scholarly journals Dual pair correspondence in physics: oscillator realizations and representations

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Thomas Basile ◽  
Euihun Joung ◽  
Karapet Mkrtchyan ◽  
Matin Mojaza
Keyword(s):  
Closed Form ◽  
Fock Space ◽  
Dual Pair ◽  
Irreducible Representations ◽  
Systematic Treatment ◽  
Dual Pairs ◽  
Howe Duality ◽  
Casimir Operators ◽  
General Aspects ◽  
Dual Pair Correspondence

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 particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.

scholarly journals Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Ivan Todorov
Keyword(s):  
Lie Algebras ◽  
Gauge Group ◽  
Fock Space ◽  
Scalar Fields ◽  
Dual Pair ◽  
Two Dimensions ◽  
Operator Product ◽  
Positive Energy ◽  
Dual Pairs ◽  
Irreducible Components

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.

scholarly journals Dual pairs of sequence spaces

2001 ◽  
Vol 28 (1) ◽  
pp. 9-23 ◽  
Author(s):  
Johann Boos ◽  
Toivo Leiger
Keyword(s):  
Sequence Space ◽  
Dual Pair ◽  
General Concept ◽  
Sequence Spaces ◽  
Dual Pairs ◽  
The Third ◽  
Convex Sequence ◽  
Starting Point ◽  
Alpha Beta ◽  
Locally Convex

The paper aims to develop for sequence spacesEa general concept for reconciling certain results, for example inclusion theorems, concerning generalizations of the Köthe-Toeplitz dualsE×(×∈{α,β})combined with dualities(E,G),G⊂E×, and theSAK-property (weak sectional convergence). TakingEβ:={(yk)∈ω:=𝕜ℕ|(ykxk)∈cs}=:Ecs, wherecsdenotes the set of all summable sequences, as a starting point, then we get a general substitute ofEcsby replacingcsby any locally convex sequence spaceSwith sums∈S′(in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair(E,ES)of sequence spaces and gives rise for a generalization of the solid topology and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality(E,Eβ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and generalized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Große-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justified by four applications with results around different kinds of Köthe-Toeplitz duals and related section properties.

ON COMMUTANTS, DUAL PAIRS AND NON-SEMISIMPLE ALGEBRAS FROM STATISTICAL MECHANICS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 675-705 ◽  
Author(s):  
PAUL P. MARTIN ◽  
DAVID S. MCANALLY
Keyword(s):  
Dual Pair ◽  
Quantum Spin ◽  
Spin Chains ◽  
Quantum Spin Chains ◽  
Complex Vector ◽  
Dual Pairs ◽  
Morita Equivalent ◽  
Finite Dimensional ◽  
Semisimple Algebras ◽  
Special Case

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that Hn(q), Uq(sl2) act as a dual pair on the tensored vector representation for all q.

ON THE FOCK SPACE REPRESENTATION OF THE WITTEN VERTEX

1994 ◽  
Vol 09 (06) ◽  
pp. 465-477
Author(s):  
RAINER DICK
Keyword(s):  
Closed Form ◽  
Fock Space ◽  
Closed String ◽  
Space Representation ◽  
Operator Representations

The bosonic overlap conditions for operator representations of the Witten vertex and its closed string analog are solved in closed form for arbitrary many external strings. This is accomplished by the use of transformed operator bases of the strings. In particular, the bosonic factor of the Witten vertex for three closed strings is realized in Fock space.

scholarly journals SEIBERG–WITTEN THEORY OF RANK TWO GAUGE GROUPS AND HYPERGEOMETRIC SERIES

1998 ◽  
Vol 13 (18) ◽  
pp. 3121-3144 ◽  
Author(s):  
TAKAHIRO MASUDA ◽  
TORU SASAKI ◽  
HISAO SUZUKI
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Dual Pair ◽  
Exceptional Group ◽  
Dual Pairs ◽  
Gauge Groups ◽  
Witten Theory ◽  
Type C ◽  
Appell Function ◽  
Appell Functions

