The AdS/CFT correspondence was introduced by Maldacena 20 years ago . Soon important contributions were made by Gubser-Klebanov-Polyakov  and by Witten . We recall the two ingredients of the AdS/CFT correspondence [433, 293, 572]: 1. the holography principle, which is very old, and means the reconstruction of some objects in the bulk (which may be classical or quantum) from some objects on the boundary; 2. the reconstruction of quantum objects, like 2-point functions on the boundary, from appropriate actions on the bulk. Here we give a group-theoretic interpretation of the AdS/CFT correspondence as a relation of a representation equivalence between representations of the conformal group describing the bulk AdS fields ø, their boundary fields ø0 and the boundary conformal operators O coupled to the latter. We use two kinds of equivalences. The first kind is the equivalence between the representations describing the bulk fields and the boundary fields and it is established here. The second kind is the equivalence between conjugated conformal representations related by Weyl reflection, e. g. the coupled fields ø0 and O. Operators realizing the first kind of equivalence for special cases were actually given by Witten and others-here they are constructed in a more general setting from the requirement that they are intertwining operators. The intertwining operators realizing the second kind of equivalence are provided by the standard conformal two-point functions. Using both equivalences we find that the bulk field has in fact two boundary fields, namely, the coupled fields ø0 and O, the limits being governed by the corresponding conjugated conformal weights. In this chapter we give a group-theoretic interpretation of relativistic holography as equivalence between representations in three cases: 1. the Euclidean conformal (or de Sitter) group; 2. the anti de Sitter group SO(3,2); 3. the Schrödinger group. In each case we give explicitly boundary-to-bulk operators and we show that these operators and the easier bulk-to-boundary operators are intertwining operators. Furthermore, we show that each bulk field has two boundary (shadow) fields with conjugated conformal weights. These fields are related by another intertwining operator, given by a two-point function on the boundary.
We give a review of some group-theoretical results related to non-relativistic holography. Our main playgrounds are the Schrödinger equation and the Schrödinger algebra. We recall the fact that there is a hierarchy of equations on the boundary, invariant w. r. t. the Schrödinger algebra. The derivation of this hierarchy uses a mechanism introduced first for semisimple Lie groups and adapted to the non-semisimple Schrödinger algebra. This requires development of the representation theory of the Schrödinger algebra, which is reviewed in some detail. In Section 2.1 the Schrödinger equation is reviewed as an invariant differential equation in the (1 + 1)-dimensional case. On the boundary this was done in  (extending the approach in the semisimple group setting ), constructing actually an infinite hierarchy of invariant differential equations, the first member being the free heat/Schrödinger equation). In Section 2.1.4 the extension of this construction is reviewed to the bulk combining techniques from  and . In Section 2.2 the Schrödinger equation is reviewed as an invariant differential equation in the general (n + 1)-dimensional case following [8, 190]. The general situation is very complicated and requires separate study of the cases n = 2N and n = 2N + 1. In Section 2.3 the (3 + 1)-dimensional case is reviewed separately and in more detail, since it is most important for physical applications. In Section 2.4 the q-deformation is reviewed of the Schrödinger algebra in the (1 + 1)-dimensional case; cf. . In Section 2.5 the difference analogues of the Schrödinger algebra in the (n + 1)-dimensional case are reviewed; cf. .
