scholarly journals Quantum Current Algebra Symmetries and Integrable Many-Particle Schrödinger Type Quantum Hamiltonian Operators

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 975
Author(s):  
Dominik Prorok ◽  
Anatolij Prykarpatski
Keyword(s):  
Representation Theory ◽  
Current Algebra ◽  
Fock Space ◽  
Integrable Quantum ◽  
Quantum Hamiltonian ◽  
Quantum Symmetries ◽  
Sutherland Model ◽  
Quantum Current ◽  
Hamiltonian Operators

Based on the G. Goldin’s quantum current algebra symmetry representation theory, have succeeded in explaining a hidden relationship between the quantum many-particle Hamiltonian operators, defined in the Fock space, their factorized structure and integrability. Interesting for applications quantum oscillatory Hamiltonian operators are considered, the quantum symmetries of the integrable quantum Calogero-Sutherland model are analyzed in detail.

W-ALGEBRA REALIZATIONS AND W-GRAVITY ANOMALIES

1991 ◽  
Vol 06 (32) ◽  
pp. 2995-3003 ◽  
Author(s):  
C. M. HULL ◽  
L. PALACIOS
Keyword(s):  
Path Integral ◽  
Current Algebra ◽  
Gravity Anomalies ◽  
Normal Ordering ◽  
Path Integral Quantization ◽  
Transformation Rules ◽  
Higher Derivative ◽  
Quantum Current ◽  
Boson Model ◽  
Background Charge

The coupling of scalars fields to chiral W3 gravity is reviewed. In general the quantum current algebra generated by the spin-two and three currents does not close when the "natural" regularization (corresponding to the normal ordering with respect to the modes of ∂ϕi) is used, and the non-closure reflects matter-dependent anomalies in the path integral quantization. We consider the most general modification of the current, involving higher derivative "background charge" terms, and find the conditions for them to form a closed algebra in the "natural" regularization. These conditions can be satisfied only for the two-boson model. In that case, it is possible to cancel all the matter-dependent anomalies by adding finite local counter terms to the action and modifying the transformation rules of the fields.

scholarly journals Two Maps on Affine Type $A$ Crystals and Hecke Algebras

2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Nicolas Jacon
Keyword(s):  
Representation Theory ◽  
Quantum Groups ◽  
Fock Space ◽  
Hecke Algebras ◽  
Type A ◽  
Crystal Basis ◽  
Affine Type

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type $A$ and in the representation theory of Hecke algebras of type $G(l,l,n)$. 

scholarly journals A FREE FIELD REPRESENTATION OF THE SCREENING CURRENTS OF $U_q (\widehat{{\rm sl}(3)})$

1995 ◽  
Vol 10 (04) ◽  
pp. 561-578 ◽  
Author(s):  
A. H. BOUGOURZI ◽  
ROBERT A. WESTON
Keyword(s):  
Current Algebra ◽  
Free Field ◽  
Quantum Affine Algebra ◽  
Affine Algebra ◽  
Field Representation ◽  
Independent Screening ◽  
Quantum Current ◽  
Quantum Analog ◽  
Wakimoto Realization ◽  
Infinite Sum

We construct five independent screening currents associated with the [Formula: see text] quantum current algebra. The screening currents are expressed as exponentials of the eight basic deformed bosonic fields that are required in the quantum analog of the Wakimoto realization of the current algebra. Four of the screening currents are "simple," in that each one is given as a single exponential field. The fifth is expressed as an infinite sum of exponential fields. For reasons which we will discuss, we expect that the structure of the screening currents for a general quantum affine algebra will be similar to the [Formula: see text] case.

scholarly journals Quantum current algebra for the AdS 5 × S 5 superstring

2010 ◽  
Vol 2010 (8) ◽  
Author(s):  
O. A. Bedoya ◽  
D. Z. Marchioro ◽  
D. L. Nedel ◽  
B. Carlini Vallilo
Keyword(s):  
Current Algebra ◽  
Quantum Current

scholarly journals DERIVATIONS OF THE TRIGONOMETRIC BCn SUTHERLAND MODEL BY QUANTUM HAMILTONIAN REDUCTION

2010 ◽  
Vol 22 (06) ◽  
pp. 699-732 ◽  
Author(s):  
L. FEHÉR ◽  
B. G. PUSZTAI
Keyword(s):  
Laplace Operator ◽  
Coupling Constants ◽  
Hamiltonian Reduction ◽  
Real Form ◽  
Quantum Hamiltonian Reduction ◽  
Quantum Hamiltonian ◽  
Sutherland Model ◽  
The Laplace Operator ◽  
Vector Valued Functions ◽  
Vector Valued

The BCn Sutherland Hamiltonian with coupling constants parametrized by three arbitrary integers is derived by reductions of the Laplace operator of the group U(N). The reductions are obtained by applying the Laplace operator on spaces of certain vector valued functions equivariant under suitable symmetric subgroups of U(N) × U(N). Three different reduction schemes are considered, the simplest one being the compact real form of the reduction of the Laplacian of GL(2n, ℂ) to the complex BCn Sutherland Hamiltonian previously studied by Oblomkov.

