scholarly journals A TOPOLOGICAL INVARIANT OF RG FLOWS IN 2D INTEGRABLE QUANTUM FIELD THEORIES

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2927-2932 ◽  
Author(s):  
R. CARACCIOLO ◽  
F. GLIOZZI ◽  
R. TATEO
Keyword(s):  
Renormalization Group ◽  
Large Class ◽  
Three Dimensional ◽  
Topological Invariant ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Integrable Quantum ◽  
Quantum Integrable Models ◽  
Ansatz Approach

We construct a topological invariant of the renormalization group trajectories of a large class of 2D quantum integrable models, described by the thermodynamic Bethe ansatz approach. A geometrical description of this invariant in terms of triangulations of three-dimensional manifolds is proposed and associated dilogarithm identities are proven.

scholarly journals QUANTUM INTEGRABILITY OF COUPLED N=1 SUPER SINE/SINH–GORDON THEORIES AND THE LIE SUPERALGEBRA D(2,1;α)

1999 ◽  
Vol 14 (16) ◽  
pp. 2551-2580 ◽  
Author(s):  
JONATHAN M. EVANS ◽  
JENS OLE MADSEN
Keyword(s):  
Quantum Theory ◽  
Lie Superalgebra ◽  
Field Theories ◽  
Continuous Family ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Classical Level ◽  
Quantum Integrability ◽  
Integrable Quantum ◽  
Conserved Currents

We discuss certain integrable quantum field theories in 1+1 dimensions consisting of coupled sine/sinh–Gordon theories with N=1 supersymmetry, positive kinetic energy, and bosonic potentials which are bounded from below. We show that theories of this type can be constructed as Toda models based on the exceptional affine Lie superalgebra D(2,1;α)(1) (or on related algebras which can be obtained as various limits) provided one adopts appropriate reality conditions for the fields. In particular, there is a continuous family of such models in which the couplings and mass ratios all depend on the parameter α. The structure of these models is analyzed in some detail at the classical level, including the construction of conserved currents with spins up to 4. We then show that these currents generalize to the quantum theory, thus demonstrating quantum-integrability of the models.

Integrable Quantum Field Theories

2020 ◽  
pp. 575-621
Author(s):  
Giuseppe Mussardo
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Conformal Theory ◽  
Integrable Quantum ◽  
Set Up ◽  
Integrable Quantum Field Theory

Chapter 16 covers the general properties of the integrable quantum field theories, including how an integrable quantum field theory is characterized by an infinite number of conserved charges. These theories are illustrated by means of significant examples, such as the Sine–Gordon model or the Toda field theories based on the simple roots of a Lie algebra. For the deformations of a conformal theory, it shown how to set up an efficient counting algorithm to prove the integrability of the corresponding model. The chapter focuses on two-dimensional models, and uses the term ‘two-dimensional’ to denote both a generic two-dimensional quantum field theory as well as its Euclidean version.

Estimates on Renormalization Group Transformations

1998 ◽  
Vol 50 (4) ◽  
pp. 756-793 ◽  
Author(s):  
D. Brydges ◽  
J. Dimock ◽  
T. R. Hurd
Keyword(s):  
Renormalization Group ◽  
Gaussian Measure ◽  
Ultra Violet ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Quantum Fields ◽  
Scalar Quantum ◽  
Polymer Expansion ◽  
Renormalization Group Flows

AbstractWe consider a specific realization of the renormalization group (RG) transformation acting on functional measures for scalar quantum fields which are expressible as a polymer expansion times an ultra-violet cutoff Gaussian measure. The new and improved definitions and estimates we present are sufficiently general and powerful to allow iteration of the transformation, hence the analysis of complete renormalization group flows, and hence the construction of a variety of scalar quantum field theories.

scholarly journals THERMODYNAMIC BETHE ANSATZ AND THREE-FOLD TRIANGULATIONS

1996 ◽  
Vol 11 (22) ◽  
pp. 4051-4064 ◽  
Author(s):  
F. GLIOZZI ◽  
R. TATEO
Keyword(s):  
Bethe Ansatz ◽  
Unknown Function ◽  
Functional Equations ◽  
Simple Solution ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Thermodynamic Bethe Ansatz ◽  
Ansatz Approach ◽  
General Explanation

In the thermodynamic Bethe ansatz approach to 2D integrable, ADE-related quantum field theories, one derives a set of algebraic functional equations (a Y system) which play a prominent role. This set of equations is mapped onto the problem of finding finite triangulations of certain 3D manifolds. This mapping allows us to find a general explanation of the periodicity of the Y system. For the AN related theories, and more generally for the various restrictions of the fractionally supersymmetric sine—Gordon models, we find an explicit, surprisingly simple solution of such functional equations in terms of a single unknown function of the rapidity. The recently found dilogarithm functional equations associated to the Y system simply express the invariance of the volume of a manifold for deformations of its triangulations.

scholarly journals RENORMALIZATION WITHOUT INFINITIES

2005 ◽  
Vol 20 (06) ◽  
pp. 1336-1345 ◽  
Author(s):  
GERARD 'T HOOFT
Keyword(s):  
Renormalization Group ◽  
Conceptual Understanding ◽  
Feynman Diagrams ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Renormalization Group Equations

Most renormalizable quantum field theories can be rephrased in terms of Feynman diagrams that only contain dressed irreducible 2-, 3-, and 4-point vertices. These irreducible vertices in turn can be solved from equations that also only contain dressed irreducible vertices. The diagrams and equations that one ends up with do not contain any ultraviolet divergences. The original bare Lagrangian of the theory only enters in terms of freely adjustable integration constants. It is explained how the procedure proposed here is related to the renormalization group equations. The procedure requires the identification of unambiguous "paths" in a Feynman diagrams, and it is shown how to define such paths in most of the quantum field theories that are in use today. We do not claim to have a more convenient calculational scheme here, but rather a scheme that allows for a better conceptual understanding of ultraviolet infinities.

scholarly journals TOWARDS THE CONSTRUCTION OF WIGHTMAN FUNCTIONS OF INTEGRABLE QUANTUM FIELD THEORIES

2004 ◽  
Vol 19 (supp02) ◽  
pp. 34-49 ◽  
Author(s):  
H. BABUJIAN ◽  
M. KAROWSKI
Keyword(s):  
Ising Model ◽  
Form Factors ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Operator Equations ◽  
Quantum Operator ◽  
Integrable Quantum ◽  
Wightman Functions ◽  
Distance Limit

The purpose of the "bootstrap program" for integrable quantum field theories in 1+1 dimensions is to construct a model in terms of its Wightman functions explicitly. In this article, the program is mainly illustrated in terms of the sine-Gordon and the sinh-Gordon model and (as an exercise) the scaling Ising model. We review some previous results on sine-Gordon breather form factors and quantum operator equations. The problem to sum over intermediate states is attacked in the short distance limit of the two point Wightman function for the sinh-Gordon and the scaling Ising model.

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