scholarly journals NONPLANAR INTEGRABILITY: BEYOND THE SU(2) SECTOR

2011 ◽  
Vol 26 (26) ◽  
pp. 4553-4583 ◽  
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.

scholarly journals NONPLANAR INTEGRABILITY AND PARITY IN ABJ THEORY

2013 ◽  
Vol 28 (12) ◽  
pp. 1350043 ◽  
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.

scholarly journals Relation between large dimension operators and oscillator algebra of Young diagrams

2015 ◽  
Vol 12 (04) ◽  
pp. 1550047 ◽  
Author(s):  
Hai Lin
Keyword(s):  
Large Dimension ◽  
Harmonic Oscillators ◽  
Oscillator Algebra ◽  
Young Diagrams ◽  
Schur Polynomials ◽  
Large N

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.

scholarly journals Rotating restricted Schur polynomials

2017 ◽  
Vol 32 (25) ◽  
pp. 1750150 ◽  
Author(s):  
Nicholas Bornman ◽  
Robert de Mello Koch ◽  
Laila Tribelhorn
Keyword(s):  
Field Theory ◽  
Symmetry Algebra ◽  
Structural Features ◽  
Schur Polynomials ◽  
Small Perturbations ◽  
Yang Mills ◽  
Dilatation Operator ◽  
The One ◽  
Symmetry Generators ◽  
Super Yang Mills Theory

Large [Formula: see text] but nonplanar limits of [Formula: see text] super-Yang–Mills theory can be described using restricted Schur polynomials. Previous investigations demonstrate that the action of the one loop dilatation operator on restricted Schur operators, with classical dimension of order [Formula: see text] and belonging to the [Formula: see text] sector, is largely determined by the [Formula: see text] [Formula: see text] symmetry algebra as well as structural features of perturbative field theory. Studies presented so far have used the form of [Formula: see text] symmetry generators when acting on small perturbations of half-BPS operators. In this paper as a first step towards going beyond small perturbations of the half-BPS operators, we explain how the exact action of symmetry generators on restricted Schur polynomials can be determined.

scholarly journals The integrable (hyper)eclectic spin chain

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Changrim Ahn ◽  
Matthias Staudacher
Keyword(s):  
Spin Chain ◽  
Nearest Neighbor ◽  
Inverse Scattering Method ◽  
Spin Chains ◽  
Scattering Method ◽  
Yang Mills ◽  
Deformation Parameters ◽  
Dilatation Operator ◽  
The One ◽  
State Spin

Abstract We refine the notion of eclectic spin chains introduced in [1] by including a maximal number of deformation parameters. These models are integrable, nearest-neighbor n-state spin chains with exceedingly simple non-hermitian Hamiltonians. They turn out to be non-diagonalizable in the multiparticle sector (n > 2), where their “spectrum” consists of an intricate collection of Jordan blocks of arbitrary size and multiplicity. We show how and why the quantum inverse scattering method, sought to be universally applicable to integrable nearest-neighbor spin chains, essentially fails to reproduce the details of this spectrum. We then provide, for n=3, detailed evidence by a variety of analytical and numerical techniques that the spectrum is not “random”, but instead shows surprisingly subtle and regular patterns that moreover exhibit universality for generic deformation parameters. We also introduce a new model, the hypereclectic spin chain, where all parameters are zero except for one. Despite the extreme simplicity of its Hamiltonian, it still seems to reproduce the above “generic” spectra as a subset of an even more intricate overall spectrum. Our models are inspired by parts of the one-loop dilatation operator of a strongly twisted, double-scaled deformation of $$ \mathcal{N} $$ N = 4 Super Yang-Mills Theory.

scholarly journals LAURENT POLYNOMIAL LANDAU–GINZBURG MODELS FOR COMINUSCULE HOMOGENEOUS SPACES

2021 ◽  
Author(s):  
PETER SPACEK
Keyword(s):  
Homogeneous Space ◽  
Homogeneous Spaces ◽  
Laurent Polynomial ◽  
Complete Intersections ◽  
Laurent Polynomials ◽  
Young Diagrams ◽  
Similar Shape ◽  
Algebraic Torus ◽  
The One ◽  
Polynomial Potentials

AbstractIn this article we construct Laurent polynomial Landau–Ginzburg models for cominuscule homogeneous spaces. These Laurent polynomial potentials are defined on a particular algebraic torus inside the Lie-theoretic mirror model constructed for arbitrary homogeneous spaces in [Rie08]. The Laurent polynomial takes a similar shape to the one given in [Giv96] for projective complete intersections, i.e., it is the sum of the toric coordinates plus a quantum term. We also give a general enumeration method for the summands in the quantum term of the potential in terms of the quiver introduced in [CMP08], associated to the Langlands dual homogeneous space. This enumeration method generalizes the use of Young diagrams for Grassmannians and Lagrangian Grassmannians and can be defined type-independently. The obtained Laurent polynomials coincide with the results obtained so far in [PRW16] and [PR13] for quadrics and Lagrangian Grassmannians. We also obtain new Laurent polynomial Landau–Ginzburg models for orthogonal Grassmannians, the Cayley plane and the Freudenthal variety.

scholarly journals Semi-Analytic Evaluation of 1, 2 and 3-Electron Coulomb Integrals with Gaussian Expansion of Distance Operators W= RC1-nRD1-M, RC1-Nr12-M, R12-Nr13-M