scholarly journals 3D Current Algebra and Twisted K Theory

2018 ◽  
Vol 30 (07) ◽  
pp. 1840011
Author(s):  
Jouko Mickelsson
Keyword(s):  
Current Algebra ◽  
Fock Space ◽  
Abelian Extension ◽  
Fredholm Operators ◽  
Gauge Transformations ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Hilbert Space Operators ◽  
Hilbert Bundle ◽  
K Theory

Equivariant twisted K theory classes on compact Lie groups [Formula: see text] can be realized as families of Fredholm operators acting in a tensor product of a fermionic Fock space and a representation space of a central extension of the loop algebra [Formula: see text] using a supersymmetric Wess–Zumino–Witten model. The aim of the present paper is to extend the construction to higher loop algebras using an abelian extension of a 3D current algebra. We have only partial success: Instead of true Fredholm operators we have formal algebraic expressions in terms of the generators of the current algebra and an infinite dimensional Clifford algebra. These give rise to sesquilinear forms in a Hilbert bundle which transform in the expected way with respect to 3D gauge transformations but do not define true Hilbert space operators.

scholarly journals A QUANTUM ANALOG OF THE BOSON–FERMION CORRESPONDENCE

1995 ◽  
Vol 10 (07) ◽  
pp. 923-942 ◽  
Author(s):  
A. HAMID BOUGOURZI ◽  
LUC VINET
Keyword(s):  
Current Algebra ◽  
Quantum Case ◽  
Fermionic Field ◽  
Classical Correspondence ◽  
Quantum Current ◽  
Quantum Analog ◽  
Level 2

We review the classical boson-fermion correspondence in the context of the [Formula: see text] current algebra at level 2. This particular algebra is ideal for exhibiting this correspondence because it can be realized either in terms of three real bosonic fields or in terms of one real and one complex fermionic field. We also derive a fermionic realization of the quantum current algebra [Formula: see text] at level 2 and by comparing this realization with the existing bosonic one we extend the classical correspondence to the quantum case.

scholarly journals ITÔ–WIENER CHAOS AND THE HODGE DECOMPOSITION ON AN ABSTRACT WIENER SPACE

2013 ◽  
Vol 16 (01) ◽  
pp. 1350008
Author(s):  
YUXIN YANG
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Fock Space ◽  
Hodge Decomposition ◽  
Wiener Space ◽  
Space Representation ◽  
Abstract Wiener Space ◽  
Wiener Chaos ◽  
De Rham ◽  
Symplectic Connections

Using the structure of the Boson-Fermion Fock space and an argument taken from [P. Bieliavsky, M. Cahen, S. Gutt, J. Rawnsley and L. Schwachhofer, Symplectic connections, Int. J. Geom. Meth. Mod. Phys.3 (2006) 375–420], we give a new proof of the triviality of the L2 cohomology groups on an abstract Wiener space, alternative to that given by Shigekawa [De Rham–Hodge–Kodaira's decomposition on an abstract Wiener space, J. Math. Kyoto. Univ.26(2) (1986) 191–202]. We apply the representation theory of the symmetric group to characterize the spaces of exact and co-exact forms in their Boson-Fermion Fock space representation.

STRUCTURE OF IRREDUCIBLE SU(2) PARAFERMION MODULES DERIVED VIA THE FEIGIN-FUCHS CONSTRUCTION

1992 ◽  
Vol 07 (06) ◽  
pp. 1233-1265 ◽  
Author(s):  
PAUL A. GRIFFIN ◽  
OSCAR F. HERNÁNDEZ
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Correlation Functions ◽  
Fock Space ◽  
Coulomb Gas ◽  
Vertex Operators ◽  
Chiral Algebra ◽  
Zero Modes ◽  
Highest Weight ◽  
Brst Cohomology

We show how the Feigin-Fuchs Coulomb-gas construction, with two free Gaussian bosons, can be used to derive the representation theory of the SU(2) parafermion models. We identify the generators of the chiral algebra within the bosonic Fock space and derive the chiral algebra of the finitely reducible models, which correspond to the SU(2) and SU(1, 1) parafermion algebras. We then focus on the SU(2) highest-weight modules in the remainder of the paper. Unitarity of the modules requires that the states of the parafermion theory be independent of the zero modes of two fermionic vertex operators of the bosonic theory. The expressions for the Virasoro highest weights of the models are doubly degenerate in the bosonic Fock space. We formulate the correlation functions of these operators in the parafermion Hilbert space, and in particular, the fusion rules for the Virasoro highest weights are derived in an elegant way. Finally, the irreducible parafermion characters are derived. We discuss the connection between our analysis and previous work on representation theory based on BRST cohomology.

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