Starting with the fundamental paper  Belavin-Polyakov-Zamolodchikov (BPZ) there has been great interest in the two-dimensional conformal theories [245, 195]. Although the underlying Virasoro algebra was introduced a long time ago by Gelfand-Fuchs  in mathematics1 and independently, by Virasoro  in physics, was extensively used in dual string theories (cf. e. g. ) the lack of a developed representation theory obviously hindered its further applications. It is no wonder that the BPZ paper appeared after the papers of Kac  and of Feigin-Fuchs . Even a decade long work on the Liouville theory by Gervais-Neveu was better understood and reshaped (cf. , and also ). The above developments inevitably made contact with the Kac-Moody symmetry approach to current algebras [570, 134, 124]. That contact was initiated by Segal  who showed that every highest weight module (HWM) of a Kac-Moody algebra may be extended to the semi-direct product with the Virasoro algebra. Consequently, the elements of the latter can be realized as bilinear combinations (normally ordered) of elements of the Kac-Moody algebras (i. e. as elements of the latter universal enveloping algebras) [243, 533]. In the physical literature such constructions can be traced back to  but were reintroduced in the study of completely integrable models by Sato and others . In the current algebra context they were mentioned by Polyakov  and used in full force in  (see also [277, 554]). In this chapter, first, following  and  we give the multiplet classification of all reducible HWMs over the Virasoro and N = 1 super-Virasoro algebras. Along with a simple explicit parametrization of all reducible Verma modules, we show all the possible embedding maps between them. These embedding maps correspond one-to-one to all possible singular vectors of the reducible Verma modules. We recall that any singular vector becomes a homogeneous polynomial and, in a function space realization, it becomes a linear differential operator of an order equal to the degree of the corresponding polynomial. These degrees are given explicitly for the singular vectors corresponding to all noncomposition embedding maps and, thus, for all singular vectors. Further, following  we give the characters of all irreducible highest weight modules over the Virasoro algebra and N = 1 super- Virasoro algebras (Neveu-Schwarz superalgebra and Ramond superalgebra). These are given in an explicit and unified form for all three (super-)algebras incorporating all previously known results. Using the character formulae we introduce a Weyl group for Virasoro and N = 1 super-Virasoro algebras . Further, following  we present formulae for singular vectors c < 1 Fock modules over the Virasoro algebras. These we present in terms of Schur polynomials generalizing the c = 1 expressions of Goldstone. We reproduce the known formulae for the singular vectors in (1, n), (m, 1) modules and give new formulae in the cases (2, n), (m, 2), (3, 3). Furthermore, we present the characters of the unitarizable highest weight modules over the N = 2 superconformal algebras following mainly  and . Finally, following  and  we consider modular invariants for theta-functions with characteristics and the twisted N = 2 superconformal and su(2) Kac-Moody algebras and the classification of the corresponding modular invariant partition functions.
The theory and applications of Kac-Moody algebras thrived in the last several decades on the interface between mathematics and physics. Kac-Moody algebras were identified first in the stationary Einstein-Maxwell equations , in two-dimensional (2D) principal- (or super-) chiral models , in 2D sigma models on coset spaces , in 2D Heisenberg model , in the (anti-) self-dual sector on pure Yang-Mills theory , in supersymmetric (N = 4) Yang-Mills theory , in completely integrable systems , in Toda systems  and also as current algebras in two- [570, 124, 439] and three-dimensional  models. Naturally, in these pioneer attempts little was used from the representation theory of these algebras (cf.  and the references therein). One motivation for our research came from the remarkable papers by the Sato followers  which use for the hierarchies of completely integrable systems the most basic representations-the so-called fundamental modules. We recall that every fundamental module appears as the irreducible subquotient of an indecomposable Verma module, the latter being a member of an infinite set of other partially equivalent indecomposable Verma modules. Such a set is called as in the semisimple Lie algebra (SSLA) case a multiplet . However, there are other multiplets besides those containing the fundamental modules. Using an approach developed earlier for semisimple Lie groups or algebras  and adapted here for Kac-Moody algebras we classify and parametrize all such multiplets for A(1)ℓ and give them explicitly for A(1)1 and A(1)2 . One motivation for this was the search for new hierarchies of completely integrable systems arising from multiplets which do not contain the fundamental modules. This seems very natural since the setting we describe contains infinitely many differential invariant operators; namely, the intertwining operators which give the partial equivalences in the multiplets become differential operators when we realize the HWM as spaces of functions. Every invariant operator between function spaces gives rise to an invariant equation. These invariant operators and equations were very useful as we know from previous chapters and volumes. Such differential equations also led to the successful treatment of some statistical physics models by Belavin, Polyakov and Zamolodchikov  by considering another infinite-dimensional Lie algebra, the Virasoro one. We mention this not only because such a line of research is another of our motivations but also because the representation theory of the Virasoro algebra and the Kac-Moody A1(1)are closely related. The exact correspondence is discussed. A more general relation is that every HWM of a Kac-Moody algebra may be extended to the semidirect product with the Virasoro algebra . The elements of the latter can be realized as bilinear combinations (normally ordered in some sense) of elements of the Kac-Moody algebras, i. e. as elements of the latter universal enveloping algebras. This in turn provides explicit constructions of the Kac-Moody fundamental modules . Thus there is rich ground for interplay between the differential operators invariant under the two algebras. Possible applications of such interplays, e. g. for finding exact formulae for anomalous dimensions and differential equations for the correlation functions were first mentioned in lectures of Polyakov in Shumen (Bulgaria) in August 1984  (see also Todorov ). Furthermore, in Knizhnik-Zamolodchikov  the simplest one was realized involving a first order linear differential equation. The Kac-Moody algebras encountered by Witten  (for even central charge) and by Craigie-Nahm  were also good starting points for such applications.