Dual Pair Correspondence in Physics: Oscillator Realizations and Representations Thomas Basilea Euihun Jounga Karapet Mkrtchyanb Matin Mojazac aDepartment of Physics, Kyung Hee University, Seoul 02447, Korea bScuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126 Pisa, Italy cAlbert-Einstein-Institut, Max-Planck-Institut f¨ur Gravitationsphysik, 14476 Potsdam, Germany E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: 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, R), GL(N, R)), (GL(M, C), GL(N, C)), ∗ ∗ (U (2M),U (2N)), (U(M+,M−),U(N+,N−)), (O(N+,N−),Sp(2M, R)), (O(N, C),Sp(2M, 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 particu- arXiv:2006.07102v2 [hep-th] 2 Sep 2020 lar, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established. Contents 1 Introduction 1 2 Generalities of the reductive dual pair correspondence 7 2.1 Metaplectic representation 8 2.2 Irreducible reductive dual pairs 13 2.3 Seesaw pairs 14 3 Dual pairs of type (GLM , GLN ) 17 3.1 GL(M, R), GL(N, R) Sp(2MN, R) 17 ⊂ 3.2 GL(M, C), GL(N, C) Sp(4MN, R) 19 ⊂ 3.3 U (2M),U (2N) Sp(8MN, R) 20 ∗ ∗ ⊂ 3.4 U(M+, M ),U(N+,N ) Sp(2(M+ + M )(N+ + N ), R) 20 − − ⊂ − − 4 Dual pairs of type (ON ,Sp2M ) 22 4.1 O(N+,N ),Sp(2M, R) Sp(2M(N+ + N ), R) 22 − ⊂ − 4.2 O(N, C),Sp(2M, C) Sp(4NM, R) 24 ⊂ 4.3 O∗(2N),Sp(M+, M ) Sp(4N(M+ + M ), R) 25 − ⊂ − 5 Representations of compact dual pairs 26 5.1 U(M),U(N) 28 5.2 U(M+, M ),U(N) 29 − 5.3 O(N),Sp(2M, R) 32 5.4 O∗(2N),Sp(M) 35 6 Representation of exceptionally compact dual pairs 37 – i – 6.1 O∗(2),Sp(M+, M ) 38 − 6.2 Sp(2N, C), O(1, C) 39 7 Representations of the simplest non-compact dual pairs 41 7.1 GL(N, R), GL(1, R) and GL(N, R), GL(M, R) 41 7.2 (GL(N, C), GL(1, C)) and (GL(N, C), GL(M, C)) 47 7.3 U ∗(2N),U ∗(2) and U ∗(2N),U ∗(2M) 50 7.4 Sp(2N, R), O(1, 1) and Sp(2N, R), O(M, M) 53 7.5 Sp(2N, C), O(2, C) and Sp(2N, C), O(2M, C) 57 8 Branching properties 61 8.1 Restriction to maximal compact subgroup 61 8.2 Irreducibility under restriction 63 9 Casimir relations 66 9.1 Duality (GLM , GLN ) 66 9.2 Duality (ON ,Sp2M ) 69 A Representations of compact groups 72 B Real forms of classical Lie groups and Lie algebras 81 C Seesaw pairs diagrams 86 D Derivation of Casimir relations 88 D.1 Relating different choices of Casimir operators for the same group GLN 88 D.2 Details on the Casimir relations for (ON ,Sp2M ) 90 – ii – 1 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 1 momentum operators in terms of two pairs of ladder operators, a, a† and b, b†, as follows 1 J3 = (a†a b†b) , J+ = a† b , J = b† a , (1.1) 2 − − thereby finding a gateway to construct all irreducible representation of su(2) in terms of two uncoupled quantum harmonic oscillators. The spin-j representations then simply become the states with excitation number 2j : N = a† a + b† b , N Ψj = 2j Ψj . (1.2) | i | i When su(2) is realized as above, there emerges another Lie algebra, u(1) — generated by the number operator N — which is not a subalgebra of the su(2). Uplifting this to Lie groups, we find that the spin-j representations of SU(2) are in one-to-one correspondence with 1In this section, we mention several important works which went before the dual pair correspondence. Our intention, however, is not to select the most important contributions, but rather to illustrate the historical development of oscillator realizations and the dual pair correspondence in physics. – 1 – 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 [19–21], 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 math- ematicians contributed to the development of the subject (see e.g. reviews [26–28] 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 [31–33]. In the 80-90’s, G¨unaydin 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 [40–42]). 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 2These representations are closely related to Majorana’s equation for infinite component spinor carrying a unitary representation of the Lorentz group, published in 1932 [12].