Ω JK = δ K I . As a consequence, Ω AB = −Ω AB and Ω IJ = −Ω IJ . Upon defining the oscillators, the generators X A B and R I J can be expressed as where we used Ω AB and Ω IJ (and their inverse) to raise and lower indices as in (1.11). The common center R + of U * (2M ) and U * (2N ) is generated by The Lie algebras of the maximal compact subgroups Sp(M ) ⊂ U * (2M ) and Sp(N ) ⊂ U * (2N ) are generated respectively by Again, one can see that the generators of these compact subgroups have the form of (creation) × (annihilation).
To single out the real forms U (M + , M − ) and U (N + , N − ), we impose where η AB and η IJ are diagonal matrices of signature (M + , M − ) and (N + , N − ) respectively, and where X A B and R I J are defined in (3.3). This can be achieved by requiring the ω operators to obey the reality condition, To introduce the oscillator realization, we split the A and I indices into (a, a) and (i, i) and set η AB = (δ ab , −δ ab ) and η IJ = (δ ij , −δ ij ). Here, a and a take respectively M + and M − values, whereas i and i take respectively N + and N − values. Upon defining we end up with (M + +M − )(N + +N − ) canonical pairs of creation and annihilation operators, as these oscillators obey with all other commutators vanishing. In terms of the a, b, c and d oscillators, the generators The maximal compact subgroups U (M + ) × U (M − ) ⊂ U (M + , M − ) and U (N + ) × U (N − ) ⊂ U (N + , N − ) are respectively generated by X a b and X a b , and R j i and R j i . Note also that when N − = 0 or M − = 0, that is, when one of the group, say G, in the pair (G, G ′ ) is compact, the Lie algebra of the dual group G ′ can be decomposed into eigenspaces of the total number operator with eigenvalues +2, 0, −2. In other words, they do not have the mixed form of (creation) 2 +(annihilation) 2 . This property will prove useful when decomposing the Fock space into irreducible representations. and which satisfy where E AB and Ω IJ are symmetric and antisymmetric invertible matrices of dimension N and 2M , respectively. Then, the reductive subgroups O N and Sp 2M are generated respectively by where U W is the metaplectic representation, and M AB and K IJ are the matrices with Note here that we have not specified the signature of the flat metric E AB . Clearly, at the level of the complex Lie algebra, different E AB can be all brought to the form δ AB by suitably redefining the y-operators. We leave the ambiguity of E AB at this stage since there is a preferable choice of E AB in each real form of O N to obtain a compact expression of their generators.
Various real forms of O N and Sp 2M can be chosen by assigning different reality conditions to the operators y I A . Such conditions can be straightforwardly deduced from the anti-involutionσ associated with each real form by asking See Appendix B for the expression ofσ in each real forms. In the following, we describe all such real forms by introducing their oscillator realizations.
To identify the real forms O(N + , N − ) and Sp(2M, R), we set E AB = η AB , the flat metric of signature (N + , N − ), and require that the generators M AB and K IJ satisfy the reality conditions, where the matrix J IJ satisfy J IJ = J JI , J IJ J JK = δ IK and J IJ Ω JK = −Ω IJ J JK . 12 The above can be realized by requiring the y I A operators to obey the reality condition, In order to introduce the oscillator realization, we split the A and I indices into A = (a, a) and I = (+i, −i) where a and a take respectively N + and N − values. In this convention, η AB = (δ ab , −δ ab ) , Ω +i −j = δ ij and J +i −j = δ ij , and the condition (4.10) becomes We therefore introduce 13) and the canonical commutation relation (4.14) In terms of the canonical pairs a and b, the generators M AB and K IJ read and Since O(N + , N − ) has 4 connected components (2 components when N + or N − vanishes), we need to take into account the reflections, where R i a and R i a flip the sign of a i a and b i a respectively.
Notice that M ab and M ab generate the maximal compact subgroup When N − = 0, no generator of Sp(2M, R) has the mixed form of (creation) 2 +(annihilation) 2 . Again, this will prove useful in decomposing the Fock space into irreducible representations. 12 Note here that we could use the more familiar reality condition, which can be realized by the operators satisfying Even though the above seems simpler, it actually leads to slightly more involved and unfamiliar expression for K IJ in terms of oscillators. That is why we chose another, but equivalent, reality condition for K IJ .

4.2
The real Lie groups O(N, C) and Sp(2M, C), are obtained by starting with complex operators y with complex conjugates y * , obeying the commutation relations, We require the reality condition, and hence the generators of O(N, C), which are given by and M + AB and K IJ + generate the subgroups O(N ) and Sp(2M, R) of O(N, C) and Sp(2M, C) respectively . To define the oscillator realization, we introduce In terms of these oscillators, the generators M AB and K IJ read and Since O(N, C) has two connected components, corresponding to elements of determinant ±1 respectively, we need to take into account the reflections, where R I A flips the sign of a I A . The generators of the maximal compact subgroup while the generators of Sp(M ) ⊂ Sp(2M, C) are given by Once again, we can notice that the maximal compact subgroups O(N ) and Sp(M ) are generated by operators of the form (creation) × (annihilation).

4.3
To identify the real forms O * (2N ) and Sp(M + , M − ), we begin with the metric E AB = J AB where J AB is the symmetric off-diagonal matrix defined, in terms of A = (+a, −a), as The reality conditions to be imposed on M AB and K IJ are where Ψ IJ is an antisymmetric matrix given by The conditions (4.32) can be realized by requiring the y operators to obey the reality condition, Splitting again the A and i indices into A = (+a, −a) and i = (r, r) so that Ω +a −b = δ ab and η ij = (δ rs , −δ rs ), the reality condition reads By repairing the symplectic indices as R = (+r, −r) and R = (+r, −r), we introduce where Ω −r +s = δ rs and Ω −r +s = δ rs . These 2N (M + + M − ) pairs of oscillators satisfy In terms of a, b oscillators, the generators M AB read and K IJ has altogether 3 types of components given by