In SU(2) Seiberg–Witten theory, it is known that the dual pair of fields are expressed by hypergeometric functions. As for the theory with SU(3) gauge symmetry without matters, it was shown that the dual pairs of fields can be expressed by means of the Appell function of type F4. These expressions are convenient for analyzing analytic properties of fields. We investigate the relation between the Seiberg–Witten theory of rank two gauge group without matters and hypergeometric series of two variables. It is shown that the relation between gauge theories and Appell functions can be observed for other classical gauge groups of rank two. For B2 and C2, the fields are written in terms of Appell functions of type H5. For D2, we can express fields by Appell functions of type F4 which can be decomposed to two hypergeometric functions, corresponding to the fact SO (4)~ SU (2)× SU (2). We also consider the integrable curve of type C2 and show how the fields are expressed by Appell functions. However in the case of exceptional group G2, our examination shows that they can be represented by the hypergeometric series which does not correspond to the Appell functions.

scholarly journals Correspondences of the Gelfand invariants in reductive dual pairs

2003 ◽  
Vol 75 (2) ◽  
pp. 263-278 ◽  
Author(s):  
Minoru Itoh
Keyword(s):  
Dual Pair ◽  
Explicit Description ◽  
Universal Enveloping Algebras ◽  
Dual Pairs ◽  
Enveloping Algebras ◽  
Central Elements ◽  
Reductive Dual Pairs

AbstractFor each complex reductive dual pair introduced by R. Howe, this paper presents a formula for the central elements of the universal enveloping algebras given by I. M. Gelfand. This formula provides an explicit description of the correspondence between the ‘centers’ of the two universal enveloping algebras.

scholarly journals Reductive Dual Pair Correspondence for Complex Groups

1995 ◽  
Vol 132 (1) ◽  
pp. 1-42 ◽  
Author(s):  
J. Adams ◽  
D. Barbasch
Keyword(s):  
Dual Pair ◽  
Dual Pair Correspondence

scholarly journals Orbifold Zeta Functions for Dual Invertible Polynomials

2016 ◽  
Vol 60 (1) ◽  
pp. 99-106 ◽  
Author(s):  
Wolfgang Ebeling ◽  
Sabir M. Gusein-Zade
Keyword(s):  
Abelian Group ◽  
Homogeneous Polynomial ◽  
Zeta Functions ◽  
Dual Pair ◽  
Dual Pairs

AbstractAn invertible polynomial innvariables is a quasi-homogeneous polynomial consisting ofnmonomials so that the weights of the variables and the quasi-degree are well defined. In the framework of the construction of mirror symmetric orbifold Landau–Ginzburg models, Berglund, Hübsch and Henningson considered a pair (f, G) consisting of an invertible polynomialfand an abelian groupGof its symmetries together with a dual pair. Here we study the reduced orbifold zeta functions of dual pairs (f, G) andand show that they either coincide or are inverse to each other depending on the numbernof variables.

scholarly journals Characterizations of the dilation of frame generator dual pairs for group

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Liang Li ◽  
Pengtong Li
Keyword(s):  
Unitary Representation ◽  
Hilbert Spaces ◽  
Dual Pair ◽  
Special Structure ◽  
Dual Pairs ◽  
Special Operator

Abstract In this paper, we are interested in the dilation problem on frame generator dual pairs for a unitary representation in Hilbert spaces. We show the existence of a Riesz generator dilation dual pair of a frame generator dual pair in Hilbert spaces. Then we reveal the uniqueness of such dilations in the sense of similarity and give a characterization of the dilation of frame generator alternate dual pairs by that of the canonical dual pair in terms of a special operator. We also exhibit that the corresponding operator between two dilations of a frame generator dual pair is in a special structure.

Sign in / Sign up

Export Citation Format

Share Document