scholarly journals Conformal quantum mechanics & the integrable spinning Fishnet

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Sergey Derkachov ◽  
Enrico Olivucci
Keyword(s):  
Conformal Symmetry ◽  
Scaling Limit ◽  
Principal Series ◽  
Conformal Group ◽  
Baxter Equation ◽  
Special Choice ◽  
Conformal Field ◽  
Quantum Magnet ◽  
Supersymmetric Theories ◽  
Conformal Quantum Mechanics

Abstract In this paper we consider systems of quantum particles in the 4d Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series ∆ = 2 + iν for any left/right spins ℓ,$$ \dot{\ell} $$ ℓ ̇ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral star-triangle identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose k-th site hosts a particle in the representation (∆k, ℓk,$$ \dot{\ell} $$ ℓ ̇ k) of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories [1]. For the special choice of particles in the scalar (1, 0, 0) and fermionic (3/2, 1, 0) representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the γ-deformed $$ \mathcal{N} $$ N = 4 and $$ \mathcal{N} $$ N = 2 supersymmetric theories.

scholarly journals Instability of complex CFTs with operators in the principal series

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Dario Benedetti
Keyword(s):  
Principal Series ◽  
Conformal Group ◽  
Point Of View ◽  
Field Theories ◽  
Operator Product ◽  
Quantum Field Theories ◽  
Conformal Field Theories ◽  
Conformal Field ◽  
Primary Operator ◽  
Tensor Models

Abstract We prove the instability of d-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension h = $$ \frac{d}{2} $$ d 2 + i r, with non-vanishing r ∈ ℝ. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdSd+1; we derive it here directly for generic d-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group to perturbations of the conformal solution in the two-particle irreducible (2PI) effective action. Some explicit examples are discussed, such as melonic tensor models and the biscalar fishnet model.

NONARCHIMEDEAN CONFORMAL FIELD THEORIES

1989 ◽  
Vol 04 (18) ◽  
pp. 4877-4908 ◽  
Author(s):  
EZER MELZER
Keyword(s):  
Correlation Functions ◽  
Central Charge ◽  
Conformal Symmetry ◽  
Conformal Group ◽  
Field Theories ◽  
Dimensional Euclidean Space ◽  
Conformal Field Theories ◽  
Linear Transformations ◽  
Associative Algebras ◽  
Conformal Field

We present a general formalism for conformal field theories defined on a non-Archimedean field. Such theories are defined by complex-valued correlation functions of fields of a [Formula: see text]-adic variable. Conformal invariance is imposed by requiring the correlation functions to be unchanged under fractional linear transformations, the latter forming the full analogue of the conformal group in two-dimensional, euclidean space-time. All fields in the theory can be taken to be "primary", under the "non-Archimedean conformal group". The conformal symmetry fixes completely the form of all correlation functions, once we are given the weight-spectrum of the theory and the OPE coefficients (which must be the structure constants of certain commutative, associative algebras). We explicitly construct non-Archimedean CFT's having the same weight spectrum as that of Archimedean models of central charge c < 1. The OPE coefficients of these "local" Archimedean and non-Archimedean models are related by adelic formulae.

scholarly journals Anisotropic scaling of the two-dimensional Ising model II: surfaces and boundary fields

2020 ◽  
Vol 8 (3) ◽  
Author(s):  
Hendrik Hobrecht ◽  
Fred Hucht
Keyword(s):  
Ising Model ◽  
Boundary Conditions ◽  
Scaling Limit ◽  
Finite Size ◽  
Square Lattice ◽  
Conformal Field ◽  
Anisotropic Scaling ◽  
Finite Lattices ◽  
Antiferromagnetic Ising Model ◽  
Boundary Fields

Based on the results published recently [SciPost Phys. 7, 026 (2019)], the influence of surfaces and boundary fields are calculated for the ferromagnetic anisotropic square lattice Ising model on finite lattices as well as in the finite-size scaling limit. Starting with the open cylinder, we independently apply boundary fields on both sides which can be either homogeneous or staggered, representing different combinations of boundary conditions. We confirm several predictions from scaling theory, conformal field theory and renormalisation group theory: we explicitly show that anisotropic couplings enter the scaling functions through a generalised aspect ratio, and demonstrate that open and staggered boundary conditions are asymptotically equal in the scaling regime. Furthermore, we examine the emergence of the surface tension due to one antiperiodic boundary in the system in the presence of symmetry breaking boundary fields, again for finite systems as well as in the scaling limit. Finally, we extend our results to the antiferromagnetic Ising model.

scholarly journals NONCOMMUTATIVE CONFORMAL FIELD THEORY IN THE TWIST-DEFORMED CONTEXT

2008 ◽  
Vol 23 (39) ◽  
pp. 3307-3315 ◽  
Author(s):  
FEDELE LIZZI ◽  
PATRIZIA VITALE
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Noncommutative Geometry ◽  
Conformal Symmetry ◽  
Two Dimensional ◽  
Moyal Product ◽  
Conformal Field ◽  
Noncommutative Plane

We discuss conformal symmetry on the two-dimensional noncommutative plane equipped with Moyal product in the twist deformed context. We show that the consistent use of the twist procedure leads to results which are free from ambiguities. This lends support to the importance of the use of twist symmetries in noncommutative geometry.