Representations of compact dual pairs
In the previous sections, we have introduced oscillator realizations for various dual pairs, and made two observations: 1. Generators of compact (sub)groups K ⊆ G of dual pairs (G, G ′ ) are of the form (creation)×(annihilation). In other words, the Lie algebra of K is the eigenspace of the total number operator with eigenvalue 0.
2. When K = G, the dual group G ′ does not have any generators of the mixed form, (creation) 2 +(annihilation) 2 . 13 In other words, the Lie algebra of G ′ can be decomposed into eigenspaces of the total number operator with eigenvalues +2, 0, −2: The subalgebra g ′ 0 corresponds to the Lie algebra of the maximal compact subgroup To analyze the representations of the dual pairs, we first construct the Fock space W using the oscillators in the usual manner: Here, I is a collective index which can be specified in each dual pairs, and the number n of W n stands for the total oscillator number. Since all the states of W are positive definite, all irreducible representations appearing in the decomposition of W are unitary. Obtaining this decomposition becomes particularly simple when K = G due to the properties 1 and 2. From now on, we assume that G is compact.
From the property 1, we know that G commutes with the total number operator, and hence the eigenspace W n can be decomposed into several irreducible representations π G (ζ): where the first sum runs over a finite number of labels ζ, m ζ n is the multiplicity of π G (ζ) in W n , and V ζ n,i is the representation space of π G (ζ) in W n with the multiplicity label i. The restriction to π G (ζ) in W n therefore reads Consequently, a finite-dimensional irreducible representation π G (ζ) is realized in infinitely many subspaces W n : where N ζ is the set of integers n such that π G (ζ) appears in W n , or equivalently the possible numbers of oscillators with which a state in π G (ζ) can be realized. All V ζ n,i with n ∈ N ζ and i = 1, . . . , m ζ n have to be related to one another by the dual group G ′ , since the decomposition of W under G × G ′ is multiplicity-free (i.e. G ′ acts irreducibly on W| G ζ ). In particular, the space m ζ n i=1 V ζ n,i carries a m ζ n -dimensional representation of the maximal compact subgroup K ′ of G ′ .
When G ′ = K ′ , its generators also commute with the total number operator, and hence they preserve the subspaces W n . Consequently, the set N ζ has only one element and the corresponding m ζ n -dimensional representation ζ ′ of G ′ is irreducible. In this way, we find the duality between the representations ζ and ζ ′ for the dual pairs of the type (K, K ′ ). In fact, the only such case is the (U (M ), U (N )) duality.
When G ′ = K ′ , the set N ζ will have the form, since the generators G ′ are quadratic in the oscillators. Here, n ζ is the minimum oscillator number with which ζ is realized. Now, from the property 2, we find that the Lie algebra of G ′ admits the decomposition (5.1) and the lowering operators in g ′ −2 annihilate the states V ζ n ζ ,i . In this way, we demonstrate that the dual representation ζ ′ of G ′ is a lowest weight representation. 14 In the following, we will review the correspondences between the representations of dual pairs, in which at least one member admits a compact real form. This correspondence was originally derived in [79] (see e.g. [28] for a review). Here, we rederive the same results using the oscillator realizations presented above. Notice that a similar treatment can be found in Appendix B of [55], though the discussion was at the level of complex Lie algebras. Let us also mention that in [34][35][36][37][38][39][40][41][42], representations of U (N + , N − ), Sp(2N, R), O * (2N ) are studied using the same oscillator realizations as the ones used in this paper. However, in these references, the role of the dual pairs and the correspondences of their representations were not explored.
Let us begin the analysis with the simplest case, the (U (M ), U (N )) duality. In this case, the groups U (M ) and U (N ) are respectively generated by where a, b = 1, . . . , M and i, j = 1, . . . , N . The Fock space W is generated by polynomials inã a i , namely Any state |Ψ ∈ W L can be expressed as in terms of a tensor Ψ which are pairwise symmetric: Ψ ···i k ···i l ··· ···a k ···a l ··· = Ψ ···i l ···i k ··· ···a l ···a k ··· . The upper and lower indices of Ψ carry tensor representations ⊗L of U (M ) and U (N ) respectively, where denotes their fundamental representation.
Let us first consider the U (M ) representation: ⊗L can be decomposed into irreducible representations corresponding to the Young diagrams with height not greater than M , and we pick up the irreducible representation corresponding to the Young diagram, where ℓ k is the length of the k-th row and p is the height h(ℓ) of ℓ, and L = |ℓ| := ℓ 1 +· · ·+ℓ p . Since the U (M ) generators can be realized in a Fock space with any constant shift, such as N 2 δ a b in (5.7), we need to specify it together with a Young diagram when labeling the U (M ) representations. Note that this shift only affects the representation of the diagonal U (1) in U (M ), but not the SU (M ) part. Taking this into account, let us label the U (M ) representation as where |ℓ| + N M/2 is the eigenvalue of the U (1) generator. Recell that the shift constant N/2 can be modified to an arbitrary half-integer by choosing a different embedding of . In this way, we can produce all U (M ) representations using an oscillator realization. The highest weight of [ℓ, N 2 ] U (M ) simply reads Now, let us see to which U (N ) representation the [ℓ, N 2 ] U (M ) representation corresponds. We can do that by picking up a particular state in the [ℓ, N 2 ] U (M ) representation -which consists of multiple states |Ψ in W L -and reading off the U (N ) representation from such states |Ψ . Let us pick up the lowest weight state in the [ℓ, N 2 ] U (M ) representation. By acting with the lowering operators (X a b with a < b) of U (M ) , we bring |Ψ to the lowest weight state of [ℓ, N 2 ] U (M ) , corresponding to the Young tableau, (5.14) Such states |Ψ satisfy To identify the solution space of the above condition, it will be convenient to first re-express the state |Ψ or the tensor Ψ as .
Then, the condition (5.15) is translated to To rephrase the above condition in words, the symmetrization of the ℓ a indices in the a-th group with one index i b in the b-th group vanishes identically. This is nothing but the definition of the same Young diagram ℓ, but this time it designates U (N ) representation, Therefore, we find the correspondence of the representations: where the height h(ℓ) of the Young diagram should be bounded by both M and N : h(ℓ) ≤ min{M, N }.
where a, b = 1, . . . , M + , a, b = 1, . . . , M − and i, j = 1, . . . , N . The Fock space W is generated by polynomials in two families of oscillators, namely Recall for a dual pair (G, K ′ ) with K ′ a compact Lie group, the realization of a finitedimensional irrep of K ′ with the minimal number of oscillators in W also forms an irreducible representation of the maximal compact subgroup K ⊂ G.
According to the seesaw duality, A lowest weight state |Ψ of (5.24) can be expressed as where the upper and lower indices of Ψ have the symmetries of the Young diagram ℓ and m respectively, with h(ℓ) = p and h(m) = q. Such a state is annihilated by X b a , as these generators decrease the total oscillator number, and |Ψ is a state with the lowest number of oscillator in the irreducible representations of U (M + , M − ) and U (N ). This condition is translated to In words, the Ψ tensor is traceless in the sense that any contraction between upper and lower indices vanishes identically. This defines the irreducible representation [ℓ⊘ m, See Appendix A for more details on general U (N ) representations. In this way, we see In the end, we find the following correspondence, For M + = M − = 1, due to the isomorphism between U (1, 1) and U (1) × SL(2, R) representations, we can write Let us briefly comment on an example, which is important in physics. The doublecover of the four-dimensional conformal group SO The SO + (2, 4) representations appearing in this duality describe CFT 4 operators or equivalently fields in AdS 5 . In light of the previous dictionary, it will be convenient to parametrize with k, n ∈ N and s i satisfying Moreover, s 1 and s 2 are either both integers or bother half-integers. Note that s 1 is positive while s 2 can be positive or negative. The correspondence (5.30) then reads For low values of N , the U (N ) labels (5.33) are restricted and the dual SO + (2, 4) representations correspond to particular fields: • N = 1: We should require k = n = 0 and s 1 = ±s 2 , so that the correspondence reads These SO + (2, 4) representations describes four-dimensional massless fields of helicity ±s, also known as singletons.
• N = 2: There are two types of U (2) irreps that can occur.
The first one characterized by n = k = 0, and for which the correspondence reads These SO + (2, 4) representation correspond to CFT 4 conserved current of spin (s 1 , s 2 ), or equivalently to massless fields in AdS 5 with the same spin.
The second type is characterized by either k = 0 and s 1 = s 2 = s, or n = 0 and s 1 = −s 2 = s, and the correspondence reads These SO + (2, 4) representations correspond to all possible CFT 4 operators with spin (s, ±s) and twist τ = ∆ − s 1 ≥ 2, or equivalently massive AdS 5 fields with the same spin.
As expected, these are all the possible representations appearing in the decomposition of the tensor product of two singletons, which was given in [80].
• N ≥ 4: No restriction is to be imposed on the U (N ) labels (5.33) and the dual representations correspond to massive AdS 5 fields of any spin or CFT 4 operators with twist τ ≥ N .
Notice that in [35,42] the oscillator realization of SU (2, 2|4) was obtained and some tensor product decompositions of singletons were analyzed. This dual pair correspondence was also used in the context of AdS 5 /CFT 4 for higher spin gauge theory in [51]. More recently its relevance as a possible tool for computing scattering amplitudes was pointed out in [81,82].
where a, b = 1, . . . N and i, j = 1, . . . M , and the Fock space W is spanned by polynomials in one family of oscillators, namely Similarly to the previous section, we can use the seesaw duality, This means that by considering the lowest weight state of [ℓ, 0] U (M ) like (5.14), the state |Ψ can be expressed as meaning that Ψ is a traceless tensor. The traceless condition imposes ℓ 1 + ℓ 2 ≤ N besides ℓ 1 ≤ N . The traceless Young diagram ℓ specifies a unique irreducible representation of O(N ): where ℓ 1 ≤ min(M, N ) and ℓ 1 + ℓ 2 ≤ N .
For M = 1, we can use the isomorphism Sp(2, R) ∼ = SL(2, R) and find the discrete series, The N = 1, 2 and 3 cases cover all available discrete series representations. Compared to (U (1, 1), U (M )) dual pairs, the (Sp(2, R), O(N )) pairs also give the irreps with half-integral lowest weights for odd N , which represent the double cover SL(2, R).
Notice that the above correspondence remains valid even for N = 1, where the O(1) ∼ = Z 2 representation just reduces to a sign [±1] Z 2 , and we find the duality, wheren = 0 or 1. In terms of highest weight, the former gives which is in accordance with the discussion of Section 2.1, and in particular (2.8).
Let us briefly comment on an example, which is important in physics. The double cover of the three dimensional conformal group SO + (2, 3) is isomorphic to Sp(4, R). Consequently, the lowest weight representation of D Sp(4,R) (ℓ 1 , ℓ 2 ), N 2 ] corresponds to the SO + (2, 3) representation D SO + (2,3) (∆; s) with conformal dimension/minimal energy ∆ and spin-s given by with s ∈ 1 2 N and n ∈ N, so that the correspondence reads For low values of N , not all two-row Young diagrams define a representation of O(N ): • N = 1: We find the correspondence, 54) where the SO + (2, 3) representations are respectively isomorphic to the spin-0 singleton (or Rac) and spin-1 2 singleton (or Di). Originally discovered by Dirac [15], they describe a free conformal scalar and spinor respectively, in three dimensions.
• N = 2: The O(2) irreps are labelled by a Young diagram which is either a single row of length 2s (i.e. n = 0), or a single column of height two (i.e. s = n = 1). For a single row diagram, we have and the SO + (2, 3) representation describes a spin-s conserved current in three dimensions or a spin-s massless field in AdS 4 , for s > 1 2 . For s = 0 (resp. 1 2 ), this representation describes a scalar (resp. spinor) CFT 3 operator / AdS 4 field with conformal dimension / minimal energy 1 (resp. 3 2 ). For the single column diagram, we have 56) and the SO + (2, 3) representation describes a scalar CFT 3 operator / AdS 4 field with conformal dimension / minimal energy 2. As expected, this spectrum is that of the decomposition of the tensor product of two singletons, first derived in [83] (later revisited and extended to arbitrary dimensions in [53,80,84]).
• N = 3: In this case, we can only consider n = 0 or n = 1 so that the correspondence reads In other words, all operators of twist τ = 3 2 and τ = 5 2 / massive fields with spin-s and minimal energy s + 3 2 or s + 5 2 appear.
• N ≥ 4: No restriction is to be imposed on the O(N ) labels and the corresponding SO + (2, 3) representations describe all operators with twist τ = k + N 2 / massive fields with spin-s and minimal energy s + k + N 2 .
Notice that the spectrum of the N -th tensor product of three-dimensional singletons was already obtained in [85] using the oscillator realization used here (see also [36,86] and references therein for earlier works).
where a, b = 1, . . . , n and R, S = 1, . . . , 2M . The Fock space W is spanned by polynomials in one family of oscillators, namely We use again the seesaw duality, where U (N ) and U (2M ) are generated by M +a−b andã R a a a S respectively. The dual pair (U (N ), U (2M )) gives the correspondence, This means that by considering the lowest weight state of [ℓ, M ] U (N ) like (5.14), the state |Ψ can be expressed as meaning that the tensor Ψ is traceless in the symplectic sense. The symplectic traceless condition imposes the stronger condition ℓ 1 ≤ M than the usual one ℓ 1 ≤ 2M . The symplectic traceless Young diagram ℓ specifies a unique irreducible representation of Sp(M ), hence we find the correspondence, Let us consider the N = 1, 2 cases. When N = 1, we have the correspondence, is the special subgroup part of the maximal compact subgroup U (2), and SL(2, R) is generated by with the usual Lie brackets, In terms of SU (2) × SL(2, R) irreps, the O * (4) representation reduces to (2) is the (ℓ 1 − ℓ 2 + 1)-dimensional representation, and with the understanding that ℓ 2 = 0 when M = 1.
Let us briefly comment on an example, which is important in physics. The doublecover of the six-dimensional conformal group SO + (2, 6) and O * (8) are isomorphic. The lowest weight representation O * (8) corresponds to that of SO + (2, 6) as where n, s i satisfy and either n, s 1 , s 2 , s 3 are all integers or all half-integers. The ± sign represents two possible isomorphisms between O * (8) and SO + (2, 6) , which reflects the possibility of parity transformation. Note however that an oscillator realization of SO + (2, 6) provide representations with + or − sign but not both.
For low values of M , a few representations are allowed for Sp(M ): • M = 1: Only one row Young diagram is allowed for ℓ, and we find the correspondence, The SO + (2, 6) representations describe the six-dimensional spin-s singletons. Notice that only positive helicity singletons appear in the decomposition of the Fock space.
• M = 2: The height of ℓ is at most two, and we find the correspondence, The SO + (2, 6) representation describes a conserved current with spin (s 1 , s 2 ) in six dimensions or a massless field with the same spin in AdS 7 . Here again, this coincides with the spectrum of the tensor product of two higher-spin singletons (of the same chirality), which was worked out in [80].

