scholarly journals A note on commutation relation in conformal field theory

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Lento Nagano ◽  
Seiji Terashima
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Minkowski Space ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Scalar Fields ◽  
Expectation Value ◽  
Conformal Field ◽  
Free Scalar ◽  
Distance Limit

Abstract In this note, we explicitly compute the vacuum expectation value of the commutator of scalar fields in a d-dimensional conformal field theory on the cylinder. We find from explicit calculations that we need smearing not only in space but also in time to have finite commutators except for those of free scalar operators. Thus the equal time commutators of the scalar fields are not well-defined for a non-free conformal field theory, even if which is defined from the Lagrangian. We also have the commutator for a conformal field theory on Minkowski space, instead of the cylinder, by taking the small distance limit. For the conformal field theory on Minkowski space, the above statements are also applied.

scholarly journals Scale-invariance, dynamically induced Planck scale and inflation in the Palatini formulation

2021 ◽  
Vol 2105 (1) ◽  
pp. 012005
Author(s):  
Ioannis D. Gialamas ◽  
Alexandros Karam ◽  
Thomas D. Pappas ◽  
Antonio Racioppi ◽  
Vassilis C. Spanos
Keyword(s):  
Scale Invariance ◽  
Planck Scale ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Scalar Fields ◽  
Scale Invariant ◽  
Expectation Value ◽  
Hilbert Action

Abstract We present two scale invariant models of inflation in which the addition of quadratic in curvature terms in the usual Einstein-Hilbert action, in the context of Palatini formulation of gravity, manages to reduce the value of the tensor-to-scalar ratio. In both models the Planck scale is dynamically generated via the vacuum expectation value of the scalar fields.

scholarly journals RELAXATION MECHANISMS: FROM STRING MODULI TO AXIONS

1995 ◽  
Vol 10 (39) ◽  
pp. 2993-2999 ◽  
Author(s):  
C.E. VAYONAKIS
Keyword(s):  
String Theory ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Scalar Fields ◽  
Relaxation Mechanism ◽  
Expectation Value

The relaxation mechanism of Damour-Polyakov for fixing the vacuum expectation value of certain scalar fields (moduli) in string theory could provide a convenient framework for the Peccei- Quinn relaxation mechanism and remove the narrow “axion window”.

scholarly journals A CONFORMAL FIELD THEORY DESCRIPTION OF THE PAIRED AND PARAFERMIONIC STATES IN THE QUANTUM HALL EFFECT

2000 ◽  
Vol 15 (27) ◽  
pp. 1679-1688 ◽  
Author(s):  
GERARDO CRISTOFANO ◽  
GIUSEPPE MAIELLA ◽  
VINCENZO MAROTTA
Keyword(s):  
Field Theory ◽  
Wave Function ◽  
Conformal Field Theory ◽  
Quantum Hall Effect ◽  
Ground State ◽  
Quantum Hall ◽  
Scalar Fields ◽  
Neutral Component ◽  
Conformal Field ◽  
Twisted Boundary Conditions

We extend the construction of the effective conformal field theory for the Jain hierarchical fillings proposed in Ref. 1 to the description of a quantum Hall fluid at nonstandard fillings [Formula: see text]. The chiral primary fields are found by using a procedure which induces twisted boundary conditions on the m scalar fields; they appear as composite operators of a charged and neutral component. The neutral modes describe parafermions and contribute to the ground state wave function with a generalized Pfaffian term. Correlators of Ne electrons in the presence of quasi-hole excitations are explicitly given for m=2.

scholarly journals Defect conformal blocks from Appell functions

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Ilija Burić ◽  
Volker Schomerus
Keyword(s):  
Field Theory ◽  
Correlation Function ◽  
Conformal Field Theory ◽  
Hypergeometric Functions ◽  
Conformal Symmetry ◽  
Scalar Fields ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Additional Defect ◽  
Appell Functions

Abstract We develop a group theoretical formalism to study correlation functions in defect conformal field theory, with multiple insertions of bulk and defect fields. This formalism is applied to construct the defect conformal blocks for three-point functions of scalar fields. Starting from a configuration with one bulk and one defect field, for which the correlation function is determined by conformal symmetry, we explore two possibilities, adding either one additional defect or bulk field. In both cases it is possible to express the blocks in terms of classical hypergeometric functions, though the case of two bulk and one defect field requires Appell’s function F4.