scholarly journals INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6175-6213 ◽  
Author(s):  
T. TJIN
Keyword(s):  
Lie Groups ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Baxter Equation ◽  
Product Decomposition ◽  
R Matrix ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Lie Groups And Algebras ◽  
Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

scholarly journals Spontaneous Symmetry Breaking in 5D Conformally Invariant Gravity

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Taeyoon Moon ◽  
Phillial Oh
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Conformal Symmetry ◽  
Brane World ◽  
Spontaneous Breaking ◽  
Warp Factor ◽  
Translational Symmetry ◽  
Tensor Perturbation ◽  
Conformally Invariant ◽  
Conformal Quantum Mechanics

We explore the possibility of the spontaneous symmetry breaking in 5D conformally invariant gravity, whose action consists of a scalar field nonminimally coupled to the curvature with its potential. Performing dimensional reduction via ADM decomposition, we find that the model allows an exact solution giving rise to the 4D Minkowski vacuum. Exploiting the conformal invariance with Gaussian warp factor, we show that it also admits a solution which implements the spontaneous breaking of conformal symmetry. We investigate its stability by performing the tensor perturbation and find the resulting system is described by the conformal quantum mechanics. Possible applications to the spontaneous symmetry breaking of time-translational symmetry along the dynamical fifth direction and the brane-world scenario are discussed.

IMPORTED CONFORMAL SYMMETRY OF ANYONS IN A UNIFORM MAGNETIC FIELD

1992 ◽  
Vol 06 (08) ◽  
pp. 1229-1242 ◽  
Author(s):  
T. AWAJI ◽  
M. HOTTA
Keyword(s):  
Magnetic Field ◽  
Quantum Mechanics ◽  
External Magnetic Field ◽  
Uniform Magnetic Field ◽  
Conformal Symmetry ◽  
Dynamical Symmetry ◽  
Conformal Group ◽  
Symmetry Consideration

We analyze an N-body quantum mechanics of anyons in an external magnetic field. It is pointed out that an SO(2, 1) dynamical symmetry, which is related to the conformal group, plays an important role in anyon dynamics. It is shown that the two-body spectrum is fully reproduced solely by the symmetry consideration. Moreover, we discuss some constraints on missing states from the symmetry consideration.

scholarly journals Fock representations of multicomponent (particularly non-Abelian anyon) commutation relations

2019 ◽  
Vol 32 (05) ◽  
pp. 2030004
Author(s):  
Alexei Daletskii ◽  
Alexander Kalyuzhny ◽  
Eugene Lytvynov ◽  
Daniil Proskurin
Keyword(s):  
Linear Operator ◽  
Fock Space ◽  
Multicomponent System ◽  
Baxter Equation ◽  
Special Choice ◽  
Bounded Linear ◽  
Creation And Annihilation Operators ◽  
Commutation Relations ◽  
System A ◽  
Fock Representations

Let [Formula: see text] be a separable Hilbert space and [Formula: see text] be a self-adjoint bounded linear operator on [Formula: see text] with norm [Formula: see text], satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator [Formula: see text] determines a [Formula: see text]-deformed Fock space [Formula: see text]. We start with reviewing and extending the known results about the structure of the [Formula: see text]-particle spaces [Formula: see text] and the commutation relations satisfied by the corresponding creation and annihilation operators acting on [Formula: see text]. We then choose [Formula: see text], the [Formula: see text]-space of [Formula: see text]-valued functions on [Formula: see text]. Here [Formula: see text] and [Formula: see text] with [Formula: see text]. Furthermore, we assume that the operator [Formula: see text] acting on [Formula: see text] is given by [Formula: see text]. Here, for a.a. [Formula: see text], [Formula: see text] is a linear operator on [Formula: see text] with norm [Formula: see text] that satisfies [Formula: see text] and the spectral quantum Yang–Baxter equation. The corresponding creation and annihilation operators describe a multicomponent quantum system. A special choice of the operator-valued function [Formula: see text] in the case [Formula: see text] determines non-Abelian anyons (also called plektons). For a multicomponent system, we describe its [Formula: see text]-deformed Fock space and the available commutation relations satisfied by the corresponding creation and annihilation operators. Finally, we consider several examples of multicomponent quantum systems.

scholarly journals Conformal group theory of tensor structures

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Ilija Burić ◽  
Volker Schomerus ◽  
Mikhail Isachenkov
Keyword(s):  
Correlation Functions ◽  
Conformal Group ◽  
Wave Functions ◽  
Dimensional Euclidean Space ◽  
Theoretic Approach ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Local Operators ◽  
Indispensable Tool ◽  
Systematic Derivation

Abstract The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are functions of cross ratios only, and the correlation functions that depend on insertion points in the d-dimensional Euclidean space. Here we develop an entirely group theoretic approach to tensor structures, based on the Cartan decomposition of the conformal group. It provides us with a new universal formula for tensor structures and thereby a systematic derivation of crossing equations. Our approach applies to a ‘gauge’ in which the conformal blocks are wave functions of Calogero-Sutherland models rather than solutions of the more standard Casimir equations. Through this ab initio construction of tensor structures we complete the Calogero-Sutherland approach to conformal correlators, at least for four-point functions of local operators in non-supersymmetric models. An extension to defects and superconformal symmetry is possible.

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