Representation of exceptionally compact dual pairs
In this section, we consider two dual pairs wherein one member becomes compact or discrete for an exceptional reason: the Lie groups O * (2N ) and O(N, C) become compact and discrete, respectively, only for N = 1. As we will discuss next, these cases require to consider an outer automorphism of the embedding symplectic group in order to make sense of the reductive dual pair correspondence.   To analyze the relevant representations, we begin with the seesaw duality, and can therefore result from the tensor product From the seesaw pair (6.4) and correspondence (6.5), we therefore deduce that the dual with Fock states, As a consequence, we find the correspondence, The SO + (1, 4) representation describes a massless field of helicity ±s in dS 4 [88][89][90]. In fact, this representation lies in the discrete series: it corresponds to the irrep π ± p,q with p = q = s of [91] and the irrep D ± ℓν with ℓ = s − 1 and ν = 1 in [92].
The groups of the dual pair Sp (2N, C), O(1, C) ⊂ Sp(4N, R) are generated by and its complex conjugate 15 (K IJ ) * with I, J = 1, · · · , 2N , and the reflection, Again the centralizer of O(1, C) is not Sp(2N, C) but Sp(4N, R) itself. We can resolve this problem like the previous case of (Sp(p, q), O * (2)) by enlarging the embedding group Sp(4M, R) with the Z 2 action, In this way, we get the dual pair (Sp (2N, C), Let us consider the seesaw duality,  (1)) gives and we find the correspondence, Notice that when N = 1, we have the isomorphism Sp(2, C) ∼ = SO + (1, 3) , and the irreps π Sp(2,C) (0) and π Sp(2,C) (1) correspond to the following irreps in the classification [9] of Harish-Chandra: They sit in the complementary and the principal series, describing respectively a conformally coupled scalar and spinor in dS 3 . Their tensor product will be considered in the next section, where we analyze the duality (O(2, C), Sp (2N, C)). These representations are also the relevant ones in Majorana's infinite component spinor equation [12].