scholarly journals Looking at shadows of entanglement wedges

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Yuya Kusuki ◽  
Yuki Suzuki ◽  
Tadashi Takayanagi ◽  
Koji Umemoto
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
De Sitter ◽  
Conformal Field ◽  
Large N ◽  
Higher Dimensional ◽  
Round Ball ◽  
Wedge Structure ◽  
Free Scalar ◽  
Reduced Density

Abstract We present a new method of deriving shapes of entanglement wedges directly from conformal field theory (CFT) calculations. We point out that a reduced density matrix in holographic CFTs possesses a sharp wedge structure such that inside the wedge we can distinguish two local excitations, while outside we cannot. We can determine this wedge, which we call a CFT wedge, by computing a distinguishability measure. We find that CFT wedges defined by the fidelity or Bures distance as a distinguishability measure coincide perfectly with shadows of entanglement wedges in anti-de Sitter (AdS)/CFT. We confirm this agreement between CFT wedges and entanglement wedges for two-dimensional holographic CFTs where the subsystem is chosen to be an interval or double intervals, as well as higher-dimensional CFTs with a round ball subsystem. On the other hand, if we consider a free scalar CFT, we find that there are no sharp CFT wedges. This shows that sharp entanglement wedges emerge only for holographic CFTs owing to the large-$N$ factorization. We also generalize our analysis to a time-dependent example and to a holographic boundary conformal field theory (AdS/BCFT). Finally, we study other distinguishability measures to define CFT wedges. We observe that some of the measures lead to CFT wedges which slightly deviate from the entanglement wedges in AdS/CFT, and we give a heuristic explanation for this. This paper is an extended version of our earlier letter (arXiv:1908.09939 [hep-th]) and includes various new observations and examples.

scholarly journals A nilpotency index of conformal manifolds

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Zohar Komargodski ◽  
Shlomo S. Razamat ◽  
Orr Sela ◽  
Adar Sharon
Keyword(s):  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Generic Point ◽  
Expectation Value ◽  
Conformal Field ◽  
Chiral Ring ◽  
Nilpotency Index ◽  
Conformal Manifolds

Abstract We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index.

scholarly journals A TWISTED CONFORMAL FIELD THEORY DESCRIPTION OF THE QUANTUM HALL EFFECT

2000 ◽  
Vol 15 (08) ◽  
pp. 547-555 ◽  
Author(s):  
GERARDO CRISTOFANO ◽  
GIUSEPPE MAIELLA ◽  
VINCENZO MAROTTA
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Ground State ◽  
Quantum Hall ◽  
Scalar Fields ◽  
Wave Functions ◽  
Charged Scalar ◽  
Conformal Field ◽  
Charged Scalar Field ◽  
Twisted Boundary Conditions

We construct an effective conformal field theory by using a procedure which induces twisted boundary conditions for the fundamental scalar fields. That allows one to describe a quantum Hall fluid at Jain hierarchical filling, [Formula: see text], in terms of one charged scalar field and m - 1 neutral ones. Then the resulting algebra of the chiral primary fields is U(1)×[Formula: see text]. Finally the ground state wave functions are given as correlators of appropriate composite fields (a-electrons).

INFINITE DIMENSIONAL LIE ALGEBRAS IN 4D CONFORMAL FIELD THEORY

2008 ◽  
Vol 05 (08) ◽  
pp. 1361-1371
Author(s):  
IVAN TODOROV
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Recent Work ◽  
Lie Algebras ◽  
Scalar Fields ◽  
Operator Product ◽  
Positive Energy ◽  
Infinite Dimensional ◽  
Conformal Field ◽  
Compact Gauge Group

It is known that there are no scalar Lie fields in more than two space-time dimensions [4]. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. Recent work, [2, 3], is reviewed, in which we classify such algebras and their unitary positive energy representations in a theory of a system of scalar fields of dimension two. The results are linked to the Doplicher–Haag–Roberts theory of superselection sectors governed by a (global) compact gauge group.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

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