Relation between large dimension operators and oscillator algebra of Young diagrams

The operators with large scaling dimensions can be labeled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large N but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu–Wigner contraction of the u(2) algebra inside of the u(p) algebra of p giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large N limit.

Download Full-text

NONPLANAR INTEGRABILITY: BEYOND THE SU(2) SECTOR

International Journal of Modern Physics A ◽

10.1142/s0217751x11054590 ◽

2011 ◽

Vol 26 (26) ◽

pp. 4553-4583 ◽

Cited By ~ 9

Author(s):

ROBERT DE MELLO KOCH ◽

BADR AWAD ELSEID MOHAMMED ◽

STEPHANIE SMITH

Keyword(s):

Harmonic Oscillators ◽

Projection Operators ◽

Young Diagrams ◽

Schur Polynomials ◽

Recursion Relations ◽

Anomalous Dimensions ◽

Dilatation Operator ◽

Analytic Expressions ◽

The One ◽

Kravchuk Polynomials

We compute the one-loop anomalous dimensions of restricted Schur polynomials with a classical dimension Δ~O(N). The operators that we consider are labeled by Young diagrams with two long columns or two long rows. Simple analytic expressions for the action of the dilatation operator are found. The projection operators needed to define the restricted Schur polynomials are constructed by translating the problem into a spin chain language, generalizing earlier results obtained in the SU(2) sector of the theory. The diagonalization of the dilatation operator reduces to solving five term recursion relations. The recursion relations can be solved exactly in terms of products of symmetric Kravchuk polynomials with Hahn polynomials. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable, extending a similar conclusion reached for the SU(2) sector of the theory.

Download Full-text

Invariant valuations on quaternionic vector spaces

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748011000247 ◽

2012 ◽

Vol 11 (3) ◽

pp. 467-499 ◽

Cited By ~ 13

Author(s):

Andreas Bernig

Keyword(s):

Vector Space ◽

Vector Spaces ◽

Young Diagrams ◽

Schur Polynomials ◽

Combinatorial Dimension ◽

Translation Invariant

AbstractThe spaces of Sp(n)-, Sp(n) · U(1)- and Sp(n) · Sp(1)-invariant, translation-invariant, continuous convex valuations on the quaternionic vector space ℍn are studied. Combinatorial dimension formulae involving Young diagrams and Schur polynomials are proved.

Download Full-text

Shiraishi functor and non-Kerov deformation of Macdonald polynomials

The European Physical Journal C ◽

10.1140/epjc/s10052-020-08540-4 ◽

2020 ◽

Vol 80 (10) ◽

Author(s):

Hidetoshi Awata ◽

Hiroaki Kanno ◽

Andrei Mironov ◽

Alexei Morozov

Keyword(s):

Network Models ◽

Elliptic Systems ◽

Macdonald Polynomials ◽

Symmetric Polynomials ◽

Leading Order ◽

Young Diagrams ◽

Schur Polynomials ◽

Pochhammer Symbol ◽

The Family ◽

Two Kinds Of Variables

AbstractWe suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables, $$\vec {x}$$ x → and $$\vec {y}$$ y → , and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When $$\vec {y}$$ y → are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion.

Download Full-text

GENERALIZED DEFORMED HARMONIC OSCILLATORS IN THE FRAMEWORK OF UNIFIED (q; α, β, γ; ν)-DEFORMATION

Modern Physics Letters A ◽

10.1142/s0217732310033050 ◽

2010 ◽

Vol 25 (15) ◽

pp. 1239-1249 ◽

Cited By ~ 2

Author(s):

I. M. BURBAN

Keyword(s):

Coherent States ◽

Anharmonic Oscillator ◽

Harmonic Oscillators ◽

Kerr Medium ◽

Oscillator Algebra ◽

Special Values ◽

Deformation Parameters ◽

Deformed Oscillator

The aim of this paper is the study of the generalized deformed quantum oscillators in the framework (q; α, β, γ; ν)-deformed of oscillator algebra. By selecting the special values of deformation parameters, we have separated a generalized deformed oscillator connected with generalized discrete Hermite II polynomials. By these means we have constructed Barut–Girardello type coherent states of this oscillator. We have found the conditions on the (q; α, β, γ; ν)-deformation parameters at which the free (q; α, β, γ; ν)-deformed oscillator approximate the usual anharmonic oscillator in the homogeneous Kerr medium.

Download Full-text

NONPLANAR INTEGRABILITY AND PARITY IN ABJ THEORY

International Journal of Modern Physics A ◽

10.1142/s0217751x13500437 ◽

2013 ◽

Vol 28 (12) ◽

pp. 1350043 ◽

Cited By ~ 5

Author(s):

BADR AWAD ELSEID MOHAMMED

Keyword(s):

String Theory ◽

Harmonic Oscillators ◽

Schur Polynomials ◽

Gauge Invariant ◽

Abjm Theory ◽

Anomalous Dimensions ◽

Matter Theory ◽

Dilatation Operator ◽

Chern Simons ◽

Double Limit

In this paper, we study the action of the nonplanar two-loop dilatation operator in an SU(2)×SU(2) subsector of the ABJ Chern–Simons-matter theory. The gauge invariant operators we consider are the restricted Schur polynomials. As in ABJM theory, there is a limit in which the spectrum reduces to a set of decoupled harmonic oscillators, indicating integrability in the large M and N double limit of the theory. We then consider parity transformations on the gauge invariant operators. In this case the nonplanar anomalous dimensions break parity invariance. Our analysis shows that (M-N) is related to the holonomy in the string theory, confirming one of the main features of the theory and its string dual. Furthermore, in the limit where ABJ theory reduces to ABJM theory, parity invariance is restored.

Download Full-text

Discretized representations of harmonic variables by bilateral Jacobi operators

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022600000285 ◽

2000 ◽

Vol 4 (4) ◽

pp. 297-308

Author(s):

Andreas Ruffing

Keyword(s):

Quantum Optics ◽

Harmonic Oscillator ◽

Sequence Space ◽

Heisenberg Algebra ◽

Harmonic Oscillators ◽

Quantum Oscillator ◽

Oscillator Algebra ◽

Jacobi Operators ◽

Quantum Properties ◽

Weighted Sequence Space

Starting from a discrete Heisenberg algebra we solve several representation problems for a discretized quantum oscillator in a weighted sequence space. The Schrödinger operator for a discrete harmonic oscillator is derived. The representation problem for aq-oscillator algebra is studied in detail. The main result of the article is the fact that the energy representation for the discretized momentum operator can be interpreted as follows: It allows to calculate quantum properties of a large number of non-interacting harmonic oscillators at the same time. The results can be directly related to current research on squeezed laser states in quantum optics. They reveal and confirm the observation that discrete versions of continuum Schrodinger operators allow more structural freedom than their continuum analogs do.

Download Full-text