Representations of the simplest non-compact dual pairs
When both of the groups in dual pairs are non-compact, their representations have very different features compared to the case where at least one member is compact. In the latter case, we have seen that the representation space of G or G ′ is spanned by states in the Fock space with a certain excitation number, or in other words, a state is produced by the action of a homogeneous polynomial of the creation operators on the vacuum state. However, in the case of non-compact dual pairs, none of the vectors in the representation space are realized as such polynomials. They instead consists of infinite linear combinations of excitation states, i.e. they are a certain kind of coherent states, and the norms of these states are divergent implying the underlying representations are tempered ones. The groups of the dual pair GL(N, R), GL(1, R) are respectively generated by and where A, B = 1, . . . , N . For the analysis of the representations associated with this duality, let us consider the seesaw pair, where the maximal compact subgroup O(N ) is generated by with the standard Lie bracket (5.68). We begin with the correspondence, and consider the restriction SL(2, R) ↓ R + × Z 2 where the R + factor is generated by 6) and the Z 2 is the reflection group with the reflection element R acting as The restriction of SL(2, R) to the Z 2 part determines the parity of ℓ. The restriction of SL(2, R) to the R + part amounts to finding the spectrum of Z in the positive discrete series representation D SL(2,R) . To tell the conclusion first, the spectrum is the entire set of pure imaginary numbers (as Z is anti-Hermitian), and any Z-eigenstate is an infinite linear combinations of H-eigenstates, namely a coherent state. As is usual for a coherent state, Z-eigenstates in W do not have finite norms and can be considered only as tempered distributions. More detailed discussions on this point will be provided below. To summarize, for a fixed eigenvalue i ζ of the Z generator of R + with ζ ∈ R and the sign ± for Z 2 , the dual GL(N, R) representation consists of all even/odd O(N ) tensors: Note that the representation space of GL(N, R) does not depend on ζ, but only onn (which is 0 or 1). The ζ dependence can be seen only from the actions of GL(N, R). This is a generic feature of a principal (or complementary) series representation in contrast to the discrete series ones that appeared for the dual pairs involving at least one compact group.
Let us find the explicit GL(N, R) action by solving the GL(1, R) conditions, According to (7.5), a vector in the representation is realized by the following state in W, up to a normalization constant. Here, {A 1 · · · A ℓ } denotes the traceless part of the symmetrization (A 1 · · · A ℓ ). In order to solve the conditions (7.9), we consider an infinite linear combination of Fock states, namely a coherent state, corresponding to an infinite linear combination of H-eigenstates, and ask them to satisfy (7.14) As a consequence, a general state |Ψ ζ,n satisfying (7.9) will be given as a linear combination, The Z-eigenstate condition (7.14) reads The above can be translated into a differential equation 16 whose solution can be uniquely determined upon imposing h ζ,α (0) = 1 as Note that the above function has the symmetry, Let us examine the norm of the states |T A 1 ···A ℓ ζ with the simplest example of the scalar states with ℓ = 0 : The above integral can be simplified by performing the change of variables t = tanh τ and s = tanh σ, then u = σ + τ and v = (σ − τ )/2. This leads to 16 One may also try to solve the Z-eigenstate condition starting from a generic state, where Ψ ζ,0/1 ( x) is a even/odd scalar function of an N dimensional vector x. Then the condition of (7.14) reduces to the differential equation, having the form of the time independent Schrödinger equation for N -dimensional harmonic oscillator with imaginary energy. Since Ψ ζ,n ( x) is not a wave function with L 2 scalar norm but a (coherent) Fock state, the "imaginary energy" does not violate unitarity.
The interested readers may consult [93] for additional details where the N = 1 case was treated explicitly within the context of the SL(2, R) coherent states. The state |T A 1 ···A ℓ ζ has a divergent norm in the Fock space, due to the factor δ(ζ − ζ ′ ) appearing in the product of two states |T A 1 ···A ℓ ζ and |T . This reflects the fact that states in different representations are orthogonal to one another, but the Dirac distribution blows up when the two GL(1, R) representations coincides. This is a usual property of a coherent state, and from the representation point of view, this means that the relevant representation is a tempered one.
Let us now spell out the GL(N, R) action on the basis |T Using the differential equations of f ζ,ℓ (z) and the decomposition, one can derive the action of X {A B} on |T A 1 ···A ℓ ; ζ as where the coefficients p ℓ , q ℓ and r ℓ are functions of ℓ, N and ζ : In this way, we find explicit expression of the GL(N, R)-representation realized on the space of even or odd symmetric O(N ) tensors.
For N = 2, the restriction of π GL(2,R) (ζ,n) to SL(2, R) gives the principal series representations, describing scalar/spinor tachyons in AdS 2 or massive scalar/spinor in dS 2 . Together with the discrete series representations D SL(2,R) (h + 1) with h ∈ N appearing in the (U (1, 1), U (1)) pair, these are all oscillator realizations of SL(2, R) in the (GL 2 , GL 1 ) dual pairs. Note that the complementary series representations of SL(2, R) do not appear in this oscillator realization.
In fact, the representation we described above becomes much simpler by moving back to the ω A andω A operators with the reality condition (3.7), namely, the Schrödinger realization with real variables x A : In this realization, the GL(N, R) generators have the form, The exponentiation of the above gives the action of a group element g ∈ GL(N, R) as whereas the dual GL(1, R) = R × acts for an element a as In the above, x g should be understood as the right multiplication of the N × N matrix g on the N vector x, i.e. (x g) A = x B g B A , where g A B are the components of g,. The Z-eigenstate condition becomes the homogeneity condition, x | U W (a) |Ψ ζ,n = sgn(a)n |a| i ζ x |Ψ ζ,n . (7.33) Using the above condition, we can reduce the representation space to the space of functions where ψn(−x) = (−1)nψn(x) . In other words, the representation space is the space of functions on RP N −1 . The scalar product is inherited from that of L 2 (R N ) as Recall that the same delta function appears also in the Fock realization where we use the basis of symmetric O(N ) tensors. Let us also recall that the relation between a L 2 (R N ) wave function and a Fock state |Ψ = Ψ(a † ) |0 is simply and hence one can move from one realization to the other using the above expression. The Schrödinger realization of the GL(N, R) representation dual to GL(1, R) is in fact what is referred to as the most degenerate principal series representation (see e.g. [94,95]).
It is worth noting that the most degenerate principal series representation belongs to a more general class of representations realized as functions on the space of M × N real matrices x with M ≤ N , endowed with the right action,  In this notation, the most degenerate principal series is π GL(N,R) (ζ,n) = π GL(N,R) (ζ,n; 1) .
Finally, let us briefly mention that the relevance of the above representations for 3d scattering amplitude: the scattering amplitude of N + incoming and N − outgoing 3d conformal fields can be viewed as the Poincaré singlet in the tensor product of N + singletons and N − contragredient singletons, (or, N + lowest (energy) weight and N − highest (energy) weight singletons). The situation can be controlled by the seesaw pair, where GL(2, R) decomposes into the dilatation GL(1, R) and the 3d Lorentz group SL(2, R), wherein we require the latter to take the trivial representation (so that the amplitude is Lorentz invariant). This is precisely the representation dual to the determinant-homogeneous degenerate principal series representation π GL(N + +N − ,R) (ζ,n) , which should be restricted to the irrep of O(N + , N − ), whose dual representation is translation invariant. This irrep should be responsible for the form factors of the scattering amplitudes. We shall provide more detailed analysis on this point in a follow-up paper. The groups of the dual pair GL(N, C), GL(1, C) are respectively generated by and where A, B = 1, . . . , N . As GL(1, C) ∼ = R + × U (1) is Abelian, its representations are simply given by eigenvectors of Z + and Z − , generating respectively R + and U (1) . The decomposition of the GL(N, C) representation under its maximal compact subgroup U (N ) is given by the seesaw pair, and They are related to the GL(1, C) generators through Let us consider an irreducible representation of GL(1, C) in the Fock space, where ζ ∈ R and m ∈ Z. Henceforth we assume m ≥ 0 as the m < 0 case can be treated analogously. According to the correspondence As a consequence, any state |Ψ ζ,m in π GL(N,C) (ζ, m) can be written as where δ A 1 B 1 Ψ B 1 ···Bn A 1 ···A m+n = 0 and the functions h ζ,m+2n+N (z) = f ζ,m,n (z) satisfy the differential equation (7.19), and can be uniquely determined with the condition f ζ,m,n (0) = 1 . In this way, we find the coherent states carrying a GL(N, C) representation dual to [ζ, m] GL(1,C) . In the case m < 0, the representation is constructed in the exact same way, except that the a and b oscillators are exchanged.
Similarly to the (GL(N, R), GL(1, R)) case, the scalar product of the states |Ψ ζ,m and |Ψ ζ ′ ,m ′ is proportional to δ(ζ − ζ ′ ) implying that the relevant representations are tempered ones. They are again known as the most degenerate principal series representation, and we can see this either by directly computing the norm using the precise form of f ζ,m,n or by moving back to the ω A andω A operators with the reality condition (3.18). The latter amounts to using the Schrödinger realization with complex variables z A : In this realization, the Z ± generators read while the GL(N, C) generators are given by and their complex conjugate. The associated group action reads where z ∈ C N . The Z-eigenstate condition gives the homogeneity condition, Using the above condition, we can reduce the representation space to the space of functions on S 2N −1 , If we mod out this U (1) symmetry, then the representation space is reduced to the space of functions on CP N −1 ∼ = S 2N −1 /U (1) , The scalar product is inherited from that of L 2 (C N ), and reads This representation can be generalized to the space of M × N complex matrices z with M ≤ N , endowed with the right action of GL(N, C) , and the left action of GL(M, C) , which can be interpreted as a determinant-homogeneity condition (see e.g. [96]). The GL(N, C) representation given in ( In this notation, the most degenerate principal series is π GL(N,C) (ζ, m) = π GL(N,C) (ζ, m; 1) .