2019 ◽  
Author(s):  
Sandor Kristyan
Keyword(s):  
Gaussian Function ◽  
Approximate Solutions ◽  
Equation System ◽  
Least Square ◽  
Position Vector ◽  
Least Square Fit ◽  
Analytic Expressions ◽  
The One ◽  
Analytical Expressions ◽  
Coulomb Integrals

The equations derived help to evaluate semi-analytically (mostly for k=1,2 or 3) the important Coulomb integrals Int rho(r1)…rho(rk) W(r1,…,rk) dr1…drk, where the one-electron density, rho(r1), is a linear combination (LC) of Gaussian functions of position vector variable r1. It is capable to describe the electron clouds in molecules, solids or any media/ensemble of materials, weight W is the distance operator indicated in the title. R stands for nucleus-electron and r for electron-electron distances. The n=m=0 case is trivial, the (n,m)=(1,0) and (0,1) cases, for which analytical expressions are well known, are widely used in the practice of computation chemistry (CC) or physics, and analytical expressions are also known for the cases n,m=0,1,2. The rest of the cases – mainly with any real (integer, non-integer, positive or negative) n and m - needs evaluation. We base this on the Gaussian expansion of |r|^-u, of which only the u=1 is the physical Coulomb potential, but the u≠1 cases are useful for (certain series based) correction for (the different) approximate solutions of Schrödinger equation, for example, in its wave-function corrections or correlation calculations. Solving the related linear equation system (LES), the expansion |r|^-u about equal SUM(k=0toL)SUM(i=1toM) Cik r^2k exp(-Aik r^2) is analyzed for |r| = r12 or RC1 with least square fit (LSF) and modified Taylor expansion. These evaluated analytic expressions for Coulomb integrals (up to Gaussian function integrand and the Gaussian expansion of |r|^-u) are useful for the manipulation with higher moments of inter-electronic distances via W, even for approximating Hamiltonian.

scholarly journals Invariant valuations on quaternionic vector spaces

2012 ◽  
Vol 11 (3) ◽  
pp. 467-499 ◽  
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.

scholarly journals One-loop non-planar anomalous dimensions in super Yang-Mills theory

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Tristan McLoughlin ◽  
Raul Pereira ◽  
Anne Spiering
Keyword(s):  
Perturbation Theory ◽  
Integrable Systems ◽  
Chaotic Systems ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Mechanical Perturbation ◽  
Yang Mills ◽  
Anomalous Dimensions ◽  
Dilatation Operator ◽  
Scalar Products

Abstract We consider non-planar one-loop anomalous dimensions in maximally supersymmetric Yang-Mills theory and its marginally deformed analogues. Using the basis of Bethe states, we compute matrix elements of the dilatation operator and find compact expressions in terms of off-shell scalar products and hexagon-like functions. We then use non-degenerate quantum-mechanical perturbation theory to compute the leading 1/N2 corrections to operator dimensions and as an example compute the large R-charge limit for two-excitation states through subleading order in the R-charge. Finally, we numerically study the distribution of level spacings for these theories and show that they transition from the Poisson distribution for integrable systems at infinite N to the GOE Wigner-Dyson distribution for quantum chaotic systems at finite N.

scholarly journals Recursive modular modelling methodology for lumped-parameter dynamic systems

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160891 ◽  
Author(s):  
Renato Maia Matarazzo Orsino
Keyword(s):  
Dynamic Systems ◽  
Recursive Algorithm ◽  
Complex Model ◽  
Projection Operators ◽  
Novel Approach ◽  
Lumped Parameter ◽  
Modelling Methodology ◽  
Modelling Techniques ◽  
Single Complex ◽  
The One

This paper proposes a novel approach to the modelling of lumped-parameter dynamic systems, based on representing them by hierarchies of mathematical models of increasing complexity instead of a single (complex) model. Exploring the multilevel modularity that these systems typically exhibit, a general recursive modelling methodology is proposed, in order to conciliate the use of the already existing modelling techniques. The general algorithm is based on a fundamental theorem that states the conditions for computing projection operators recursively. Three procedures for these computations are discussed: orthonormalization, use of orthogonal complements and use of generalized inverses. The novel methodology is also applied for the development of a recursive algorithm based on the Udwadia–Kalaba equation, which proves to be identical to the one of a Kalman filter for estimating the state of a static process, given a sequence of noiseless measurements representing the constraints that must be satisfied by the system.

ONE-LOOP CORRECTION TO ENERGY OF SPINNING STRINGS IN AdS5×T1, 1

2012 ◽  
Vol 27 (03) ◽  
pp. 1250015
Author(s):  
HYOJOONG KIM ◽  
NAKWOO KIM ◽  
JUNG HUN LEE
Keyword(s):  
Field Theory ◽  
Mode Frequency ◽  
Harmonic Oscillators ◽  
Quantum Corrections ◽  
Gauge Field Theory ◽  
Leading Order ◽  
Loop Correction ◽  
Conformal Dimension ◽  
Normal Mode Frequency ◽  
The One

We consider circular spinning string solutions in AdS5×T1, 1 and calculate the quantum corrections to the energy at one-loop on worldsheet. The fluctuations are given as a set of harmonic oscillators and we calculate their normal mode frequency in closed form. The sum of frequency is equal to the one-loop string energy, which through AdS/CFT correspondence corresponds to the leading order correction of the conformal dimension for long operators in Klebanov–Witten conifold gauge field theory.

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