Like the (GL(N, R), GL(2, R)) case, the degenerate principal representations appearing in the dual pair (GL(N, C), GL(2, C)) is relevant for the scattering amplitudes of 4d conformal fields. Consider the seesaw pair, where we can apply exactly the same logic as the seesaw pair for 3d amplitudes (7.44). Again, more detailed analysis on this point will be provided in one of our follow-up papers.
Let us consider the seesaw dual pairs, where O * (4) is generated by the SU (2) generators R IJ , while the SL(2, R) ones are given by In this way, we find the correspondence, Let us consider an irreducible representation of U * (2) in the Fock space, The decomposition (7.76) tells us that |Ψ m ζ,m can be written as where Ω A 1 B 1 Ψ m A 1 ...A m+n ,B 1 ...Bn = 0 and the function h ζ,m+2n+2N (z) = f ζ,m,n (z) satisfies the differential equation as (7.19), and can be uniquely determined with the condition f ζ,m,n (0) = 1. Other states |Ψ k ζ,m with k = m can be obtained by successive actions of J − on |Ψ m ζ,m Again, the scalar product between the states |Ψ k ζ,m is proportional to δ(ζ − ζ ′ ) and the relevant representations are tempered one. They are the most degenerate principal series representation of U * (2N ), similarly to the GL(N, R) and GL(N, C) cases. We can see this by moving back to the ω A andω A operators with the reality condition (3.29). The corresponding Schrödinger realization makes use of 4N complex variables q εa ǫ : with the reality conditions, In terms of the above, the generators of U * (2N ) and U * (2) are the R ε ǫ action on it can be written as where p ∈ U * (2) = GL(1, H), and hence can be considered as a non-zero quaternion, and |p| = √ det p as the quaternionic norm. Also the U * (2N ) group action associated to X εa ǫb can be treated similarly, and it is more natural to view it as GL(N, H): where q g is the right multiplication of the N × N quaternionic matrix g on the Ndimensional quaternionic vector q, i.e. (q g) a = q b g b a where g b a are quaternionic components of g ∈ GL (N, H) . Note also that det g is the determinant of g seen as an element of U * (2N ) ⊂ GL (2N, C), which is a positive real number, and q |Ψ k ζ,m is a square integrable function on H N taking values in C. The Z-eigenstate condition reads H) ] , (7.88) wherep = p/|p| is a unit quaternion corresponding to an SU (2) element when viewed as a 2 × 2 matrix, and D m is the (m + 1)-dimensional representation of SU (2) . Using the above condition, we can reduce the representation space from the space of functions on If we mod out this SU (2) symmetry, the representation space is reduced to the space of functions on HP N −1 ∼ = S 4N −1 /SU (2) , The scalar product is inherited from that of L 2 (H N ) as This can be generalized to the (U * (2N ), U * (2M )) duality with M ≤ N , whose representations are realized in the space of M × N quaternionic matrices q , endowed with the right action of GL (N, H) , H) ] , (7.92) and the left action of GL(M, H) , The last equality can be interpreted as a determinant-homogeneity condition (see e.g. [97]). Note here that we do not have the additional label m besides ζ in contrast with the (GL (N, H), GL(1, H)) case. This is due to the isomorphism GL(N, H) ∼ = R + × SL (N, H), where the last factor does not reduce to SU (2) In this notation, the most degenerate principal series is π GL(N,H) (ζ, 0) = π GL(N,H) (ζ; 1) .
Analogously to the (GL(N, R), GL(2, R)) and (GL(N, C), GL(2, C)) cases, the degenerate principal representations appearing in the dual pair (GL (N, H), GL(2, H)) can be used for the scattering amplitudes of 6d conformal fields. The relevant seesaw pair is where the same logic as for the seesaw pair of the 3d amplitudes (7.44) can be applied. Again, we reserve more detailed analysis on this point for a follow-up paper. The dual pair (Sp (2N, R), O(1, 1)) is realized as where i, j = 1, . . . , N . We need to supplement O(1, 1) by two reflections R a and R b which act as (7.99) and which form a finite group Z a 2 × Z b 2 . Let us consider the seesaw pair, The maximal compact subgroup U (N ) of Sp(2N, R) is generated by K +i−j , while its dual U (1, 1) ∼ = SL(2, R) ⋊ U (1) is generated by the SL(2, R) generators, with the usual Lie bracket (5.68), and the diagonal U (1) generator, which is also the center U (1) of U (N ) ⊂ Sp(2N, R). Note that U (1) ⊂ SL(2, R) and the diagonal U (1) , generated respectively by H and Z , contain the discrete subgroups Z + 4 and Z − 4 generated by the elements, They are related to the finite subgroup Z a 2 × Z b 2 of O(1, 1) as The (U (N ), U (1, 1)) duality has the correspondence of the representations, where [µ] E is one or two dimensional space, then supplemented with the restriction to the subgroup Z 2 generated by R. The restriction SL(2, R) ↓ R + again amounts to finding a M -eigenstates in the positive discrete series representation D SL(2,R) (h). For any h and M -eigenvalue i µ with µ ∈ R, one can find such a (coherent) state. Taking into account the restriction to the Z 2 with eigenvaluen, we find that the representation space of Sp (2N, R) is where the µ-dependence does not appear explicitly. For further specification, we can solve the O(1, 1) irrep condition, by taking an infinite linear combination of U (N ) representation states as where the U (N ) irrep condition imposes δ i 1 j 1 Ψ j 1 ···jn i 1 ···im = 0 and the M -eigenstate condition is translated to the differential equation (7.19) for h ±µ,m+n+N (z) = f ±µ,m,n (z), which can be uniquely fixed by f ±µ,m,n (0) = 1 . The other state |Ψ − µ,n is, by definition, R a |Ψ + µ,n .
Note that the (Sp (2N, R), O(1, 1)) dual pair can serve also for the tensor product between a metaplectic representation and a contragredient metaplectic representation of Sp(2N, R) : The above can be understood from the seesaw pair, Conversely, we can also use the above seesaw diagram to construct the (Sp (2N, R), O(1, 1)) representations out of the (Sp (2N, R), O(1)) ones, namely the metaplectic representations. We begin with the tensor product of two metaplectic representations, that is nothing but the Fock space with doubled oscillators (the doubling is from (a i ,ã i ) to (a i , b i ,ã i ,b i )). The same result (7.111) can be obtained simply imposing the O(1, 1) irrep conditions (7.110) on this space. The difference of the construction lies on the viewpoint whether we consider the Fock space as the representation space of (U (N ), U (1, 1)) or the tensor product of the (Sp (2N, R), O(1)) representation space.
The representations that we have constructed are again the most degenerate principal series representation of Sp(2N, R) . It will become more transparent if we move to the Schrödinger realization, (7.116) Let us work first with the dual pair (Sp (2N, R), O(M + , M − )) with any M ± and metric η AB of signature (M + , M − ). In this realization, the generators take the analogous form, The A, B indices above are lowered or raised by η AB . Now consider the M + = M − = M case with off-diagonal metric η AB : setting A = (+a, −a), the metric components are η ±a±b = 0 and η +a−b = δ ab . Then, the generators become and If we perform a Fourier transformation on the variable x i −a , the Sp(2N, R) generators become homogeneous, In this way, we find the natural action of (Sp (2N, R), O(M, M )) on the space of M × 2N real matrices (x + , p − ) or 2M × N real matrices x + x − (x ± and p − are M × N real matrices with components x i ±a and p −ai ): This space is reducible and we can impose the determinant-homogeneous condition, In order to describe O(2, C), it is convenient to define where I = 1, . . . , 2N is the symplectic index. In terms of these oscillators, O(2, C) generators are given by with the reflection flipping the sign of a I 2 andã 2 I , Recall that the symplectic indices are contracted as in (1.12). The dual group Sp(2N, C) is generated by where T IJ generates the maximal compact subgroup Sp(N ). The relevant seesaw pair is where the correspondence of the (Sp(N ), O * (4)) representations is Here, the SL(2, R) and SU (2) of O * (4) are generated by and Let us fix an irrep of O(2, C) as as the coherent state, where Ω IJ Ψ I(ℓ 1 ),J(ℓ 2 ) = 0 and h µ,m+2n+2k (z) = f µ,m,n,k (z) satisfies the differential equation (7.19) and can be fixed uniquely with f µ,m,n,k (0) = 1. The other state |Ψ − µ,m can be obtained by acting with R on the above, by definition.
Note that the (Sp(2N, C), O(2, C)) dual pair can serve also for the tensor product of the irreps π Sp(2N,R) (n) appearing in the (Sp (2N, C), O(1, C)) dual pair: π Sp(2N,C) (µ, 2n + 1 − δmn) . (7.142) The above can be understood from the seesaw pair, These irreps match the entire principal series representations of SO + (1, 3) (see Harish-Chandra [9]), which describe massive spin s fields in dS 3 . Note that all these representations, that is, for any mass and any spin, can be obtained by the tensor product of two conformal representations π SL(2,C) (0) or π SL(2,C) (1) , while the latter irreps uplift to the singleton irreps of the conformal group SO + (2, 3), mentioned in Section 5.3. Recall that the tensor product of two singletons give the tower of all massless spin s fields in AdS 4 . Therefore, the direct sum of (7.144) over all masses and spins is the restriction of the aforementioned tower to dS 3 .
The representations we have constructed is again the most degenerate principal series representation of Sp (2N, C) . Let us see this point with the more general case of (Sp (2N, C), O(2M, C)) duality with any M . We first move to the Schrödinger realization, where we introduced a metric η AB for O(2M, C), which will also appear in the commutation relation of the y I A operators. Since the group is complex, different signatures of η AB are all equivalent. In other words, we can move from one to the other by a suitable redefinition of the y operators. For our purpose, it will be convenient to fix the metric η AB as the off-diagonal one with non-trivial components η +a−b = δ ab where A = (+a, −a). Then, the O(2M, C) and Sp(2N, C) generators become and The rest of generators are simply the complex conjugate of the above. We can also perform a Fourier transformation on the half of the variable z i −a to render the Sp(2N, C) generators homogeneous: Here, w −ai are the variables in the Fourier space. In the end, we can realize the dual pair (Sp (2N, C), O(2M, C)) in the space of functions of M × 2N complex matrices (z + , w − ) or (z + , z − ): The determinant-homogeneous condition, [ a ∈ GL(M, C) ] , (7.152) or equivalently,

Branching properties
The representation of G appearing in the dual pair (G, G ′ ), wherein G ′ is the smallest in its family, has several interesting branching properties. In fact, when the group G is the conformal group Sp(4, R) ∼ = SO + (2, 3), SU (2, 2) ∼ = SO + (2, 4) or O * (8) ∼ = SO + (2, 6), this representation becomes a conformal field, namely a singleton representation, which possesses a number of special properties (see e.g. [98] for a short review). Therefore, in a sense, we can consider these representations as 'generalized singletons'. In the following, we discuss their branching properties along with a comparison with those of conformal fields.

Restriction to maximal compact subgroup
The d-dimensional singleton is the special representation of SO + (2, d), whose name is a reference to a peculiarity, first derived in [99], namely that its restriction to the maximal compact subgroup SO(2) × SO(d) consists in a single direct sum where each irrep appears with multiplicity one.
In fact, all the representations treated in this paper have such properties. Let us gather the relevant formulas at this place: The only cases that we have not explored are the dual pairs (O(N + , N − ), Sp(2, R)) and (O(N, C), Sp(2, C)), whose representations are not treated in this paper. Let us mention that the scalar singleton of O(2, d) has been explored within the (O(2, d), Sp(2, R)) duality in [53] (see also [57]). It is also possible to generalize the above properties to the other branching rules where the maximal subgroup K is replaced by a group whose complexification coincides with that of K. We shall provide the analysis of these cases together with (O(N + , N − ), Sp(2, R)) and (O(N, C), Sp(2, C)) in our forthcoming paper.

Irreducibility under restriction
Other properties of the d-dimensional singletons are that • they decompose into at most two irreps, when restricted to a d-dimensional isometry group, namely the (anti-)de Sitter or Poincaré subgroup (see e.g. [17,98,100]), • and, they are unique, or are one of two possible extensions of a d-dimensional isometry irrep to the d-dimensional conformal group.
Once again, this property can be investigated using seesaw pairs, , (8.9) where the conformal group and isometry group can be placed at G andG, respectively. In order that a singleton-like irrep π G (ζ) decomposes at most into two isometry-like irreps πG(ζ), on the dual side there should exist at most two irreps πG ′ (θ(ζ)) which can branch to π G ′ (θ(ζ)). Similarly, in order for an isometry-like irrep πG(ζ) to admit an extension to at most two singleton-like irreps of G, the restriction of the dual representation πG ′ (θ(ζ)) should contain at most two G ′ -irreps. This property is guaranteed, if the dual groupsG ′ and G ′ are isomorphic, or isomorphic up to Z 2 finite group. The simplest case ofG ′ and There are four more such groups among the irreducible ones, 11) which are in fact different real forms of O 2 ⊇ GL 1 . If one takes into account the reducible pairs, there are two more options, Here, G ′ can be any group suitable for (reducible) dual pairing. In the following, we present the dual groups G andG and their representations for each of the above seven cases.  (2N, R), and we find where n ∈ Z. Notice that for N ≥ 2, the trivial representation of U (1) can be found in two O(2) representations, [(0)] O(2) and [(1, 1)] O (2) . As a consequence, the Sp(2N, R) representation dual to this latter O(2) irrep also enters the branching rule, and we find The N = 2 case corresponds to the restriction of four-dimensional singletons of helicity n 2 to massless fields of spin |n| 2 in AdS 4 and the restriction of four-dimensional scalar singleton to conformal scalar and pseudo scalar fields in AdS 4 .
3. The dual of O * (2) ↓ U (1) is the restriction of U (2N + , 2N − ) ↓ Sp(N + , N − ), and we find where ζ ∈ R andn = 0, 1. The N = 2 case is to be compared with the GL(4, R)invariant equations for massless fields in AdS 4 discussed in [101]. Note, however, that the representations here do not describe massless spin s fields, but scalar or spinor tachyons in AdS 4 .

Casimir relations
One of the consequences of the dual pair correspondence is that the Casimir operators of the two groups in a pair are related [102][103][104][105]. Since the Casimir relations can be derived at the level of the metaplectic representation of the embedding symplectic group, it is possible to study them in terms of the previously introduced ω or y operators without specifying the representations of each group. However, when a specific representation of one group is chosen, the values of its Casimir operators will be fixed. This, in turn, determines the values of the Casimir operators of the dual group, and hence can be used to identify the dual representation. In the following, we derive the relations between Casimir operators for the complex dual pairs, i.e. (GL M , GL N ) and (O N , Sp 2M ). The relations of the Casimir operators for real forms of these complex dual pairs can be obtained by imposing the relevant reality conditions. 9.1 Duality (GL M , GL N ) We will here derive the relation between the Casimir operators of GL M and GL N in a dual pair. Let us first introduce generating functions for the Casimir operators: where C n [X] and C n [R] are the Casimir operators of order n defined by Here, X and R stand for matrix-valued operators with components X A B and R I J . It is also convenient to introduce two other generating functions, where X t and R t are the transpose of X and R , and hence the operators C n [X t ] and C n [R t ] have the form, Both C n [X] and C n [X t ] are n-th order invariants of GL M , and their difference shows up starting from n = 3 : The freedom in defining the n-th order Casimir operators reflects the ambiguity in choosing a basis for the center of the universal enveloping algebra. Below we will also derive a relation between the above two choices, which we will make essential use of to derive a relation between the generating functions in (9.1).
The operator X A B and R I J contain the (ω,ω) operators in different orders: it is convenient to work with the following order, which are related to the X A B and R I J by By introducing generating functions for Tr[Y n ] and Tr[S n ] as we can find the relation between x(t) and y(t), Similarly, s(t) is related tor(t) as Now, let us relate y(t) and s(t): In the definition of Tr(Y n ), we move theω operator in the left end to the right end and find the relation, This relation can be rephrased in terms of the generating functions as which can be solved as Finally combining the results so far obtained, we find the relation, (9.14) The formula (9.14) generates the relations between C n [X] andC n [R]. Next we would like to relate the generating functions r(t) andr(t). This computation is done in Appendix D.1 and the result isr , (9.15) which can be inserted in (9.14) to arrive at The inverse relation is given by This is the same formula as (9.16) with x(t) ↔ r(t) and M ↔ N , as expected. The first few relations, generated by (9.16), read A straightforward check shows that our result for the Casimir relations at any order (9.16) reproduce the relations of [105] (Theorem A).
The Casimir operators C n [R] with n > N are given by the lower order Casimirs, and the relation can be extracted from 20) where p N (t) is an order N polynomial with p N (0) = 0. The coefficients of p N (t) parameterize the independent Casimir operators. By differentiating the above equation, we obtain the relation between p N (t) and the generating function r(t) of C n [R] : .
We can put this expression into (9.16) to obtain a rational function whose numerator and denominator have degree N + 1 and N + 2, respectively.
One can also solve forr(t) in (9.14) and get the following formula: We could do exactly the same computation as when deriving (9.14), but for the r(t) in terms ofx(t), and get a similar relation with replaced M and N : (9.23) One may notice that the relations (9.22) and (9.23) are interchanged upon (t,r(t), x(t)) ↔ (−t, r(−t),x(−t)): r(−t) depends onx(−t) exactly same way asr(t) on x(t). This can be understood in the following way: the difference in the structure of the r(t) andr(t), and correspondingly x(t) andx(t) are given via commutators, which are changing their signs when we change the sign of the generators. On the other hand, changing the signs of the commutators is equivalent to taking the opposite product rule.   it is possible to derive exact relations between the Casimir operators to all orders through their generating functions. To derive these relations we find it useful to expand the space of generating functions to the following set of two-parameter family of generating functions: (9.30) where the brackets are short-hand for α [n] β := y α γ 2 y γ 2 γ 3 · · · y γn β , α [n] β := y γ 2 α y γ 3 γ 2 · · · y β γn , (9.31) 17 The numbers N and M should not be confused with the generators N A B and M A B in this section.
with the greek indices being generic, and with α [0] β := δ α β and α [0] β := δ β α . The a, b, d system of generating functions is a direct uplift of the n, l system, as should be clear by setting t = 0 or u = 0. The usefulness of introducing these generalized generating functions is seen directly, when trying to relate n with l in a similar way, as y was related to s from (9.11) in the previous section: The equivalent expression of (9.11) here reads 32) valid for n ≥ 1, and with the last term understood as zero for n = 1. It can be equivalently written as As described in Appendix D.2, it is possible to relate the generators a, b, d with each other, through similar, but much more involved manner, and as an outcome it is possible relate n entirely in terms of l (or vice-versa). The result is that , (9.34) and as a bonus, we are also able to find expressions for a, b, d in terms of n and l, reading Notice that a(t, u) is well-defined at u = −t, and consistent with (9.33), while b and d are singular at that point (but the limit exists, as it should by definition). This singularity in b and d explains the necessity of uplifting the original problem to a larger space of functions, through which it can be solved. Now that we have established a relation between n and l it is straightforward to pass this on to a relation between m and k: By formally rewriting ∞ n=0 t n Tr(O n ) = Tr( 1 1−tO ) (cf. (9.9)) and using the relations in (9.27), the following pairwise relations between the generating functions are found: From these and (9.34) we finally establish (9.40) (9.41) We notice that the expressions for k and m are symmetric under the exchange N ↔ −2M and m(t) ↔ −k(t), while those for n and l are simply symmetric under n(t) ↔ −l(t). This fits into the picture of universality (see, e.g. [106][107][108]) and the SO N ←→ relation [109,110].
Our results for the (O N , Sp 2M ) Casimir relations do not agree with those of [105] beyond the quadratic order. Therefore, it is worth checking the relations at the lowest orders without using the generating function method. The first two within each group read From the above low-order relations it can now be checked that which is exactly what one finds directly from (9.40).

A Representations of compact groups
Finite-dimensional representations of compact Lie groups are known to be in one-to-one correspondence with integral dominant weights. The latter are weights whose components in the Dynkin basis are all positive integers. In other words, finite-dimensional representations of compact Lie groups G are highest weight representations, with highest weight of the form α = rank(G) k=1 α ω k ω k with α ω k ∈ N the Dynkin labels and ω k the fundamental weights of G. In this appendix, we will review how these highest weight representations are realized as spaces of tensors with special properties. To do so, it will be more convenient to express the weights of G in the orthonormal basis {e k } rank(G) k=1 , where the vectors e i are mutually orthogonal unit vectors in R rank(G) . We will write α = (α 1 , . . . , α N ), for α 1 , . . . , α N the components of α in this orthonormal basis.
Finite-dimensional representations of GL N Let us start with the reductive group GL N . In the orthonormal basis, the components of a GL N integral dominant weight α are all integers (not necessarily positive) and ordered, i.e.
Recall that the GL N -irreps appearing in the decomposition of V L = (C N ) ⊗L are uniquely associated with the irreps of the symmetric group S L , defined by Young diagrams ℓ, where Par(L|N ) denotes the set of Young diagrams with L boxes and at most N rows. This standard result is known as the Schur-Weyl duality, and can be seen as another instance of duality in representation theory, similar to the dual pair correspondence (see e.g. [70,71,111] and references therein). The subspace of V L carrying the S L -irrep ℓ can be singled out by the projector P ℓ : V L → ℓ, i.e. the operator that symmetrizes a given tensor T a 1 ...a L so that the resulting tensor has the symmetry of the Young diagram ℓ. In other words, the GL N representation denoted [ℓ] GL N and dual to the S L -irrep ℓ consists of tensor with the symmetry of the Young diagram ℓ.
For instance, when L = 2 there are two possible representation of S 2 to consider, namely ℓ = (2) and ℓ = (1, 1), corresponding to a symmetric and an antisymmetric tensor. The associated projectors are For L = 3, there are three possible Young diagrams that can appear (for N > 2), namely ℓ = (3), ℓ = (1, 1, 1) and ℓ = (2, 1). Only one standard Young tableau can be attached to the first two, while two for the last diagram. The associated projectors read The action of GL N on [ℓ = (ℓ 1 , . . . , ℓ p )] GL N with p ≤ N reads where g b c denote the components of the matrix g ∈ GL N . Correspondingly, the action of the Lie algebra gl N is given by Recall that these generators obey from which we can see that X a a generate a Cartan subalgebra, while X a b for a < b (resp. a > b) are raising (resp. lowering) operators. Let us show that the component T 1(ℓ 1 ),...,p(ℓp) of this tensor (i.e. the component obtained by setting all the indices of a-th group of symmetric indices to the value a) is the highest weight vector of this gl N irrep. For this particular component, the above action simplifies to a single term, when b ≤ p, and vanishes otherwise. First of all, it is an eigenvector of the Cartan subalgebra generator X a a (no summation implied), with eigenvalue ℓ a . The raising operator are the generators X a b with a < b, which implies that their action takes the form i.e. they correspond to spaces of tensors T with two types of indices: T a 1 ···a L b 1 ···b M .
First, let us have a look at V 0,1 = C N * . As a vector space, this corresponds to the dual space of the fundamental representation. As such, it also carries a representation ρ of for g ∈ U (N ), v ∈ C N and φ ∈ C N * , and where , denote the pairing between C N and C N * . More concretely, this means that 18 For instance, in the case of ℓ = (2, 1), the action of a generator X i j of gl N reads Finally, consider the case L = M = 1, i.e. the space of tensors of the form T a b , on which GL N acts as for g ∈ GL N . In particular, one can notice that the trace of T = δ b a T a b defines a onedimensional invariant subspace of V 1,1 , as ρ V 1,1 (g) T = T . This observation extends to higher rank tensors, and hence we can conclude that the traces of any tensor (in the sense of any contraction of an upper index with a lower one) define a invariant subspace. In other words, the irreducible representations of GL N in V L,M are composed of traceless tensors. Added to the previous discussions, we can conclude that highest weight representations of GL N , with highest weight Notice that the tracelessness implies that the tensors with p + q ≥ N identically vanish.
It will be useful to record the expression of the dimension of the space of tensor with symmetry ℓ, The finite dimensional representations of GL The dual tensorΨ has now two types of upper indices: the first p groups have the symmetry ℓ, while the next r groups (resulting from dualization) have the symmetry of the dualized Young diagramm = [m 1 , . . . ,m r ] withm k = N − m k . In other words, the dual tensor Ψ has the symmetry of the tensor product ℓ ⊗m, which we should try to decompose. In the symmetric basis,m = (m 1 ,m 2 , . . . ) is given by (see Figure 1): Figure 1. A Young diagram m (black) and its dualm (gray). This implies that, when applying Littlewood-Richardson rule to decompose ℓ ⊗m, no antisymmetrization of an index in ℓ with a column ofm is allowed. Such a diagram occurs only when p ≤ N − q, that is, p + q ≤ N and is unique. This diagram is obtained by simply attaching ℓ to the right side ofm: Requiring a tensor to be traceless imposes some restriction on the possible symmetry ℓ it can have. To see that, let us have a look at a rank k + l tensor with the symmetry of the Young diagram [k, l] in the antisymmetric basis (i.e. a two-column Young diagram, of respective height k and l). It has independent components, and its trace being a tensor with symmetry [k − 1, l − 1]. As a consequence, the traceless part of such a tensor has independent components. This number vanishes for k + l = N + 1, and becomes negative when k + l becomes larger. This means that a tensor T with shape [k, l] is necessarily "pure trace", i.e. of the form with some tensor U of symmetry [k − 1, l − 1]. The same conclusion applies to more general tensor: if the corresponding Young diagram ℓ = [ℓ 1 , ℓ 2 , . . . , ℓ r ] is such that ℓ 1 +ℓ 2 > N , then its traceless part identically vanishes. As a consequence, finite-dimensional representations of O(N ) correspond to traceless tensor having the symmetry of a Young diagram ℓ such that the sum of the height of its first two columns is at most N .
Highest weight of the representation. In order to show that the O(N ) irrep consisting of traceless tensor with symmetry ℓ is an highest weight module, it will be useful to work in the basis where the metric takes the form The last commutator shows that the generators M +a−b span a u(n) subalgebra. When N = 2n + 1, there are 2n additional generators, denoted N ±a = M ±a * , to take into account: where ǫ, ε = ± and all other commutators vanish. From these commutators, we can see that M +a−a generates a Cartan subalgebra 19 while M +a+a , N +c and M +a−b for a < b (resp. M −a−a and N −c and M +a−b for a > b) are raising (resp. lowering) operators. The action of so(N ) on a tensor of symmetry ℓ = (ℓ 1 , . . . , ℓ p ), with p ≤ N , reads We saw previously that if T is traceless, then ℓ 1 + ℓ 2 ≤ N . This implies that only the first column can be higher than n. When it is the case, i.e. when ℓ 1 ≥ n, we can dualize this tensor (as we did in the previous subsection): which, in particular than n rows (see Figure 2). n dualization Figure 2. Example of a Young diagram with a first column higher than n (left) and the Young diagram obtained after dualization (right).
With this in mind, we can now assume (without loss of generality) that p ≤ n, i.e. consider tensor corresponding to Young diagrams with less than n non-empty rows. Now let us show that the component T +1(ℓ 1 ),...,+p(ℓp) is the highest weight vector of the representation. First of all, we can immediately see that it is annihilated by all M +a+b and N +c . Secondly, the action of the generators M +a−b on this component simply takes the form which vanishes as a consequence of the symmetry of the tensor T . To summarized, we have proved that the O(N ) irrep defined by traceless tensor with symmetry ℓ is an SO(N ) highest weight module with highest weight ℓ if the diagram has at most n rows, or highest weightl (dualized diagram) otherwise.

Finite-dimensional representations of Sp(N )
Finite-dimensional representations of Sp(N ) can also be obtained by restriction from the tensor representations of GL 2N constructed previously. The symplectic group Sp(N ) possesses an invariant tensor, namely the canonical symplectic matrix Ω ij . Being nondegenerate, one can raise and lower indices with it, so that it is sufficient to consider tensors with one kind of indices. The restriction of [ℓ] GL 2N to Sp(N ) is not irreducible: it contains invariant subspaces generated by traces taken with respect to Ω. For instance, [(1, 1)] GL 2N which is the space of antisymmetric tensors T i,j contains two irreducible representations of Sp(N ), corresponding to its trace T = Ω ij T i,j and its traceless partT i,j = T i,j + 1 2N Ω ij T .
Requiring a tensor to be traceless imposes some restriction on the possible symmetry ℓ it can have. Consider for instance a totally antisymmetric tensor of rank k. It has 2N k independent components, and its trace being a rank k − 2 antisymmetric tensor has 2N k−2 . As a consequence, the traceless part of a rank k antisymmetric tensor has independent components. This number vanishes for k = N +1, and becomes negative when k becomes larger. This means that an antisymmetric tensor T of rank k > N is necessarily "pure trace", i.e. of the form with some rank k−2 antisymmetric tensor U . This arguments extends to tensors with more general symmetry, leading to the conclusion that tensors finite-dimensional representations of Sp(N ) correspond to traceless tensor having the symmetry of a Young diagram ℓ with at most N rows.
Highest weight of the representation. Recall that the algebra sp(N ) is spanned by symmetric generators K AB = K BA with the index A taking 2N values and subject to the commutation relations Splitting the index into A = ±a with a = 1, . . . , N , and such that the only non-vanishing components of the symplectic matrix are Ω −a+b = δ ab , the above relations now read From this commutators, we can see that K +i−i generate a Cartan subalgebra, 20 while K +j+k (resp. K +j+k ) and K +l−m for l < m (resp. for l > m) are raising (resp. lowering) operators. Notice also that the last commutation relation shows that K +i−j forms a u(N ) subalgebra in sp(N ). The action of sp(N ) on a tensor of symmetry ℓ = (ℓ 1 , . . . , ℓ p ), with p ≤ N , is given by due to the symmetry of T . We can therefore conclude that the Sp(N ) irrep carried by the space of traceless tensors with symmetry of a Young diagram ℓ with at most N rows is a highest weight module whose highest weight coincides with ℓ in the orthonormal basis.

B Real forms of classical Lie groups and Lie algebras
A real form g σ of a complex Lie algebra g can be identified as a subalgebra of the realification g R of g , namely where σ : g R → g R is an anti-involution, i.e. it satisfies σ([A 1 , A 2 ]) = −[σ(A 1 ), σ(A 2 )] and σ 2 = 1.
In the following, we summarize the list of all real forms and present the action σ on their generators. For that, let us define first the following matrices where I N is the N × N identity matrix. 20 Notice that the Killing form κ of sp(N ) is given by κ(KIJ , KKL) ∝ Ω I(K Ω L)J so that the Cartan subalgebra generators Hi = K+i−i verify κ(Hi, Hj ) = δij (upon choosing the proper normalization). In other words, the weights computed later will be expressed in the orthonormal basis.

General linear Lie groups and Lie algebras
The group of invertible n × n complex matrices, GL(N, C), is a real Lie group of dimension 2N 2 . The associated Lie algebra gl(N, C) is the commutator Lie algebra of N × N complex matrices generated by the matrices X A B with the components (X A B ) C D = δ A C δ D B . The generators X A B satisfy the commutation relation, For later use, it will be useful to note where Y A B are the components of the matrix Y .
All real forms of the complex Lie group/algebra GL N /gl N are subgroups/subalgebras of GL(N, C)/gl(N, C). There are three classes of real forms: Since (X A B ) † = X B A , the action of σ is given by where η AB are the components of the matrix η . where Ω AB and Ω AB are the components of Ω and its inverse: Ω AB Ω BC = δ C A .

Orthogonal Lie groups and Lie algebras
The where Ω AB are the components of Ω −1 .

Symplectic Lie groups and Lie algebras
The complex symplectic Lie group and Lie algebras is defined as the group of 2N × 2N complex matrices preserving the canonical symplectic form Ω, Since Ω A is a symmetric matrix, it is more convenient to consider the Lie algebra obtained by the isomorphism, where Ω AB are the components of Ω .
All real forms of Sp 2N /sp 2N are subgroups/subalgebras of Sp(2N, C)/sp(2N, C). There are two types of real forms: Due to the second identity in (B.25), the Lie algebra ρ U (sp(2n, R)) has the same Lie bracket as (B.31).
• Sp(N + , N − ) , the group of (N + + N − ) × (N + + N − ) matrices over the quaternions that preserve the metric of signature (N + , N − ), which can also be defined as and Ψ AB and Ψ AB are the components of Ψ and Ψ −1 .

C Seesaw pairs diagrams
In Section 2.3, we explained how a seasaw diagram can be obtained from a dual pair by systematically looking for the maximal compact subgroups of all groups appearing in the process. Below, we present such seasaw diagrams for all seven irreducible dual pairs listed in Table 1.

D Derivation of Casimir relations
In the following two subsections we provide the details of the derivations leading to the duality relations, presented in the main text, between all Casimir operators of the two groups in a dual pair, in terms of their generating functions.

D.1 Relating different choices of Casimir operators for the same group GL N
Here we derive the relation between the generating functions r(t) andr(t) of the Casimir operators of the same group GL N spanned by R I J , with I, J = 1, . . . , N , as introduced in (9.1) and (9.3). In order to do that, we introduce the following notations: These two types of operators differ only by ordering and therefore are related as ((R t ) n ) I J = n m=0 a n m (R m ) I J , (D.2) for some a n m that may also depend on traces of powers of R (9. The latter equation induces recursion relations for a n m in the following form: a n+1 m = a n,m−1 − N a n m (m ≥ 1) , a n+1 0 = n m=0 a n m C m [R] − N a n 0 . (D.5) It is useful to introduce a generating function for a n m in the form, a(x, y) = a(x, y) = 1 (D. 13) which can be used to get finallỹ . (D.14) The inverse relation is given by .

(D.15)
Note that these relations go to one another, if we change the sign of N . where E AB is an arbitrary symmetric flat metric with eigenvalues ±1, and Ω IJ is the 2M dimensional symplectic metric, and their inverses E AB and Ω IJ are defined by The Likewise, their derivatives evaluated at t = 0 or u = 0 are related to derivatives of l and n. But as we will see now, it is at the point u = −t that we need to understand these generating functions.
To relate the Casimirs of O N to those of Sp 2M , we first move the y operator in the left end of Tr(N n ) to its right end to find Tr(N n ) = Tr(L n ) + making it evident that another relation for a(t, −t) in terms of n and l is needed. We therefore now derive relations between a, b, and d.
By moving the y operator in the left end of I [2k + 1] A to its right end, we find Multiplying by t k u l and summing over all k ∈ N and l ∈ N 0 , as well as making use of the following relations that will be used repeatedly, Notice that u = −t is not included in the region of validity of this expression, meaning that it cannot be simply evaluated at u = −t. It turns out, however, that its limit at u = −t is well-defined and continuous, but leads to a differential expression for a(t, −t), t 1 − t + t n(t) + t 2 ∂ ∂u a(t, −t) = n(t) − b(t, −t) , (D.39) which we will not make use of. Instead we establish more relations away from u = −t to find an expression for n and l at u = 0.  − t u (t + u) 2 u 2 a(−u, u) + t u (a(t, u) + a(−u, −t)) + t 2 a(t, −t) [u a(−u, −t) + t a(t, −t)] n(−u) . (D.42)